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HANDBOOK 

FOR 

MACHINE DESIGNERS, SHOP 
MEN AND DRAFTSMEN 



McGraw-Hill DookCompai^ 

Electrical World The Engineering and Mining Journal 
Engineering Record Engineering News 

Railw^A^e Gazette American Machinist 

Signal LngiriGGr .American Engineer 

Electric Railway Journal Coal Age 

Metallurgical and Chemical Engineering Power 



HANDBOOK 

FOR 

MACHINE DESIGNERS, SHOP 
MEN AND DRAFTSMEN 



BY S' 

FREDERICK A/ HALSEY, B.M.E. 

EDITOR EMERITUS OF THE AMERICAN MACHINIST, MEMBER THE AMERICAN SOCIETY OF 
MECHANICAL ENGINEERS, AUTHOR OF "SLIDE VALVE GEARS," "THE USE 
OF THE SLIDE RULE," "wORM AND SPIRAL GEARING," "THE 
METRIC FALLACY," "METHODS OF MA- 
CHINE SHOP WORK," ETC. 



Second Edition 



McGRAW-HILL BOOK COMPANY, Inc. 

239 WEST 39TH STREET. NEW YORK 



LONDON: HILL PUBLISHING CO., Ltd. 

6 & 8 BOUVERIE ST., E.C. 

1916 






Copyright, 1913, 1916, by the 
McGraw-Hill Book Company, Inc. 



¥'^ 



DEC -5 1916 



.f 



TRB.MAPLE.PRESS.TORK.PA 



^CI,A446692 



PREFACE TO THE SECOND EDITION 

Those only who prepare books of the character of this realize the rapidity with which material 
for them accumulates. The author would not have believed, in advance, that, after an interval of 
three years, so extensive a revision as the one here presented would be necessary. Ignoring many minor 
items, some of the following subjects appear in this edition for the first time, others have been rewritten 
in the light of additional information and still others have been sufficiently expanded to justify their 
mention here. 

Thrust Bearings; Knife-edge Bearings; Roller Bearings; The Critical Speed of Shafts; Tandem or 
Riding and Steel Belts; The Geometrical Progression of Speeds; The Strength of Spur and Herringbone 
Gears; Chordal Pitch; Gaging Gear Teeth; Cutting Bevel Gears with Rbtary Cutters, including Parallel 
Depth Bevel Gears; Modified Addendum of Bevel Gears; Axial Thrust of Bevel Gears; Skew Bevel 
Gears; Practice with Friction- Gears; Worm Gears; Roller Chains; Friction Clutches; Spiral Springs of 
the Watch-spring Type; The Wire System of Measuring Screw Threads; Sizes, Properties and Strength 
of Wire; The Capacity of Horizontal Cylindrical Tanks, Full and Partly Full; Weirs, Rectangular 
and V-notch; Standard Pipe Tables; Pipe Flanges and Fittings; The Measurement of Tapers and Dove- 
tails; Forming and Other Tools; Press Fits, Straight and Taper; Balancing Revolving Parts, including 
the Technique of Running Balance; The Floating Lever; Velocity and Force Relations in Linkwork; 
Permissible Cost of Special Shop Equipment; The Weight of Solids of Revolution; The Diameter of 
Shell Blanks; The Power Consumed by Drilling Machines; Taylor's Cutting Speeds; Hardness Tests, 
including the Relation of Brinell and Scleroscope Hardness Numbers to One Another and to the 
Strength of Steel; Heat Treatment of Steel including the Relations of the Heat Treatment, the Degree 
of Temper and the Physical Properties of Carbon Steels and the Heat Treatment and Properties of 
Alloy Steels; Temperature Equivalents of Temper Colors; Materials for Steam Boilers and the Propor- 
tions of Rivetted Joints; The Discharge Capacity of Safety Valves; The Properties of Superheated 
Steam; Steam-pipe Coverings; Approximate Beams of Uniform Strength; The Strength of Columns; 
Materials and Constructions for Resisting Shock. New tables will be found in many of these additions 
and also in the last section as follows: Whole and Fractional Inches Reduced to Decimals of a Foot; 
Lengths of Circular Arcs; Cutting Speeds and Revolutions; Decimal Equivalents of Prime Number 
Fractions; Square and Cube Roots of Binary Fractions, and Chords for Spacing Circles. 

All of these additions fill gaps which needed filling, while many are fundamental and unique. The 
prominence of graphical methods has been retained, some of the added applications of graphics being 
not only time savers but, in themselves, elegant and new. All of the information contained herein, 
both old and new, is believed to be useful, definite and workable, much of it not readily accessible and 
a considerable portion of it not in existence elsewhere. The collection of design constants to be used 
with the rules and formulas is believed to be larger than any other, while special care has been taken, 
in all cases, to name the units to be used in connection with the formulas. 

In books of this character there must, of necessity, be some overlapping of contents, but the aim 
has been to include subjects which others have ignored or treated in a fragmentary or otherwise un- 
satisfactory manner, while matters of common knowledge and constructions having an academic 
interest only have been omitted. As it is the intention that the volume shall not become a museum 
of antiquities, superseded material has been rigorously removed. 

As the revision has gone on, the shop character of much of the material included, together with 
the impossibility of drawing any line of demarcation between shop and drawing office information has 
become more and more apparent, and the title of the book has, therefore, been changed to one which 
more clearly indicates its real scope. Except to make room for something better, nothing of interest 
to the designer and draftsman has been omitted, but, on the contrary, much has been added as the 
above list of subjects will show. 

Defacement of the alignment charts may be avoided by the following excellent expedient suggested 
by S. C. Bliss {Amer. Mach., Apr. 22, 1915): 

Obtain a sheet of thin, transparent celluloid about one inch smaller in each dimension than the 
book page and roughen one side with a piece of fine emery cloth. Place the sheet over the chart with 
the rough side up and rule the lines required with a soft lead pencil. The lines so ruled may be re- 
moved with a pencil eraser and the sheet thus be used indefinitely. 
October, 1916. 

V 



PREFACE TO THE FIRST EDITIOIS 

As an editor, the author's heart has often ached at the manner in which contributions to tech- 
nical journals of permanent value and usefulness form a procession to the limbo of forgotten things 
and benefit none but those under whose eyes they happen to fall at the date of publication. This 
volmne is primarily an effort to rescue from the oblivion of the out of print such contributions as are 
of direct use in the design of machinery. "The search for material has not, of course, been limited to 
periodicals but has extended to the transactions of many engineering societies, wherein information 
is nearly as effectively buried as in the back numbers of periodicals. In filling the gaps that remained 
after the search was completed, willing friends have come to the author's assistance. 

Not only is this the way in which this volume has been prepared, but the author is convinced that 
it is the only way, and, more than this, that there should be deliberate co-operation between con- 
tributors, editors and collectors, with efforts focused on books of this character as the ultimate outcome. 

To be more specific, the author is under no delusions regarding the many things that should be 
between these covers but that are not, nor of those others of which the data presented are inadequate 
but, now that a place has been provided for the preservation of information of the sort here gathered 
together, he hopes that increased activity in the preparation and the publication of such information 
will follow. He will certainly be glad to do his part toward the incorporation of such information in 
future editions. Assistance may be rendered in other ways than by preparing contributions. Wide 
as the search has been, it is not possible that all of the articles and papers that contain desirable data 
have been discovered. Those who know of such sources of information' are invited to forward memo- 
randa of the places where they may be found. 

Due credit to those who have supplied material will be found scattered through the volume. 
From the many who have given willing help it is almost invidious to make selections, but the author 
feels that it would be an injustice not to make special mention of Mr. J. A. Brown, Mr. Axel Pedersen 
and Prof. J. B. Peddle. F. A. H. 

New York, 
November, 1913. 



vu 



CONTENTS 

Page 

Mechanical Principles of Design i 

Equal Length Wearing Surfaces, i; Equalized Wearing Surfaces, 2; The Narrow Guide, 2; Tubular Torsion 
Members, 3; The Division of Functions, 3; Reducing the Overhang of Cranks, 4; The Center of Pressure of 
Bearings, 4; Frames and Supports, 5; Charts in Systematic Design, 6. 

Plain or Sliding Bearings 8 

Permissible Pressures, 8; Relation of Speed and Pressure, 10; Relation of Speed, Pressure and Temperature, 12; 
Conditions of Film Lubrication, 13; Dimensions for Film Lubrication, 15; Final Temperatures, 17; Materials, 
18; Westinghouse Practice, 21; Bearing Design, 25; Water Cooling, 25; End Play, 26; Thrust Bearings, 27; 
Schielie Curve Bearings, 29; Knife Edge Bearings, 29. 

Ball and Roller Bearings 3° 

Leading Dimensions, 30; General Conditions of Success, 30; Radial Bearing Mountings, 31; Collar Thrust 
Bearing Mountings, 34; Standardized Dimensions, 36; Dimensions and Loads of Radial Bearings, 36; Dimen- 
sions and Loads of Thrust Bearings, 37; Roller Bearings, 39; Dimensions and Loads of Roller Bearings, 40; 
Conical Roller Bearings, 42; Roller Thrust Bearings, 42. 

Shafts and Keys 43 

Strength of Shafts, 43; Hollow Shafts, 44; Torsion of Shafts, 44; Critical Speed of Shafts, 45; Friction of Line 
Shafts, 49; Keys and Key ways, 50; Improved Forms of Keys, 51. 

Belts and Pulleys 54 

Driving Power of Belts, 54; Idler Pulleys and Quarter Twist Belts, 56; Substitutes for Quarter Twist Belts, 
57; Mule Pulley Stands, 58; Tandem or Riding Belts, 59; Length of Belts, 60; Steel Belts, 60; Belt Shippers, 
61; Dimensions of Pulleys, 62; Arms of Pulleys and Sheaves, 64; Surplus Pulley Face, 64; Crown of PilUeys, 
64; Tight and Loose Pulleys, 65; Bursting Strength of Pulleys, 65. 

Fly Wheels 67 

Centrifugal Stress, 67; Bursting Speeds, 68; Bursting Tests, 69; Construction, 69; Safe Speeds, 71; Shrink 
Links, 74; Regulating Power, 75; Fly Wheels for Intermittent Work, 75. 

Cone Pulleys and Back Gears 80 

Graphical Solution of Cone Pulleys, 80; Geometrical Progression of Speeds, 80; Back Gear Ratio, 81; High 
Power Cone Pulley, 82; Slide Rule Solution of Cone Pulleys, 82; Arithmetical Solution of Cone Pulleys, 84; 
Planetary Back Gears, 89; Gear Ratios for Motor Drives, 90; Gear Box Construction, 90. 

Spur Gears '. 93 

Gear Tooth Systems, 93; Interference of Teeth, 93; Dimensions of Diametrical Pitch Teeth, 94; Approximate 
Tooth Outlines, 95; Grant's Odontograph, 95; Multipliers for Diameters, 96; Tooth Parts by Circular Pitch, 
96; Tooth Parts by Diametral Pitch, 97; Strength by Calculation, 98, 102; Strength by Graphics, 98, 102; 
Diagrams of Gear Teeth of Full Size, 99; Strength of Shrouded Teeth, 102; Strength of Bronze, Rawhide and 
Fabroil Gears, 103; Strength of Herring-bone Gears, 103; Dimensions of Arms of Gears, 105; Chordal Pitch, 
107; Gear Cutter Sets, 108; Gaging Gear Teeth, 108; Prime Factors of Members, no; Pitch Diameters from 
Circular Pitch, in. 

Bevel Gears 113 

Dimensions and Angles, 113; Profiles of Teeth, 116; Strength by Calculation, 116; Strength by Graphics, 116; 
Selection from Stock Lists, 116; Offset Method of Cutting with Rotary Cutters, 119; Parallel Depth Method of 
Cutting with Rotary Cutters, 121; Modified Addendum, 122; Skew Bevel Gears, 123. 

Friction Gears 126 

Working Loads, 126; Contact Pressure, 126; Coefiicient of Friction, 126; Practice with Friction Drives, 128; 
Variable Speed Disk Friction Drives, 129. 

Worm Gears 130 

Thread Profiles, 130; Interference of Teeth, 130; Cutting Diametral Pitch Worms, 131; Durability and EflS- 
ciency, 131; Relation of Circular and Normal Pitches, 132; Relation of Pressure and Velocity, 134; Load 
Capacity, 134; Tooth Parts, 135; Milled Worm Wheels, 135; Wire System of Measuring, 136; Angles for 
Gashing Worm Wheels, 137. 

Helical (Commonly Miscalled Spiral) Gears 138 

Helical and Worm Gears Compared, 138; Helical Gears of 45 Deg. Helix Angle on Shafts at Right Angles by 
Calculation, 138; Ditto by Graphics, 140; Cutters, 140; Helical Gears of any Helix Angle on Shafts at Right 
Angles by Calculation, 140; Ditto by Graphics, 144; Helical Gears of any Helix Angle on Shafts at any Angle 
by Calculation, 144; Real Diametral Pitches, 145; Helical Gears of Opposite Hands, 146. 

ix 



X CONTENTS 

Page 

Planetary (Epicyclic) Gears 147 

Velocity Ratios of Various Types of Planetary Gears, 147. 

Ropes 150 

American and British Practice in Rope Driving Compared, 150; Comparative First Cost of Belt and Rope 
Drives, 150; Most Economical Speed of Ropes, 150; Relation of First Cost and Speed, 150; Effect of Centrif- 
ugal Force, 150; Horse-power of Manilla Ropes, 151; Cross-sections of Sheaves, 151; Manilla Rope for Hoist- 
ing, 151; Horse-power of Cotton Ropes, 152; Splicing Manilla Ropes, 152; Knots, 153; Splicing Wire Rope, 
154; Hoisting Drums, 155; Durability of Wire Ropes, 156; Wire Rope Tables, 157; Efficiency of Rope Driving, 
163; Safe Loads on Manilla and Wire Rope Slings, 163. 

Chains . 164 

Leading Types, 164; Loads on Hoisting Chains, 164; Sprockets for Crane Chains, 164; Types, Uses and Speeds, 
165; Attachment to Hoisting Drums, 165; Ewart Chain, 166; Sprockets for Ewart Chain, 167; Correct Use of 
Ewart Chain, 168; Roller Chain, 169; Sprockets for Roller Chain, 170; Morse Silent Chain, 171; Link Belt 
Silent Chain, 173; Safe Loads on Hoisting Chain Slings, 174. 

Brakes 175 

Retarding Moment of Band Brakes, 175; Width of Band Brakes, 175; Differential Band Brakes, 175; Re- 
tarding Moment of Block Brakes, 176; Retarding Moment of Axial Brakes, 177; Ratio of Tensions in Band 
Brakes, 177; A Superior Hoisting Brake, 177; Automatic Brakes, 178; The Prony Brake, 178; Area of Prony 
Brake Surface, 179; The Rope Brake, 179; Coefficients of Friction in Brakes, 181. 

Friction Clutches 182 

Analysis, 182; Dimensions, 182; Axial Pressure on Cone Clutch, 184; Angle of Cone, 185; Multiple Disk 
Clutches, 186; Ball Friction Clutches, 186; The Claw Clutch, 187. 

Cams 188 

Laying Out Drum Cams, 188; Laying Out Face Cams, 188; Two Step Cams, 189; Making the Templet, 
191; Making the Former, 191; High Speed Cams, 193; The Cam Chart, 193; Charts with Separate Base Lines, 
194; Conjugate Cams, 194; Cam Levers, 195; Spring Adjustments, 196. 

Springs 197 

Preliminary Design, 197; Helical Springs in Tension and Compression, 197; Helical Springs in Torsion, 200; 
Conical Helical Springs, 200; Elliptic and Semi-elliptic Springs, 201; Flat Single Leaf Springs, 201; Flat Spiral 
Springs, 203; Materials and Miscellaneous Information, 208; Common Defects, 208; Steel for Springs, 208; 
Tempering Steel Springs, 209; Springs for Use in Salt Water, 209. 

Bolts, Nuts and Screws 210 

Lead and Pitch, 210; V Threads, 210; Friction of Screws, 210; Efficiency of Screws, 210; Stress Due to Initial 
and Applied Loads, 210; U. S. Standard Bolts and Nuts, 211; Tap Drills for U. S. Threads, 211; Acme Stand- 
ard, 212; British Association Standard, 212; Whitworth Standard, 212; International Metric Standard, 213; 
S. A. E. Standard, 213; A. S. M. E. Standard Machine Screws, 214; Tap Drills for Machine Screws, 214; 
A. S. M. E. Standard Machine Screw Heads, 215; S. A. E. Standard Lock Washers, 216; U, S. Standard Pipe 
Threads, 216; Baldwin Locomotive Works Standard Taper Bolts, 217; Split Nuts, 217; Set Screws, 218; 
Wire System of Measuring, 218; Tooth Parts of Acme Threads, 220; Acme and Brown and Sharpe Threads 
Compared, 220. 

Wire and Sheet Metal Gages 221 

Westinghouse Method of Abandoning Gages, 221; Gage Sizes in Decimals, 222; List of Gages used in the U. S., 
223; Standard Decimal Gage, 224; U. S. Standard Gage for Sheet and Plate Iron and Steel, 224; Sizes and 
Properties of Wire, 225; Strength of Wire, 229. 

Hydraulics and Hydraulic Machinery 230 

Hydraulic Constants, 230; Pressure and Feet Head Compared, 230; Capacity of Tanks, 230; Flow of Water, 
230; Spouting Velocity, Discharge and Power of Jets, 231; Spouting Velocity and Discharge of Jets, 234; 
Loss of Head by Friction, 237; Resistance of Pipe Fittings, 238; Discharge over Weirs, 239; Effective Fire 
Streams, 240; Hydraulic Press Cylinders and Rams, 240; Hydraulic Packing, 242; High Pressure Hydraulic 
Valves and Fittings, 244; Air Chamber Charging Devices, 246; Air Chambers for Suction Pipes, 247; The 
Siphon, 248; Power and Capacity of Pumps, 249; The Hydraulic Ram, 250. 

Pipe and Pipe Joints 251 

Dimensions of Commercial Drawn Pipe, 251; of Extra and Double Extra Heavy Pipe, 251; of Standard 
Square Pipe, 252; of Seamless Brass and Copper Pipe, 252; of Lead Pipe, 252; of Block Tin Pipe, 252; of Large 
O. D. Pipe, 253; of Cold Drawn Seamless Tubing, 253; of Rectangular Pipe, 253; Bursting and Collapsing 
Strength of Pipe, 251; British Standard Pipe Threads, 254; Length of Pipe per Square Foot of Surface, 254; 
Test and Bursting Pressures of Pipe, 255; Equation of Pipes, 256; Bursting Strength of Elbows and Tees, 257; 
Cast Iron, Rivetted and Copper Pipe, 257; A. S. M. E. Standard Flanges and Fittings. 258; Pipe Joints, 259; 
Spiral Rivetted Pipe, 260; Screwed Pipe Fittings, 265; Pipe Markings, 267. 



CONTENTS xi 

Page 

Minor Machine Parts 271 

Standard Tapers, 271; Originating Tapers, 273; Tapers and Corresponding Angles, 273; S. A. E. Standard 
Cotter Pins, 274; Taper Pins, 274; Total Taper from Taper per Foot, 274; Dovetails and T-Slots, 276; Shaft 
Couplings, 277; Jig and Fixture Details, 278; Wrenches, 279; Handles and Hand Wheels, 280; Plug Cocks, 
280; Ball Handles, 280; Forming Tools, 281; Rack for Bar Stock, 283; Foundation Bolt Washers, 283; Punches 
and Dies, 284, 286; Knuckle Joints, 286; Yoke and Eye Rod Ends, 287, 288; Studs, Cam Rolls and Levers, 
287, 288; Fluted Reamers, 289; Series Taps for Acme and Square Threads, 289; Counterbores, 289. 

Press and Running Fits 290 

Tolerances and Allowances for Running Fits, 290; Straight Press Fits, 290; Taper Press Fits, 290; Suitable 
Lubricants for Press Fits, 290; Suitable Pressures for Press Fits, 290; Allowances for Press Fits, 292, 294; Hoop 
Stresses Due to Press Fits, 293, 295; Gaging Taper Press Fits, 296; Practice of Various Manufactures with 
Allowances and Tolerances for Fits, 296-299. 

Balancing Machine Parts 300 

Standing Balance Fixtures, 300; Westinghouse Balancing Machine, 300; The Riddell Balancing Machine, 302; 
Depth of Drilling to Remove a Given Weight, 302; Theory of Running Balance, 303; Balancing Long Drums, 
303; The Norton Balancing Machine, 305; The Akimoff Balancing Machine, 305; The Technique of Running 
Balance, 305; Balancing Reciprocating Parts, 305. 

Miscellaneous Mechanisms, Constructions and Data 308 

The Hooke Universal Coupling, 308; The Baush Universal Coupling, 308; The Floating Lever, 308; Velocity 
and Force Relations in Linkwork, 310; The Geneva Stop, 313; Rock Arms and Linkwork, 314; The Ball 
Expansion Drive Stud, 315; Balance Diaphragms, 315; Effective Pressure Area of Poppet Valves, 316; Cast- 
iron Floor Plates, 316; Approximate Ellipses, 317; Arcs of Large Circles, 317; Length of Circular Arcs, 318; 
Permissible Cost of Special Stop Equipment, 318; Weight of Solids of Revolution, 319; Diameters of Shell 
Blanks, 320; Addition of Binary Fractions, 321; Standard Cross-sections, 321; Filing Notes and Clippings, 
321; Blue Print Solution, 322; Metallic Indicator Paper, 322; American Railroad Clearances, 323. 

Performance AND Power Requirements OF Tools 324 

Power Constants for Lathe Tools, 324; for Twist Drills, 324; Pressure on Lathe Tools, 325, 326; Power Con- 
sumed by Drilling Machines, 327; Power Constants for Milling Machines, 330; Speeds of Twist Drills, 331; 
Power Consumed by Milling Machines, 333, 334; Sizes of Motors for Machine Tools, 335; for Wood Working 
Machines, 339; for Power Presses, 339; for Planers, 342; Standard Motor Ratings, 343, 345, Power Requirements 
of Machine Tools in Groups, 346; Power Constants for Punching and Shearing, 348; for Centrifugal Fans, 350; 
for Moving Heavy Loads, 351; Measuring the Energy of Hammer Blows, 352; Cutting Capacity of Power 
Presses, 354; Taylor's Tool Forms, 356; Taylor's Cutting Speeds, 357, 358; Average Cutting Speeds, 359; 
Speeds for Tapping and Threading, 359; Milling Machine Cutters, 360. 

Cast Iron 361 

Constituents of Cast Iron and Their Influence, 361; Chemical Composition of Cast Iron for Various Purposes, 
362; Tests of Malleable Cast Iron, 365; Stress Strain Diagrams of Malleable Iron, 366. 

Steel 368 

Composition and Properties of Carbon Steels, 368; Forging Steel with Press and Hammer, 369; Representa- 
tive Specifications for Steel, 370; Characteristics of Steel Forgings and Castings for the U. S. Navy, 371; 
Composition of Steel for Cutting Tools, 372; Principles of the Heat Treatment of Steel, 374; Hardness Tests 
of Steel, 376; Brinell Hardness Numerals, 376; Relation of Brinell and Scleroscope Hardness Numerals, 377; 
Properties of Steel as Affected by Heat Treatment, 379; S. A. E. Specifications for Carbon Steels, 378; for 
Nickel Steels, 381; for Nickel Chromium Steels, 382; for Chromium Steels, 384; for Chromium Vanadium 
Steels, 385; for Silico-manganese Steels, 386; Relation of Properties and Temper of Spring Steel, 387; S. A. E. 
Standard Heat Treatments, 387; Temperature Equivalents of Temper Colors, 389. 

Alloys 390 

Composition and Strength of Copper-Tin-Zinc Alloys, 390; Standard Sheet Brass, 390; Low Brass, 391; Brazing 
Brass, 391; Fine Cutting Brass, 391; Red Metal or Commercial Bronze, 391; Gilding Metal, 391; Copper 
Sheets and Strips, 391; German Silver, 391; Brass Rods, 391; Tobin Bronze, 392; Brass Casting Metals, 392; 
Cast Manganese Bronze, 392; Aluminum AUoys, 392; Design of Parts of Aluminum Alloys, 392; Pouring 
Aluminum Alloys, 393; Casting Alloys for the U. S. Navy, 394; Fusible Alloys, 394. 

Weight of Materials 395 

Specific Gravity and Weight of Metals and Wood, 395; Weights of Iron, Brass and Copper Wire, 395; of 
Seamless Brass Tubing, 396; of Steel Hexagon and Octagon Bars, 396; of Spheres of Various Metals, 396; of 
Sheet Iron, Steel, Copper and Brass, 397; of Flat Sizes of Steel, 397; of Round and Square Steel Bars, 398; 
of Brass, Copper and Aluminum Bars, 400; of Steel Plates, 400; of Flat Rolled, Hoop or Band Steel, 400. 



Heat 



401 

Defects of the Centigrade Thermometer Scale, 401; Conversion Formulas between Fahrenheit and Centigrade 
Scales, 401; Conversion Table, Centigrade to Fahrenheit, 401; Fahrenheit to Centigrade, 402; British Thermal 
Units to Kilogram Calories, 403; Melting Points of Various Substances, 404; Coefficients of Expansion of 
Various Substances, 404; Transmission of Heat through Metallic Tubes, 404. 



Xll CONTENTS 

Page 

Steam Boilers 405 

Horse-power and Heating Surface per Horse-power, 405; Ultimate Strength of Materials, 405; Minimum 
Thickness of Plates and Tubes, 405; Specifications for Steel Plates, 405; Standard Test Piece, 406; Specifica- 
tions for Rivets, 406; for Stay Bolt Steel, 406; for Tubes, 407; Formula for Working Pressure, 407; Efficiency 
of Joints, 408; Ligaments, 410; Braced and Stayed Surfaces, 411; Allowable Loads on Stays and Stay Bolts, 
412; Net Areas of Segments of Heads, 412; Angles for Staying Segments of Heads, 413; Tube Sheets, 414; 
Circular Furnaces and Flues, 414; Manholes, 415; Safety Valves, 416; Percentage of Plate Strength, 418; 
Percentage of Rivet Strength, 418; Saving Due to Heating Feed Water, 419; Loss of Coal Due to Scale, 419; 
Analysis and Heating Value of Coals, 420; Hanging or Supporting Boilers, 420; Horse-power of Chimneys, 421; 
Properties of Standard Tubes, 421. 

The Steam Engine 422 

Steam and Coal Consumption, 422; Properties of Saturated Steam, 422; of Superheated Steam, 423; Effi- 
ciencies of Pumping Engines, 424; Calculating Expected Steam Consumption, 424; Power Calculations, 426; 
Construction and Dimensions of Parts, 427; Cylinder Cover Joints, 433; Cylinder Head Bolts, 434; Areas 
of Ports and Pipes, 435; Drop of Pressure in Pipe Lines, 437; Steam Pipe Coverings, 437; Laying Out the 
Slide Valve, 439; the Link Motion, 440; Friction of Slide Valves, 441; Poppet Valves, 441; Condensing Water, 
442. 

The Gas Engine 444 

Dimensions of Parts, 444; Probable Brake Horse-power, 444; Weight of Fly-wheels, 444. 

Compressed Air 449 

Pneumatic Constants, 449; Barometric Pressure at Various Altitudes, 449; The Various Efficiencies, 449; 
Power Calculations, 449; Compound Compression, 450; Efi'ect of Altitude, 451; Graphic Power Calculations, 
451; Index of Compression Curve, 452; Friction of Compressed Air in Pipes, 453; Plotting the Compression 
Curve, 454; Intercoolers, 457; Reheating, 457; Constructive Details, 458; Consumption of Compressed Air by 
Pneumatic Tools, Hoists and Pumps, 459; The Air Lift Pump, 461; Discharge of Compressed Air through 
Orifices, 461. 

Mechanics 464 

Mechanical Advantage of Mechanical Powers, 464; Relations of Velocity, Height of Fall and Time, 464; 
The Toggle Joint, 465; Laws of Falling Bodies, 465; of Accelerated Motion, 466; Stored Energy, 466; Centrif- 
ugal Force, 466; Center of Gravity of Plane Figures, 466; Moment of Inertia of Irregular Figures, 467; Areas of 
Irregular Figures, 469. 

Strength of Machine Parts 472 

Factor of Safety, 472; Strength of Leading Materials, 472; Shrinkage of Castings, 472; Beams, 472; Moments 
of Inertia of Sections, 473; I-Beams with Reinforcing Plates, 474; Properties of I-Beams, 477; Standard Test 
Pieces, 478; Beams of Uniform Strength, 478; Columns, 482; Flat Plates, 483; Combined Tension and Shear, 
485; Punch and Shear Frames, 485; Hoisting Hooks and Lifting Eyes, 486; India Rubber, 489; Materials and 
Constructions for Resisting Shock, 491; Resilience of Steel, 492. 

Weights and Measures 495 

American and British Compared, 495; The Case against the Metric System, 495; British-American-Metric 
Conversion Factors of Units of Length, Weight and Capacity, 496; of Fractional Dimensions of Length, 498; 
of Compound Units, 499; of Units of Pressuxe, 499; of Units of Value, 500; of Units of Work and Power, 500. 

Mathematical Tables 501 

Extending the Range of Tables, 501; Factors and Relations of ir, 501; Logarithms, 502; Antilogarithms, 503; 
Hyperbolic Logarithms, 505; Trigonometric Functions, Their Relations, 506; Their Signs in the Four Quad- 
rants, 507; Natural, Table of, 506; Factoring Method of Extracting Roots, 523; Largest Squares from Round 
Stock, 523; Whole and Fractional Inches Reduced to Decimals of a Foot, 524; Lengths of Circular Arcs, 524; 
Cutting Speeds and Revolutions per Minute, 525; Decimal Equivalents of Prime Number Fractions, 526; 
Lengths of Circular Arcs to a Radius of One Inch, 526; Squares of Mixed Numbers, 526; Areas, Perimeters, etc., 
of Regular Polygons, 531; Spacings of Circles, 531; Areas of Circular Segments, 532; Sides and Diagonals of 
Squares, 533; Circles and Squares of Equal Area, 533; Square and Cube Roots of Binary Fractions, 533; 
Lengths of Chords for Dividing Circles, 534; Decimal Equivalents of Binary Fractions, 535; of other than 
Binary Fractions, 535; Surfaces and Volumes of Spheres, 536; Squares, Cubes, Square and Cube Roots, 
Reciprocals, Circumferences and Circular Areas of Integers, 537; Areas and Circumferences of Circles, Decimal 
Divisions, 544; Areas, Circumferences, Squares, Cubes and Fourth Powers of Binary Fractions, 548; Areas 
and Circumferences of Circles, Binary Divisions, 549; Areas of Circles of Wire Gage Diameters, 550. 



I 



WANTED 

CORRECTIONS— SUGGESTIONS — CONTRIBUTIONS 

T)LANS have been made for keeping this book continuously up to date. 
The general make-up, with each section beginning on a fresh page, 
while the illustration and table numbers of each section begin with No. i, 
as well as the fact that the type forms have been preserved, will make it easy 
to drop old and introduce new material anywhere. 

In this plan for continuous development the co-operation of all users of 
the book is solicited. Suitable contributions will be published first in the 
American Machinist and will be paid for as contributions to that paper. 

Contributions should be sent to the editor of the American Machinist, 
505 Pearl Street, New York. Other communications should be sent to the 
author at the same address. 

Since future additions to this book will be published first in the 
American Machinist, it is only necessary for each user of the book to 
follow the American Machinist in order to keep his copy continuously 
up to date. 



HANDBOOK FOR 
MACHINE DESIGNERS AND DRAFTSMEN 



MECHANICAL PRINCIPLES OF DESIGN 



"When a thing is wholly right it is pretty sure to look right, 
though it may be pretty bad and appear to be fairly good or be 
absolutely bad and not appear so to the casual observer. . . . 
When a thing is known to be bad and it looks right to an observer, 
it is time for him to cultivate his taste. . . . When a thing is 
bad if carried to an extreme, it cannot be right when carried only 
part way" (Professor Sweet). 

Equal Length Wearing Surfaces 
"The thing that does not tend to wear out of truth does not 



wear much. 



The thing that wears out of truth is never 



machine wears in the same way and for the same reason. In both 
cases the conditions favor local wear. Were the stationary and 
moving pieces of the same length, neither would wear hollow, and 
truth in both cases would be indefinitely prolonged. With this con- 
struction, local wear being impossible, the form, and hence the fit, 
are preserved indefinitely. 

Applications of the principle are shown in Figs. 1-5. Fig. i shows 
the equal length guide and platen of a Becker milling machine and 
Fig. 2 the head stock of the Newall measuring machine, in which 
latter the principle is especially important. The measuring screw 
a and its nut are of the same length, by which local wear, which 




mm 




Fig. I. 



Fig. 



h Valve Rod 






a 



Valve Rod 



'^r—r 



® 



Eccentrlo 
Rod 





Fig. 3. 



Fig. 4. 
Figs, i to s- — Examples of equal length wearing surfaces. 



Fig. s. 



right long and never gets fixed until it is too bad to use " {Professor 
S^'eet). 

Next to extent of wearing surface, the chief essential of dura- 
bility is fit. Whatever destroys fit limits durability. The chief 
enemy of fit is local wear, because local wear means change of 
shape and hence loss of fit. Conditions that favor local wear favor 
short life. 

The stationary cross rail of a planer wears hollow at the center 
because it is most used there. The moving platen of a milling 



would destroy the accuracy of the machine, is prevented. Figs. 
3, 4 and 5 are by Professor Sweet (4wer.ikracA.,iVow. 17, 1904), who 
originated the principle. Fig. 3 shows the usual and bad construc- 
tion of steam-engine valve-rod guides and Fig. 4 the correct construc- 
tion in which the sliding surfaces are of equal length. Forty or 
fifty engines made in this way showed no wear after twelve or fifteen 
years of use while, should they wear, the slack can be taken up with- 
out refitting the wearing surfaces. Fig. 5 shows an application to 
the slide valve of a common steam engine. As commonly made, these 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



valves have the seat so long that the valve overruns but a short 
distance, the construction being due to the impression that in- 
creased surface gives increased life. This is bad practice, as the seat 
always wears concave. If it is designed to have the valve cut off at 
three-quarter stroke, the lap of the valve will be one-quarter the 
travel. If the ports and bridges are also one-quarter the travel, 
then by cutting away the valve face until it is only as long as the 
valve, as shown in Fig. 5, there will be the same wearing surface on 
the seat as on the valve and the two will remain straight and keep 
tight much longer than if made the common way. 

There are cases to which the principle does not apply, an example 
being the beds and tables of planing machines. Here the chief load 
on the V's is the weight of the table which is not very stiff vertically. 
Were the bed and table of equal length, the flexibility of the table 
would lead, as it overruns, to a concentration of pressure at the ends 
of the bed and near the center of the table, leading to wear of the bed 
into a convex and of the table into a concave shape. In this and 



and that is not apt to be done; while, if cut away too much, the result 
will be no worse than if cut away too little by the same amount, 
that is, it will still be better than if not cut away at all. Like the 
factor of safety, the amount to be cut away is a matter of judgment 
at first and of experience later. 

In all cases the wearing surface of the guide should be cut away 
so that the slide shall overrun at the ends. 

Assuming that the use of the head at different locations is pro- 
portional to its distance from the center of the rail, which is not far 
from average conditions, the correct method of laying out the 
widths of the parts to be recessed and of those to be left as lands 
is shown in the diagram below the cross rail. Draw the diagonal 
line across the guide surface. Locate the edge of the first recessed 
portion at a, when the distance from a to h gives the width c of the 
space to be recessed and the distance from d to e gives the width / 
of the land. Similarly, gh and ij give the width of the next space 
and land and so on to the end of the rail. The recesses are, of course, 



m'/'-mwrn v-/- 



m^ 



mm 




m 




\ 


f 


9 

I 


















'~7 




A 


— 




— 


- .. 







Fig. 6. — Bearing surfaces in proportion to use. 



similar cases the bed should be the longer, F. A. Pratt's rule for 
planers under average conditions being that the length of the sur- 
faces should be in the ratio of 17 to 12. 

Equalized Wearing Surfaces 

There are other cases in which the principle cannot be applied 
for obvious reasons, examples being the cross rails and saddles of 
planing and planer-type milling machines. In many such cases, 
including these examples, an alternative construction (also due to 
Professor Sweet) is available. As he has put it, "when conditions 
are known, flat guides may be made to stay flat and when condi- 
tions are not known, common practice may be improved." The 
principle here is to make the wearing surfaces proportional to their 
use. Fig. 6 shows this principle applied to the cross rail of a travers- 
ing machine, which is primarily a vertical spindle milling machine 
of the planer type. In the case of a planer the head on the cross 
rail does not move under the cut, whereas, in this case the head 
travels across under the cut. Hence, it was found that the cross 
rail wore out in the middle, and the cross rail was recessed as shown 
by the shaded sections, thus reducing the wearing surface where the 
head is used the least and equalizing the wear. This principle can be 
applied to great advantage wherever the wear is unequal. The 
exact extent to which it can be carried is undeterminable, because 
it is impossible to know how much the machine will be used with the 
head in the center or at the ends, but, as the surface is cut away, the 
result will be progressive improvement until too much is cut away 



shallow — the principle is to get rid of the wearing surface where it 
does harm. 

In Professor Sweet's traversing machine, as first made without 
the cut-away feature, the rail required refitting after two years' 
use, while, after it was cut away as above, it ran six years before re- 
fitting was necessary. Similarly a shaper slide, as first made without 
the cuts, required refitting after two or three years' use, while after 
being cut away it ran fifteen years. 

Another illustration of the principle that things that do not tend 
to wear out of truth do not wear much, is found in the Schiele curve 
bearing, which see. The principle of this construction is uniformity 
of wear and it has remarkable durability. There is no question 
that its merits are not adequately appreciated. 

The Narrow Guide 

The narrow guide was first used by Professor Sweet on his travers- 
ing machine in 1886. The cross rail of that machine is shown in 
Fig. 7. Its merits, as contrasted with those of the usual construc- 
tion, may best be realized by imagining the usual construction ex- 
aggerated in height, when its weakness against side tilting and its 
tendency toward local wear at the ends of the short guiding surfaces 
will be realized. Just as the usual construction is better than the 
exaggerated illustration, so the narrow guide is better than the usual 
form. The construction is such that there is vertical clearance at 
the top of the cross rail, the weight of the head being carried by the 
gib at the bottom. 



MECHANICAL PRINCIPLES OF DESIGN 



Fig. 8 shows the narrow guide as applied to a lathe bed by John 
Lang and Sons, while Fig. g shows the natural development of the 
principle as adapted to the American V guide, the illustration being 
an end view of the Brown and Sharpe grinding machine. Professor 
Sweet advocated and practised the single V guide for lathe con- 
struction as early as 1876. Another application is found in the cross 
rail of the BuUard boring mill, Fig. 10, and still another in the arm 
of the Cincinnati-Bickford radial driU arm, Fig. 13. 

Tubular Torsion Members 

"The box section is the best form metal can be put into to resist 
the various strains machine frames are subjected to" {Professor 
Sweet). 

The readiest way for the designer to learn the value of the tubular 
section as a torsion member is to compare, by twisting in his hands, 
two pieces of common pasteboard mailing tube, one complete and 



The weakness of the slit tube is due to the absence of any pro- 
vision for the longitudinal shearing stress. If, to meet structural or 
operative conditions, it becomes necessary to cut holes through the 
tube, it may be done without serious harm provided ample metal 
is left for the shearing stress. 

The torsional stress on that member would make the bed of a lathe 
an ideal place for the tubular section, but for the necessity for get- 
ting rid of the chips, which makes the application of the complete 
tube impracticable. The Tangye lathe bed. Fig. 14, illustrates how 
the practical necessities can be met and most of the rigidity of the 
tubular section be preserved. Note that, in a partial tube of rec- 
tangular section, wide flanges aa are highly important. 

In planer beds, unlike lathe beds, there is nothing to prevent the 
use of the tubular section with such openings as are required for the 
gears. The continued use of the ladder bed for planers is due to 
nothing more creditable than custom and precedent. 




Fig. 7. 




Fig. 10. 




Fig. 8. 




Fig. 9. 



Figs. 7 to 10. — Examples of the narrow guide. 



the other slit down its length as shown in Fig. 11. The difference, 
which is simply startling and can scarcely be expressed in figures, 
is not a matter of the material but of the construction. Relatively 
speaking, the same difference exists between a cut and an uncut 
tube of iron. No possible addition of material can make up the 
loss due to slitting a tube. Next to a lath, the very common I- 
beam section, while ideal to resist bending, is about the worst pos- 
sible distribution of metal to resist torsion. 

An excellent example of the use of tubular sections in appro- 
priate places is seen in Fig. 12, by the Beaman and Smith Co., in 
which both bed and upright are tubular. Another example is seen 
in Fig. 13, which is a section of the arm of a radial drilling machine 
by the Cincinnati-Bickford Tool Co. In the latter case the tubular 
section is combined with another correct construction — the narrow 
guide. 



The Division of Functions 

Many cases of improved design, when analyzed, are cases of the 
division or separation of the functions. The principle is of consider- 
able application and deserves to be recognized. 

The most common application of the principle is the well-known 
Pratt and Whitney pattern of turret lathe index ring and latch bolt, 
Fig. 15. Were the notches and bolt made of truncated V form, both 
sides would be equally concerned in the functions of locating and 
moving the plate, both must be made with equal accuracy and both 
would be subject to wear. In the construction shown, the functions 
are divided, the radial side doing the locating and the inclined side 
the moving to position. The result is that the radial side only need 
be of refined accuracy, while the wear is chiefly on the inclined side 
where it does no harm. 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Another case is found in the loose center piece snap gage, Fig. 
i6. In the usual form of snap gage, one piece of metal determines 
the size of the gage and of the piece of work measured. In the form 
shown, the functions are divided, the center piece determining the 
size of the gage, while the jaws determine the size of the work. Wear 
is confined to the jaws and, after it occurs, the gage may be brought 
back to its original size by removing the jaws and lapping them flat. 
Note that, if a limit gage is to be made, further division of function 
must be made if the full advantage of the construction is to be real- 
ized. One jaw must be divided as in Fig. 17, the limits being made 
on the center piece, by which plan both limits once made, are per- 




easily be reduced by reversing the position of the hub as in Fig. 19, 
by Professor Sweet {Amer. Mack., Dec. 8, 1904). The improve- 
ment is obvious and it costs nothing. 

The Center of Pressure should be at the Center of a Bearing 

The neglect of this principle is almost universal and leads to the 
bell-mouthed wear of the bearings — that is to local wear which 
always leads to short life. The correct construction is not always 
possible, although it is possible in many cases in which it is not 
used. A common case of bad construction is that of the rock shaft 
introduced to provide for the offset of the eccentric and valve rods 



m i— i n r~i 






Fig. 12. 
Figs, i i to 13. — Examples of the tubular torsion member. 



Fig. 15. Fig. 17. 

Figs. 15 to 17. — Examples of the division of functions. 




Lfco 



Fig. 14. — Example of a compromise tubular torsion member. 



manent the lapping of the jaws flat being all that is necessary to 
remove the effects of wear. 

Another case is found in the Newall measuring machine head, 
Fig. 2. The functions of traversing and of carrying the weight of 
the parts are here divided, the screw a doing nothing but the travers- 
ing, bearings be being provided to carry the weight. The chief 
cause of wear of the most essential piece — the screw — is thus removed. 

All these are cases of obvious improvement and they are all 
applications of the principle of the division of functions. 

Reducing the Overhang of Cranks 

The common method of making overhung cranks with the hub on 
the rear side, Fig. 18, leads to an amount of overhang which may 



of slide-valve steam engines. In the common construction. Fig. 20, 
the tendency is to oscillate back and forth around a vertical center 
line, wear the hole bell-mouthed at both ends and wear off the shaft 
in like manner. Fixing the rocker to the shaft as in Fig. 21 is better, 
as it not only throws the bearings farther apart but they are better 
lubricated as the oil can be introduced on the slack side. Such 
rocker arms are best when cast of hollow box section, as that form is 
best to resist torsion. 

Where a form such as shown in Fig. 22 can be used, it is a great 
improvement if a line drawn from the center of one wrist to the center 
of the other passes centrally through the main bearing. The form 
shown in Fig. 23 is better still, for the reason that Fig. 21 is better 
than 20. 



MECHANICAL PRINCIPLES OF DESIGN 



The same principle is embodied in the form shown in Fig. 24, 
which, however, requires ball connections for the eccentric rod, 
although it requires much less of a projection for the supporting 
bracket {Professor Sweet, Amer. Mach., Dec. 8, 1904). 

Frames and Supports. 

"Whenever inconsistent or useless things are stuck on to improve 
the appearance, they always fail. To be good, a design must be 
consistent" {Professor Sweet). 

Any machine frame standing on three legs is free from twisting 
stress and from the resulting distortion. When machines have con- 
siderable height, as in the case of a lathe, the omission of one of the 




Fig. I 



Fig. 19. 



Figs. 18 and 19. — Reducing the overhang of cranks. 




Fig. 20. 




Fig. 21. 






_ Fig. 24. 

Fig. '22. Fig. 2.^. 

Figs. 20 to 24. — Correct and incorrect constructions of rocker arms. 



of form. In the planer bed. Fig. 26, the distance between the sup- 
ports is no greater than in Fig. 27, and as the center of the load in 
planing would, in no case, overhang the supports more than a slight 
distance, the construction shown in Fig. 26 is quite as well supported 
as the other, and if the iron in the legs and the work to fit them were 
put into the casting, the bed could be brought down to the floor as 
in Fig. 28, greatly improving the structure. 

Were the bed made of tubular section, with one leg under the back 
of each housing and one under the middle toward the other end, the 
results would be still better. Such a planer could be set anywhere 
on anything solid, and that is all that need or can be done {Professor 
Sweet, Amer. Mach., Mar. 9, 1910). 

An example of correct frame design and support on these principles 
is seen in Fig. 29, which shows a Norton grinding machine. The 
underneath view shows the arrangement of the three points of 
support and of the connecting ribs. 

One of the main points in designing frames is not to expose thin 
sections, as in the case of holes through plates and webs. In a stand- 
ard or column 12 ins. on the sides, or in diameter, the exposed sec- 
tions should not be less than 15 ins.; or, to make a rule, the exposed 
sections should be equal to an eighth of the extreme faces, as in Figs. 
30 and 31. External beads should not be employed, because they 
convey an idea of thin sections unless their width corresponds to 
the flange e, or to the base flanges of the frame. 




Fig. 25. — Example of a pivotted tailstock lathe leg and tubular bed. 



customary four legs would lead to a lack of stability, but this condi- 
tion can be met by swivelling the leg to the frame as in Fig. 25, which 
shows the construction used at this point in Professor Sweet's Artisan 
speed lathe, which also has a tubular bed. 

The customary location of supports under the extreme ends of 
machine frames leads to an unnecessary increase of span and of 
spring, while the placing of a third pair of legs under the center 
should be a last resort. Machines should be complete in themselves, 
whenever possible, and not depend on foundations for maintenance 



Another feature that has a good deal to do with the symmetry of 
frames is the thickness of base flanges. These should follow the rule 
of exposed sections and equal an eighth of the faces or the diameter 
of the trunk above when the latter is either round or rectangular. 
This is required not only to produce harmony of dimensions, but to 
insure against accident by fracture. 

The base flanges for frames or pedestals larger than 10 ins. in di- 
ameter should be cored out beneath, as shown in Fig. 32. The top 
corners of base flanges, when of the proportions named, should be 



TT 



s: 



Fig. 26. 



Tz: 



Fig. 27. 



TT 



3 



V^T 




■7 V" 



Fig. 28. 



Figs. 26 to 28. — Correct and incorrect supports for planers. 



6 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



rounded to a radius of about one-fourth the thickness so as to 
avoid the contour indicated in Fig. 33, which is a monstrosity of 
the " ogee " order of architecture. 

Struts are diflScult things to bring into harmony with machine 
framing, especially when connected to cyUndrical or rectangular 
sections as in Figs. 34, 35 and 36. When the frame is cored as in 
Fig- 35t the best way is to use a solid section for the strut as at a, 





Fig. 29. — Example of correct frame design and support. 



Figs. 34 and 36, the corners being slightly rounded so as to harmonize 
with the other members but never made semicircular. This is 
always wrong and looks so. 

Struts, ties and braces should be in straight lines, unless set in 
curved intersections for reinforcement, as at e, in Fig. 37. If the 
corner at a is of short radius, as in pipe flanges, the brace e should be 
straight. 

In rib sections, of which Fig. 38 shows examples, there is the com- 



mon mistake of considering the web a as a principal member and the 
flange e a reinforcement. This leads to a thicker section for the plate, 
and is a waste of material. The web a is no more than a brace or 
tie, and should always be made as thin as it can be cast without cool- 
ing strains, usually not to exceed one-half as thick as the members e. 

The curved form for bracing ribs as in Fig. 39 is still adhered to in 
most cases by habit, and because we reluctantly abandon the old 
idea of curves and ornament, but we are fast reaching the point when 
the shape shown in Fig. 40 will be substituted for the curves. 

Fig. 41 shows a cored section which is especially suitable for large 
frames, and conveys an idea of an indented surface rather than of 
ribs, and is a means of relieving broad, flat surfaces that always look 
"skinny " unless perfectly flat and smooth. It also forms a reinforce- 
ment of corners, which are the weakest part, and for any machine 
of fine character the corners can be finished (John Richards, Amer. 
Mack., June 8, 1899). 

The outline sketch. Fig, 42, shows an appropriate base form from 
which to derive suitable frames for a great variety of purposes. 
Appropriate modifications to provide a base and attachments for 
bearings are shown in Fig. 43. 

It will be a matter of astonishment to those who have not pre- 
viously considered the matter, to discover the extent to which this 
form of the projecting beam or bracket enters into machine-tool 
framing. In that type called vertical machines, such as those for 
drilling, slotting, and planing, nearly aU have this feature, and it 
has beside a wide place in horizontal supports that project from the 
main standards, such as tables for drilling, or other purposes. It is, 
therefore, well worth considering as a distinctive feature in design. 

This form of standard is often spoiled by inharmonious bolted- 
on parts, such as have a ribbed section when the main member is 
hoUow or cored. This is an incongruous thing, too common in 
practice. There is nothing saved and generally something lost by 
attaching ribbed parts to box framing. Good practice demands that 
all bearings requiring positive alignment be cast integral with the 
main frame and in harmony therewith {John Richards, Amer. Mach., 
May 25, 1899). 

Charts in Systematic Design 

The use of charts in systematic machine design is illustrated by a 
very simple case in Figs. 44-47 by H. S Britt {Amer. Mach., Mar. 
22, 1906). 




F1G.37. 



Fig. 38. 
Figs. 30 to 41. — Examples of correct and incorrect machine frames. 



Fig. 41, 



MECHANICAL PRINCIPLES OF DESIGN 



Assuming a line of sizes of any part of a type in which judgment 
is the chief element in the design to be in contemplation, two sizes 
near the extremes are first designed, after which the intermediate 
sizes are taken direct from a diagram, Fig. 46, which is first laid 
down to the sizes already designed. 

For example, suppose it is desired to get up a line of boxes, such 
as are shown, for shaft sizes from xi in- to 2^^^ ins. inclusive. 
First the if-in. size. Fig. 44, and the 2l|-in. size. Fig. 45, would 
each be laid out, the design and proportions being determined 
by the judgment of the designer. The chart. Fig. 46, is then con- 
structed by plotting the values of each dimension for the large and 
small sizes and connecting the plotted points by straight lines, when 
the ordinates corresponding to the intermediate sizes determine that 
particular dimension for those sizes. The letters showing to what 
dimension each line refers correspond to those in Fig. 47. Part of 




Fig. 42^ Fig. 43. 

Figs. 42 and 43.— Correct frame construction. 

the lines are laid ofi to the scale on the right side of the chart. These 
are distinguished from the remaining lines, which are to the scale on 
the left, by being dotted. From the chart the table, Fig. 47, is 
fiUed out. 

For instance, suppose it is desired to find the width D for the i^z 
in. size. On the chart the intersection of the vertical line marked 
j-ps and the inclined line D is found to be close to the horizontal 
line corresponding to 25 in. on the right-hand scale. The dimension 
thus obtained is entered in the table. 

It will be noticed that there are no lines on the chart for dimensions 
F and G. A line is plotted for the distance from the center of the bolt 
holes to the outside of the metal around the bearing, or ^{F — B), 
and from this F is determined, the B column having previously been 
filled out. A line is also plotted for the distance from the center of 
the bolt holes to the ends of the bases or |(G — F), the line in this 
case coinciding with the line for E. ^{G — F) having been obtained 
from the chart, G is found from F by addition in the same way as 
F was previously found from B. 

The reason for determining these two dimensions in this indirect 
manner is that these dimensions depend partly upon B and partly 
upon the size of the bolts, and for that reason will not increase in a 
regular manner, the increase being greater whenever the size of bolt 
changes. In this particular, as in many others, judgment and dis- 
cretion are necessary in the use of such a method. 

In general, the lines thus found will not pass through the origin 
but above it. After some experience with the method, judgment 
will enable one to use it with only one originally designed size if the 



new sizes are not too much larger than it. The dimensions of one 
size being plotted, the lines are drawn through the plotted points 
and at such distances above the origin as experience indicates. 



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<&, 


^''M 


i-M. 


1 


1% 


% 


3M 


4« 


% 


"/,0 


y> 


l'/16 


iM 


IX 


•iH 


% 


4M 


5% 


% 


y. 


% 


1V,B 


2H 


ly^ 


■M 


'A 


4M 


t% 


%. 


y. 


H 


I'^ie 


3« 


iji 


3 


y. 


6 


m 


% 


y. 


X 


1'5/iB 


3M 


\Y> 


m 




bVi 


-h 


yi 


1 


y. 


2M« 


i^A 


2H 


3H 


1 


6« 


8« 


Y, 


\H 


1 


2V16 


4K 


2?i 


i% 


iH 


7 


Vi 


1 


IX 


IH 


2'ho 


4K 


1% 


m 


i% 


■i7t 9!i 


1 


i« 


IX 


2>5'10 


5« 


2K 


5 


m 


8 10 Vj 


\% 


134 


IM 



Fig. 47. 

Figs. 44 to 47. — Chart method for the systematic design of machine 

parts. 

This method is safer if the new sizes are smaller than the original. 
When larger, caution should be used by comparing the resulting parts 
with the chart and correcting the latter if found desirable. 



PLAIN OR SLIDING BEARINGS 



For additional information on steam-engine bearings see Bear- 
ings for Steam Engines. 

For additional information on gas-engine bearings see Bearings for 
Gas Engines. 

For the fit allowances of bearings see Press and Running Fits. 

Table i. — Permissible Pressures on Bearings eor Steam 
Engines and Other Machines 



Kind of bearing and condition of operation 



Bearings for very low speeds and intermittent 
service as in turntables and bridges. 



Allowable bearing 

pressure in pounds 

per square inch of 

projected area 



7000 to 9000 



American Railroad Practice 



Locomotive cross-head pin bearings 

Locomotive crank pin bearings 

Locomotive driving wheel journal bearings. 

Car axle bearings 

Tender axle bearings 



3000 to 4000 
1500 to 1700 

to sso 

300 to 325 
to 42s 



British Railway Practice 



Locomotive crank pin bearings 

Locomotive cross-head pin bearings. 
Locomotive driving axle bearings. . . 
Car axle bearings 



... to 1400 
, . . to 2000 

250 to 300 
to 330 



United States Naval Practice 



Main engine bearings 

Main engine crank pin bearings 

Steam turbine bearings (for weight alone) . 
Thrust bearings for torpedo boats 



27s to 400 
400 to 500 
. ... to 8s 
, ... to so 



Merchant Marine Practice 



Main engine bearings 

Main engine crank pin bearings. 
Thrust bearings 



400 to soo 

400 to SCO 

70 



High-speed Stationary Engine Practice 



Main bearings (for dead load) 

Main bearings (for steam load) 

Crank pin bearings, overhung crank. 

Crank pin bearings, center crank 

Cross-head pin bearings 



60 to 120 

ISO to 2S0 

900 to I soo 

400 to 600 

1000 to 1800 



Slow-speed Stationary Engine Practice 



Main bearings (for dead load) . . 
Main bearings (for steam load). 

Crank pin bearings 

Cross-head pin bearings 



80 to 140 

200 to 400 

800 to 1300 

1000 to 1500 



Air Compressor Practice^ 



Straight line, steam-driven, 100 lb. steam and air 



Main bearings 

Crank pin bearings 

Cross-head pin bearings. 



160 to 237 
S6s to 700 
628 to 820 



Straight line, belt-driven, center crank, 100 lb. steam and air 



Main bearings 

Crank pin bearings 

Cross-head pin bearings. 



122 to 220 
244 to 402 
400 to 78s 



Table i. — Permissible Pressures on Bearings for Steam 
Engines and Other Machines. — (Continued) 



Kind of bearing and condition of operation 



Allowable bearing 

pressure in pounds 

per square inch of 

projected area 



Straight line, belt-driven, side crank, 100 lb. steam and air 



Main bearings 

Crank pin bearings 

Cross-head pin bearings. 



178 to 227 
628 to 825 
628 to 825 



Straight line, steam-driven, side crank, 100 lb. steam and air 



Main bearings 

Crank pin bearings 

Cross-head pin bearings. 



198 to 227 
462 to 82s 
462 to 825 



Duplex, Meyer cut-off, steam-driven, 100 lb. steam and air 



Main bearings 

Crank pin bearings 

Cross-head pin bearings. 



157 to 200 
644 to 8ss 
8so to 1370 



Duplex Corliss valve gear, steam-driven, 100 lb. steam and air 



Main bearings 

Crank pin bearings 

Cross-head pin bearings 

Direct-connected, motor-driven main bearings. . 



1x5 to 141 

513 to 708 

732 to 1150 

. ... to 70 



Gas Engine Practice 



Main bearings 

Crank pin bearings 

Cross-head pin bearings. 



500 to 700 
I soo to 1800 
1500 to 2000 



Electrical Machinery Practice 



Generator and motor bearings 

Main engine bearings, driving generators. . 

Horizontal steam turbine bearings 

Vertical steam turbine steps 



30 to 80 

40 to 80 

40 to 60 

200 to 1000 



Rolling Mill Practice^ 





Rubbing velocity. 






Ft. per Min. 




Pinion housing bearings. . . . 


3SO to 600 


3o.to so^ 


Roll housing bearings 


350 to 600 


100 to 2000^ 


Table roller bearings 


ISO 


30 to so 


Table line-shaft bearings... . 


ISO 


30 to so 


Main bearings of shears. . . . 


SO to 6s 


1800 to 2500 



Miscellaneous Practice 



Bearings for slow-speed and intermittent load as 
in punch presses, shears, and the like. 
Main bearings of slow-speed pumping engines.... 
Heavy line-shaft bearings, bronze or babbitt lined . 

Light line-shaft bearings, cast-iron 

Heavy slow-speed step bearings 

Drill press thrust collars 

Angular-thrust bearing for boring mill tables. . . . 



3000 to 4000 

.... to 600 

100 to ISO 

IS to 2S 

.... to 2000 

to 32s 

....to 7s' 



'Canadian IngersoU-Rand Company 



* Mesta Machine Company, Pittsburg, Pa. 

^ These factors are of value as showing good practice, not for purposes of 
design. The diameters and lengths of the bearings are determined by the 
requirement of strength in the pinion and roll necks and their housings. 

^ Practice of Bullard Machine Tool Company. 



PLAIN OR SLIDING BEARINGS 



9 



"Whenever it is possible to give journals end play, it will be found 
that they polish rather than cut and endure rather than wear out. 
The wearing surfaces should be of equal length in the box and shaft 
and be so controlled that the overrun will be equal at each end" 
(Professor Sweet). 

Much of what follows is taken from the exhaustive treatise, Bear- 
ings and Their Lubrication, by L. P. Alford, to which the reader is 
referred for much additional information not to be found elsewhere. 

The lack of a complete theory connecting the pressures, velocities 
and temperatures of bearings, until it was supplied by Axel K. Ped- 



ERSEN {Amer. Mach., Oct. lo, 1912) and given below, has made it im- 
possible to determine pressure factors of general application. Never- 
theless, the factors in common use, within the fields from which 
they were obtained and to which they are intended to apply, are use- 
ful and adequate. 

Tables i to 8 give such pressure factors. Tables 2 to 8 are by G. 
W. Lewis and A. G. Kessler (Amer. Mack., Aug. 31, Sept. 14, Nov. 
9, 191 1 ). They are the result of an extended investigation and 
correctly represent modern practice. 



Table 2. — Main Bearing Pressures for Stationary Gas Engines 



Horizontal 


D 


4 


8 


12 


16 


20 


Assumed 


Dmb 

Lmb 

Amb 


1-3 
2.6 

3-4 


3-1 

6.75 
20.9 


4.8s 
10.8 


6.6 
14.9 
98. 5 


8.4 
19. 1 

160.5 


DmbXLmb 


Pm 






250 






Assumed 


Kmb 


462 


300 


270 


255 


244 




Pm 






300 






Assumed 


Kmb 


553 


360 


324 


307 


293 




Pm 






35° 






Assumed 


Kmb 


647 


420 


377 


358 


342 




Pm 






400 






Assumed 


Kmb. 


738 


503 


430 


408 


391 





D = cylinder diameter, ins. 

£>w6 = main bearing diameter, ins. 

Lmb =main bearing length, ins. 

.4 w6 = projected area main bearing (one) = DmbXLmb. 



Vertical 


D 


4 


8 


12 


16 


20 


Assumed 


Dmb 

Lmb 

Amb 


. I* 

. 3i 
■ 5-25 


3i 

61 

23.6 


5^ 
10 

55 


7i 
13 
97-5 


9i 
16 
152 


DmbXLmb 


Pm 






250 






Assumed 


Kmb 


• 300 


267 


258 


258 


258 




Pm 






300 






Assumed 


Kmb 


•1 359 


320 


310 


310 


310 




Pm 






350 






Assumed 


Kmb 


■1 415 


373 


360 


360 


360 




Pm 






400 






Assumed 


Kmb 


■ 481 


414 


414 


414 


414 





Pm = maximum explosion pressure, lbs. per sq. in. of piston face. 
Kmb = maximum unit bearing pressure, lbs. per sq. in. considering 
explosion to occur on dead center. 



Table 3. — Crank-pin Bearing Pressures For Stationary Gas Engines 



Horizontal 


D 


4 


8 


12 


16 


20 


Assumed 


Dcp 

Lcp 

Acp 


If 
2.44 


3i 

3i 

10.15 


4i 

4l 
23.2 


61 

6^ 

41.75 


8i 
81 
68 


DcpXLcp = Acp 


Pm 






250 






Assumed 


Kcp 


1290 


1240 


1220 


1210 


1150 


From equation A 


Pm 






300 






Assumed 


Kcp 


1550 


148s 


1450 


1450 


1390 


From equation A 


Pm 






350 






Assumed 


Kcp 


1800 


1730 


1710 


1690 


1620 


From equation A 


Pm 






400 






Assumed 


Kcp 


2060 


1980 


1950 


1930 


1850 


From equation A 



D = cylinder diameter, ins. 

Pm = maximum explosion pressure, lbs. per sq. in. of piston face. 

I>c/i = bearing diameter of crank pin, ins. 

Lcp = bearing length of crank pin, ins. 



Vertical 


D 


4 


8 


12 


16 


20 


Assumed 


Dcp 

Lcp 

Acp 


If 
if 
2 . 64 


3i 

3f 

II. 8 


4l 

5f 

27.8 


6i 

7f 
49-75 


8A 
9f 

78.75 


DcpXLcp 


Pm 






250 






Assumed 


Kcp 


1 190 


1065 


1035 


1015 


995 




Pm 






300 






Assumed 


Kcp 


1430 


1280 


1240 


1215 


1200 




Pm 






350 






Assumed 


Kcp 


1660 


1490 


1440 


1420 


1400 




Pm 






400 






Assumed 


Kcp 


1920 


1720 


1660 


1620 


1600 





jRTc/) = maximum unit bearing pressure, lbs. per sq. in. 
Acp=DcpXLcp, sq. ins. 

Max. Kcp = 71 — Pm^ (DcpXLcp). (A) 



[ 



10 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 4. — Wrist or Piston-pin Bearing Pressures for Stationary Gas Engines 



Horizontal 


D 


4 


8 


12 


16 


20 




Dwp 

Lwp 

Awp 


°-93 

1.6 

1.49 


1.62 
2.8 

4-54 


2.76 

4-77 
13.2 


4-36 

7-52 
32.8 


6.42 
II. 15 

71-5 


From equation (A) 
From equation (B) 
DwpXLwp 


Pm 






250 






Assumed 


Kwp 


2100 


2760 


214s 


1530 


1 100 




Pm 






300 






Assumed 


Kwp 


2530 


3320 


2570 


1840 


1320 




Pm 






350 






Assumed 


Kwp 


2950 


3880 


3000 


2150 


1 1540 


1 


Pm 






400 






Assumed 


Kwp 


3370 


442s 


3430 


2455 


1760 


1 



Dwp= .0143 Z)^+.7 in. (A) 
Lwp =1.75 Dwp (B) 

D — cylinder diameter, ins. 

Z'm'^ = bearing diameter of piston pin, ins. 

Lwp = bearing length of piston pin, ins. 



Vertical 


D 


6 


8 


12 


16 


20 




Dwp 

Lwp 

Awp 


If 

2| 
4.67 


If 

3f 
6.33 


2i 

4i 
11.25 


3f 

64 

20.65 


4i 

84 

36.6 


Equation (C) 
Equation (D) 
DwpXLwp 


Pm 






250 






Assumed 


Kwp 


I5IO 


1990 


2520 


2430 


2I4S 




Pm 






300 






Assumed 


Kwp 


181O 


2380 


3020 


1 2920 


2580 




Pm 






350 






Assumed 


Kwp 


2120 


2780 


3520 


3410 


3010 




Pm 






400 






Assumed 


Kwp 


2420 


3190 


4030 


3900 


3440 





Dwp= .00795 I>'+ifin. (C) 

Lwp= 1.S2 Dwp (D) 
.4 01/1 = projected area piston pin, sq. ins. 
Pm = maximum unit explosion pressure. 
Kwp = maximum unit bearing pressure, lbs. per sq. in. 



Table 5. — Main Bearing Pressures for Automobile Engines 



Center bearings 



D 


4 


4i 


5 


si 




Deb 
Lcb 
Acb 


3.42 


If 

2f 

4-3 


3flt 
S.7S 


2A 

3l 

7-5 


From equation (5) 
From equation (6) 
DcbXLcb 


Pm 






250 




Assumed 


Kcb 


690 


615 


620 


575 




Pm 






300 




Assumed 


. Kcb 


830 


730 


750 


690 




Pm 






350 




Assumed 


Kcb 


970 


855 


870 


810 




Pm 






400 




Assumed 


Kcb 


IIOO 


980 


1000 


920 





D = cylinder diameter, ins. 
Dcfc = diameter o£ center bearing, ins. 
Lcb = length of center bearing, ins. 
K!;6 = maximum unit bearing pressure, lbs. per sq. in. 
Pm = maximum explosion pressure, lbs. per sq. in. of piston face. 
Z)cfc= .32 n+.3 in. (S) 
Lcb =2.8 Deb — 2.2 in. (6) 



Front bearings 



D 


4 


4^ 


5 


Si 1 


Dfb 
Lfb 
Afb 


2IJ 

4.2 


If 

2j 
5-02 


3 
5.63 


21V 
3* 
6.6 


From equation (7) 
From equation (8) 
Dfb X Lfb 


Pm 






250 




Assumed 


Kfb 


560 


595 


640 


670 




Pm 






300 




Assumed 


Kfb 


665 


710 


775 


800 




Pm 






350 




Assumed 


Kfb 


780 


830 


900 


930 




Pm 






400 




Assumed 


Kfb 


890 


945 


1030 


1075 





D = cylinder diameter, ins. 

Dfb = diameter of front bearing, ins. 

Lfb = length of front bearing, ins. 

Kfb = maximum unit bearing pressure, lbs. per sq. in. 

Pm = maximum explosion pressure, lbs. per sq. in. of piston face. 

D/i=.32D+.3 in. (7) 

Lfb = Dfb + iiir).. (8) 



Rear bearings 



Drb 
Lrb 
Arb 



4 


4i 


5 


Si 


lA 


II 


i| 


2A 


3 


4 


4f 


s\ 


4-7 


7 


8.67 


II. 6 



From equation (9) 
From equation (10) 
DrbXLrb 



Pm 






250 




Assumed 


Krb 


1 565 


495 


1 495 1 


46 s 




Pm 






300 




Assumed 


Krb 


1 660 


575 


1 575 1 


535 




Pm 






350 




Assumed 


Krb 


t 760 


660 


1 655 1 


605 




Pm 






400 




Assumed 


Krb 


1 855 


735 


1 735 1 


675 





D = cylinder diameter, ins. 
Drfr = diameter of rear bearing, ins. 
Lrb =length of rear bearing, ins. 
y4r6 = projected bearing area, sq. ins. 
Krb = maxirrmm bearing pressure, lbs. per sq. in. 
Pm = maximum explosion pressure, lbs. per sq. in. of piston face. 
Drb= .32D+ .3 in. (o).' Lrb = 5.3 Drb — 5.3 m. (10) 



Relation of Speed and Pressure 

The velocity of rubbing being, equally with the load, a factor in 
determining the work of friction which must be dissipated as heat, it 
follows that the velocity of rubbing should appear in a rational for- 
mula for the size of bearings. Such a formula has been given the 
form 

pv = C, 
in which />= pressure on projected area, lbs. per sq. in. 
V = velocity of rubbing, ft. per min. 
C = a constant determined from observation. 
Values of this constant are much less numerous than those for simple 
pressure. Table 9 gives such authentic values as the author has been 
able to find. 

The sources of these constants are as follows: (i) A. M. Bennett 
{Amer. Mach., June 17, 1909), who bases his conclusions on a large 
number of bearings which operated under a rise of temperature not 
exceeding 72 deg. Fahr.; (2) H. P. Bean {Trans. A. S. M. E., Vol. 
27); (3), (4) and (7) Jas. Christie {proc. Engrs. Club ofPhila., 1898); 

(Continued on next page, second column) 



PLAIN OR SLIDING BEARINGS 



11 



Table 6. — Wrist-pin Bearing Pressures for Automobile 
Engines 



Table 7.- 



-Crank-pin Bearing Pressures for Automobile 
Engines 



D 


4 


4i 


5 


5h 




Dwp 

Lwp 

Awp 


i 
2 

1-75 


I 

2.25 

2.25 


2f 
3.12 


If 
3A 

4.2 


From equation (i) 
From equation (2) 
DwpXLwp 


Pm 






250 




Assumed 


Kwp 


1800 


1780 


1570 


1420 


Equation (a) 


Pm 






300 




Assumed 


Kwp 


2150 


2130 


1890 


1700 


Equation (a) 


Pm 






350 




Assumed 


Kwp 


2510 


2480 


2200 


1980 


Equation (a) 


Pm 






400 




Assumed 


Kwp 


2870 


2840 


2510 


2270 


Equation (a) 



D = cylinder diameter, ins. 
Dwp = bearing diameter, ins. 
Lwp = bearing length, ins. 
^ «)/>• = projected bearing area, sq. ins. 

Pm = maximum unit explosion pressure, lbs. per sq. in. of piston 
face. 

Dwp= .34.D- .53 in. (i) 
Lwp =2.2$ Dwp (2) 

Kwp = maximum unit bearing pressure, lbs. per sq. in. 
Pm D^ .7854 , . 

^^^ = — A^^r~ ^^^ 



Table 8. — Average Rubbing Speed and Work of Friction for 
Automobile Engines 

Computed for a piston speed of 1000 ft. per min. and a mean pres- 
sure of 20 lbs. per sq. in. of piston face. 



D 


4 


4i 


5 


54 


Average 


L 


4l 


S 


si 


6 




RPM 


1370 


1200 


1090 


1000 




Rubbing speed in ft. 
per min. on bear- 
ings listed below. 


560 


SSo 


535 


540 


546 


V 


9-3S 


91S 


8.9 


9 


91 


P (me a. 71) ; . . . . 


252 


320 


395 


475 








Crank-pin / Km 


76 


77 


84 


84 


80 


bearing. \ W 


710 


70s 


745 


755 


729 


Center bear- / Km 


ss 


49.2 


49.8 


46 


50 


ing. \ W 


SIS 


4SO 


445 


415 


456 


Front bear- f Km 


44-7 


47 -S 


52 


53-4 


49-4 


ing. \ W 


418 


435 


462 


480 


449 


Rear bearing. I 


S6 


45-5 


43-5 


38 


45-7 


S^i 


415 


388 


342 


417 



D = cylinder diameter, ins. 

L = stroke, ins. 

P{mean) = mean total pressure on piston for entire cycle, lbs. (assumed). 

RPM =r.p.m. at 1,000 ft. per min. piston speed. 

Km = mean unit bearing pressure, lbs. per sq. in. 

V = rubbing speed, ft. per sec. 

W = work of friction = (KmXfV) ft. lbs. per sec. 

L =i.iZ>. 



D 


4 


4i 


5 


54 




Dcp 

Lcp 

Acp 


3-32 


2| 

4-17 


i| 
4.68 


2^ 
5-68 


From equation (3) 
From equation (4) 
Dcp X Lcp 


Pm 






250 




Assumed 


Kcp 


945 


960 


1050 


1050 


From equation (b) 


Pm 






300 




Assumed 


Kcp 


1130 


1150 


1260 


1260 


From equation (b) 


Pm 






35° 




Assumed 


Kcp 


1320 


1350 


1470 


1470 


From equation (b) 


Pm 






400 




Assumed 


Kcp 


1510 


1540 


167s 


1675 


From equation (b) 



D 

Dcp 

Lcp ■■ 
Acp 
Kcp 
Pm 



cylinder diameter, ins. 
bearing diameter, ins. 
bearing length, ins. 
projected bearing area, sq. ins. 
= maximum unit bearing pressure, lbs. per sq. in. 
= maximum unit explosion pressure, lbs. per sq. in. of piston 
face. 

Dcp =.32D+.3 in. (3) 
Lcp = 1.3s Dcp (4) 

Pm Z)2 .7854 



Kcp = 



Acp 



(b) 



(5) and (6) Fred. W. Taylor {Trans. A. S. M. E., Vol. 27), whose 
figures are based on observations on eleven bearings in an overloaded 
mill; (8) and (9) G. W. Dickie {Trans. A. S. M. E., Vol. 27); (10), 
(11), (12), (13) and (14) The Mesta Machine Co. 

In interpreting these constants regard must be had for the influence 
of reciprocating and momentary loads. The former is seen in (2) 
and (3) and the latter in (11) and (14). 

It is probable that the diversity of the constants is largely due to 
the inaccuracy of form of the equation, the probability being that the 
pressure should not be reduced in the same proportion that the speed 
is increased. A recognition of this is the basis of Edwin Reynold's 
rule for the main bearings of steam engines, which see. 

Table 9. — Product of Pressure, Lbs. per Sq. In. of Projected 
Area, and Velocity, Ft . per Min. of Bearings 
Kind of Bearings and Condition of Operation Values of C 
(i) Self-aligning ring-oiled bearings with continuous 

load in one direction 36,000-40,000 

(2) Main bearings of Corliss engines (steam load 

only) 60,000-78,000 

(3) Steam engine crank pins 200,000 

(4) Steam engine cross head slide (figured on pres- 

sure at mid-stroke) 50,000 

(5) Mill shafting with self-aligning ring-oiled bab- 

bitted bearings, highest admissible value 24,000 

(6) Mill shafting with self-aligning cast-iron bear- 

ings, sight or wick oil feed or grease cups, should 

be less than 12,000 

(7) 110,000 lbs. freight or axle journals at 10 miles 

per hour 60,000 

(8) Water-cooled thrust bearings of steamships, cus- 

tomary value 37,500 

(9) Water-cooled thrust bearings of steamships with 

extra care in water cooling 61,000 

(10) Rolling-mill pinion-housing bearings i8,oco 

(11) Rolling-mill roll-housing bearings 60,000-70,000 

(12) Rolling-mill table roller bearings 4,500-7,500 

(13) Rolling-mill table line-shaft bearings 4,500-7,500 

(14) Main bearings of rolling-mill shears 120,000 



12 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Relation of Speed, Pressure and Temperature 

The following methods of bearing design are from the practice of 
the General Electric Co. and are the results of extended experimental 
investigations: It is very desirable in laying out bearings to keep the 
diameter as small as possible, consistent with sufficient strength of 
shaft and suitable deflection of the journals both inside and outside 
the bearings as the work of friction is thereby reduced. It is also very 



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200 400 600 800 



1200 IGOO 2000 2400 2S00 3200 
Bubbing Speed, Ft. per Min. 



3600 4000 



Fig. I. — Relation between rubbing speed and safe maximum pressure 
on bearings without artificial cooling for perfect film lubrication. 

- desirable to so dimension bearings that they are fairly well loaded, in 
order to avoid bulky machines and also because the coefficient of 
friction rises quite rapidly when the load is less than 50 lbs. per sq. 
in. of projected area. 

When calculating the projected area of any bearing, especially if 
it is to be heavily loaded, the amount of space lost through the drain 



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Pressure.Lbs. per Sq. In. Proj. Area 
Fig. 2. — Temperature rise of oil-ring bearings in still air, room 
temperature 75 deg. Cent. — 77 deg. Fahr. 

grooves at both ends must be deducted. This is particularly impor- 
tant when the length of the bearing is small in proportion to the 
diameter. 

It is also necessary — and this applies to all forms of lubrication — 
that there be no sharp corners on the edges of the oil distributing 
grooves or channels, but that these be gradually cased off so that 
the oil can be drawn in between the journal and the bearing. Sharp 



corners are invariably oil wipers and often absolutely prevent proper 
lubrication. 

The heat generated in any bearing may be dissipated: 

1. By radiation from the housings and conduction by the shaft. 

2. By forcing cooled oil through the bearing. 

3. By surrounding the bearing by some form of water jacket. 
Bearings without artificial cooling are usually lubricated by oil rings 

or similar devices or by gravity feed. It is essential that an abun- 
dant supply of oil be delivered to all parts of the bearing by suitably 
arranging the channels so that a perfect film will be maintained at all 
times between the journal and bearing, and that there is no oppor- 
tunity for the oil forming this film to escape through openings or 
grooves at the points of greatest pressure and thus allow the metals 
to come in contact. 

The heat generated in bearings having no artificial cooling is con- 
ducted away and radiated by the housings. The great variation in 
the design of bearing housings and the different conditions of venti- 
lation, etc., make it extremely difficult to predetermine the ultimate 



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Pressure.Lbs, per Sq.In.Proj.Area 



Fig. 3. — Temperature rise of oil-ring bearings for well ventilated 
condition but without artificial cooling, room temperature 25 deg. 
Cent. — 77 deg. Fahr. 

temperature of such bearings with any great accuracy, and it is 
always necessary to allow a considerable margin of safety. 

Fig. I covers the range of pressure and speed ordinarily permissible 
in this type of bearing, while Figs. 2 and 3 show the ultimate tempera- 
tures for different speeds and loads. These curves were made up 
from the readings obtained from special bearings and afterward 
checked by the test records of a large number of machines — both of 
the pillow-block and shield types — which have gone through the test- 
ing department. Fig. 2 shows the temperatures to be expected under 
the most unfavorable conditions, that is, of a bearing so situated that 
no current of air can circulate about it, and therefore cooled by radia- 
tion only. There is, however, a considerable circulation of air about 
most machines, due to the fanning action of the revolving parts, and 
the ultimate temperatures to be expected in such cases are shown by 
the curves of Fig. 3. These curves apply to the great majority of 
open generators and motors, both of the pillow-block and end-shield 
types. When the machine is enclosed, or the free circulation of air 
in any way interrupted, higher temperatures will result, until finally 
the conditions of Fig. 2 are reached. 

A part of the heat of the bearings of motors and generators is usu- 
ally conducted away by the shaft and radiated by the spider and 
other revolving parts. When machines are totally enclosed or are 
connected to other machinery whose temperature is high, heat may 
be transmitted through the shaft to the bearing, thus raising the 
latter's temperature, and due allowance must be made for this. 



PLAIN OR SLIDING BEARINGS 



13 



When bearing pressures and speeds are unusually high it is often 
necessary to force oil under pressure into the bearings and advantage 
is often taken of this to keep the heating down by artificially cooling 
the oil. This method, although used to a very considerable extent, 
is usually not as efficient as a water jacket. 

For all practical purposes, it may be considered that the entire 
heat generated is taken away by the oil, and it is therefore possible 
to predetermine the bearing temperature with considerable accuracy. 
Fig. 4 shows the ultimate temperature of bearings using the quantities 
of oil most commonly pumped through, and with the assistance of 
these curves the necessary amount can be determined. Intermediate 
speeds and pressures can be easily interpolated. 

For pressures and speeds beyond the limits of this table, it is advis- 
able to resort to water-jacket cooling. 

In arranging bearings for this form of lubrication, care must be 
taken to force the oil to the point where the work is being done, as 
otherwise the oil coming from the bearing may be comparatively cool. 



water circulated per minute, and, where the conditions are unusually 
good and the jacketing carefully arranged, from lo to 12 h.p. per 
gallon can be dissipated. 

With water jackets any suitable method of lubrication may be 
used which will insure at all times a good film of oil between journal 
and bearing. 

For designs of water-jacketed bearings, see below. 

Conditions of Film Lubrication 

The experiments of Beauchamp Tower (Froc. I. M. E., 1885) demon- 
strated that, given flooded lubrication and suitable relations of speed 
and pressure, the condition of affairs between a journal and bearing 
becomes that illustrated in exaggerated form in Fig. 5. The rotating 
journal assumes an eccentric position in its bearing, and is separated 
from it by a circular wedge-shaped film of oil. The journal brings up 
more oil than can be carried around the space between journal and 





















































































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Fig. 4.- 



1500 1800 2100 

Rubbing Speed, Ft. per Min. 
Loads in lbs. per sq. in. of projected area. Oil feeds in gals, per min. 
-Relation between rubbing speed and rise in temperature for forced lubrication and three rates of oil feed, room temperature 25 deg. 

Cent. — 77 deg. Fahr. 



while the bearing itself is much too warm. In addition to this, it 
means that an excessive amount of oil must be pumped through the 
bearing requiring unnecessarily large pumps, piping, etc. Experi- 
ments with such bearings show that when oil begins to run out of the 
ends quite freely, nothing is gained by forcing through a larger 
quantity. 

A properly designed water jacket will carry away a very much larger 
amount of heat than wiU any form of forced oil lubrication, as the 
specific heat of water is very much higher. As is the case of forced 
oil lubrication the ultimate temperature of a well-designed water- 
jacket bearing can be very accurately determined. It is essential 
that the pipes or channels be located as close as possible to the surface 
of the lining where the work is being done, in order that the heat 
generated may be absorbed without danger of damage to the lining. 
Water circulated at some distance away from the lining surface is of 
comparatively little assistance, as heat may be generated so rapidly 
that the lining will be destroyed before the heat reaches the jacket. 
The water passages must also be so arranged that an even and con- 
tinuous circulation is kept up in all parts. 

With properly constructed passages, it is safe to assume that heat 
may be removed at the rate of from 3 to 5 h.p. for each gallon of 



bearing, and some oil is therefore forced out sidewise and, the film of 
oil resisting this action by virtue of its viscosity, there is set up a 
wedging action which will support the bearing away from the journal 
against considerable pressure. By drilling holes in the bearing and 
inserting pressure gages, Mr. Tower found curves somewhat like 
a' c" b' to represent the pressure at various (projected) points of the 
circumference of the journal. The film is thinnest, not at the point of 
application of external load, but at a point somewhat farther along in 
the direction of rotation. 

The summation of these pressures was found to equal the total load 
on the bearing with a surprising degree of accuracy. 

An immediate practical result of these experiments is the demon- 
stration that the oil should be introduced at the point of no pressure. 

Mr. Tower's experiments show that the action of high-speed bear- 
ings is entirely different from that of low-speed bearings. In the 
latter we have oily surfaces in actual rubbing contact. An accidental 
increase of temperature reduces the viscosity of the lubricant, which 
in turn increases the intimacy of contact, thereby bringing about ad- 
ditional cumulative increase of temperature. Such a bearing may be 
said to be, as regards temperature, in unstable equilibrium. A high- 
speed bearing, on the other hand, is in stable equilibrium. If the 



14 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



speed is sufficiently above the critical speed at which the film action is 
established to prevent the reduced viscosity from bringing the surfaces 
into actual contact, there is no reason why the heating action should 
be cumulative, and such bearings may safely be run at temperatures 
that would be unsafe below the critical speed. 

Similar difference exists in the tendency to wear. H. M. Martin 
{The Design and Construction of Steam Turbines) says that steam 
turbine bearings, after years of use, show no signs of wear. 

For these reasons the complete film system of lubrication should be 
aimed at whenever possible. Dr. Herbert F. Moore {Amer. Mach., 
Sept. lo, 1903) determined experimentally the relation of pressure and 
rubbing speed at which the film breaks down and the lubrication 
becomes of the ordinary kind between oily surfaces. Dr. Moore's 
results are represented graphically by the full line of Fig. 6, the dotted 
line being an approximation represented by the equation: 

Pmox = 7.47\/^, 

in which Pmax= limiting pressure on projected area of bearing at 
which the oil film breaks down, lbs. per sq. in. 
D = velocity of rubbing, ft. per min. 




Fig. 5. — Journal and Bearing with film lubrication. 



This equation is fundamental and is generally accepted. It forms 
the starting-point of the first complete theory connecting the pressures, 
velocities and temperatures of bearings, by Axel. K. Pedersen, 
analytical expert the General Electric Co. {Amer. Mach., Oct. 10, 
1912). 

Mr. Pedersen's remarkable deductions are based on a large number 
of widely scattered experiments, including those of Beauchamp 
Tower, and are given below. It must be remembered that they 
apply to complete film lubrication only, the bearing proportions being 
determined from the conditions for preserving a perfect film at a 
permissible final bearing temperature. 

Introducing a proper factor of safety, Mr. Pedersen obtains the 
equation: 

d^=.o6S453(^^Wy~ {a) 

in which <f = diameter of bearing, ins., 
5 = factor of safety, 

Pmax J 

actual pressure on projected area, lbs. per sq. in. 

W = total load on bearing, lbs., 

/ = length of bearing, ins., 

I 
x = - 



—d 



I . 



w = r.p.m. of journal. 
For each class of machinery, the ratio ^ = t is a well-defined quan- 
tity. Following are customary values of this ratio: 



Type of bearing • Values I -i-d 

Marine engine main bearings i to 1.5 

Stationary engine main bearings i . 5 to 2 . 5 

Ordinary heavy shafting with fixed bearings 2 to 3 

Ordinary shafting with self-adjusting bearings 3 to 4 

Generator and motor bearings 2 to 3 

Machine-tool bearings 2 to 4 

Equation (a) can readily be used for determining the diameter of 
the bearing. The factor of safety is selected by considering the 
importance of safe running. A factor of safety of i would indicate 
that the journal is running under limiting conditions, that is, that the 
oil film is on the point of breaking down. For ordinary light machin- 
ery, the factor of safety may be taken as low as 2 and for heavy 
(especially high-speed) machinery as high as 8 or even 10. As a 
good average 4 to 5 may be taken at the first trial. 



90 " '- " ip 




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^ "^ -* '' '^ 




80 - - - _^^>- 




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70 - _ _ - -y" y'^ 


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20 - s^ 


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r 


in J~ 


10 tH 


7 


r 







20 40 60 80 100 120 140 

Velocity of Rubbing, Ft. per Mln. 

Fig. 6. — Breaking-down point of perfect oil film. 

The alignment chart, Fig. 7, was designed for the prompt solution 
of equation {a). The use of the chart is explained below it. 

The diameter of the bearing being thus determined, the length is 
fixed by the selected ratio x or 

l = xd (6) 

The pressure on the projected area is 

W 



'iXd 



{c) 



The fundamental consideration, in connection with the final bear- 
ing temperature and the specific losses, deals with the laws of friction 
and the heat-radiating capacity of a bearing. From the great num- 
ber of test data available in regard to the coefficient of friction the 
following important fundamental principles may be stated: For a 
journal revolving above 500 ft. per min. (8.5 ft. per sec. approxi- 
mately) we use the formula given by Lasche: 

fp{t-32)=51.2 {d) 

This formula is a very close practical approximation; actually, the 
coefficient of friction is not independent of the rubbing speed of the 
journal, but increases slightly with the speed up to a speed of about 
2000 ft. per min.; this increase, however, may be neglected. 



PLAIN OR SLIDING BEARINGS 



15 







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Connect the ratio of length divided by diameter and the selected factor of safety and note the intersection with axis I; 
connect the intersection and the load and note the intersection with axis II; connect this intersection with the revolutions per 
minute and read the diameter and rubbing speed from the appropriate scales. The chart may be read in the opposite direction 
if desired. 

E Fig. 7. — Dimensions of bearings for film lubrication. 



16 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



For a journal revolving below 500 ft. per min. (8.5 ft. per sec. ap- 
proximately) the coefficient of friction is dependent on the velocity 
of rubbing. From many test data it has been concluded that for 
these speeds 

fp{t-32) = 2.s\/6ov (e) 

In (d) and (e) /= coefficient of friction, 

^ = pressure on projected area, lbs. per sq. in. 
/ = final bearing temperature, Fahr. 
V = rubbing speed of journal, ft. per sec. 

Formula (e) has been fully corroborated by comparison with the 
experiments of Beauchamp Tower {Proc. I. M. E., 1885), the agree- 
ment being quite remarkable {Amer. Mach., Oct. 10, 191 2). 

The heat-radiating capacity of a bearing depends mainly upon the 
iron masses contained in it and upon the surrounding air. If the 
bearing is located in a place where the surrounding air is easily moved 
(ventilated bearing), and if the bearing contains large masses of 
iron, we have the best conditions possible. On the other hand, if 
the bearing and its housing are of comparatively small dimensions 
and the air is still, the heat-radiating capacity is at a minimum. 

In the chart. Fig. 8, the first condition is represented by the point 
M for ventilated bearings, the second by the point N for still-air 
bearings. We may have a condition where the bearing contains large 
masses of iron, but is surrounded by still air; evidently, then, a point 
located approximately midway between the points M and N should 
be used. 

The following formulas are very close approximations to the 
experiments by Lasche on the heat-radiating capacity of bearings. 
These experiments are given in chart form in " Bearings and Their 
Lubrication, " by L. P. Alford. 

For ventilated bearings we have the heat-radiating capacity in ft.- 
Ibs. per sec. per sq. in. of the projected area of bearing expressed by 
{t-to+izY ^ (_^) 



and for still-air bearings 



i860 



{t-to+3iY 



is) 



3300 
in which to = temperature of room, Fahr. 

t = final bearing temperature, Fahr. 
The maximum friction loss must not be greater than the heat- 
radiating capacity of the bearing, otherwise artificial cooling must be 
resorted to. The friction loss in ft.-lbs. per sec. per sq. in. of projected 
bearing area is pfv, hence for ventilated bearings 

(t-t„ + 33)^ 



and for still-air bearings 



pfv 



pfv = 



i860 



{t-to + 33y 



(A) 



H) 



3300 

As the coefiicient of friction follows different laws whether the rub- 
bing speed is above or below 500 ft. per min., we must consider this in 
formulas (h) and («). 

For speeds above 500 ft. per min., we combine {h) and (i) with (d), 
and solving for v, we get for ventilated bearings 

{t- 32){t- to + 33V 
j,_ 

95232 

for still-air bearings 

(t- 32) (t- to + 33)^ 



(j) 



168960 ^^^ 

For speeds below 500 ft. per min., (h) and (i) are combined with 
(e); then for ventilated bearings 



(^-32) it-to + 33)' ]^ 

X tOA„ J 



2.3\/6c 



i860 



for still-air bearings 



^ I 2.3\/6o 



(i-_to^+3_3)' 
33°° 



(/) 



im) 



Equations (j), {k), (Z), and (w) are the fundamental formulas for 
plotting the chart. Fig. 8, as far as the determination of the final bear- 
ing temperature is concerned. 

The chart also gives the specific losses y, namely 

y=Pf •(») 

Hence from (d) for speeds above 500 ft. per min., or 8.5 ft. per sec, 
approximately, 

t-2,2 ' 

and from (e) for speeds below 500 ft. per min. or 8.5 ft. per sec, approxi- 
mately, 

2.3\/6cj» 



y=- 



^ 



■32 



The total friction loss in the bearing is obtained from 
Y = yvld ft.-lbs. per sec. 



550 ^ 



iP) 



(g) 



(0 



In equations {iii)-iy) 

31 = specific losses; that is, the losses due to friction in ft.-lbs. per 
sec. per sq. in. of projected bearing area for each foot of rubbing 
speed of the journal, 
Y = total friction losses in the bearing. 

The use of the chart. Fig. 8, is as follows: (a) To determine the 
final bearing temperature: Locate the proper value of the rubbing 
speed on the A A scale, connect this point with points iV or If or 
some intermediate point on the line NM, according to the conditions 
of the surrounding air and the design of the bearing. The connecting 
line locates a point on the BB axis. Trace from here horizontally 
to the curve giving the proper temperature of the room, thence 
vertically down to the temperature scale and read the final bearing 
temperature. 

(6) To determine the specific losses y: Here different methods must 
be employed, one for speeds above 8.5 ft. per sec, and another for 
speeds below 8.5 ft. per sec. 

Rubbing speeds above 8.5 ft. per sec: 

From the final bearing temperature trace parallel to BB axis to the 
dotted curve CC, thence horizontally to the right to the axis DB and 
read the value of y, the specific loss. 
Rubbing speeds below 8.5 ft. per sec: 

From the final bearing temperature trace parallel to the BB axis 
to the dotted curve CC, thence horizontally to the left to the BB axis, 
thus locating a point on this axis. Now connect this point with the 
proper value of the rubbing speed on the speed scale EE; the connect- 
ing line intersects a point on the specific-loss scale FF, where the 
specific loss y is read. The general procedure is shown by the con- 
necting lines on the chart. 

Example i. — Design a motor bearing for the following data: 
Ventilated bearing and large masses of iron; 
Ratio of length to diameter = 2; 
Factor of safety for preserving perfect oil film = 5; 
Total load on bearing =1700 lbs.; 
Revolutions per minute =650; 
Temperature of room =75 deg. Fahr. 
From the chart, Fig. 7, we get the diameter <Z = 4.S ins., approxi- 
mately, hence from equation (6) 

/ = a;X(i = 2X4.S=9 ins., the length, 

then from equation (c) 
W 
^~lXd^ 



1700 



= 42 lbs. per sq. in. 
9X45 

The chart, Fig. 7, also determines the rubbing speed 

j)=i2.7 ft. per sec, nearly. 



PLAIN OR SLIDING BEARINGS 



17 































•09< 

lilt 


3 J9d -^y; g-g 9AoqB sp99ds Suiqqnjj aoj sgssoq 

in o m © in o 

CO -^ -^ m in to 

1 II 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 


oyio9dg 

mo in o 

<D t- t- 00 

1 1 1 1 1 t 1 1 1 1 1 1 1 1 1 


Q 


Curves for Temperature 
of Room, 
Deg. Fahr. 










































1 
























205 195 185 175 165 155 145 135 125 115 105 95. 
Final Temperature of Bearing, Degrees Fahrenheit 
































































?^ 


.^ 






















































^ 


L:: 


'> 


^ 


^ 
















































^ 




^ 


y 


y' 




^ 
















































-^ 


^ 


y' 






^ 


y 










■^ 






































^ 


^ 






^ 


^ 


^ 




y' 


^ 










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^ 






























r* 




^ 




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yy 










































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X 




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1 








































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y 




y 




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A 








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S g ^ S ^ S S / S cp op tp \ «> ? \ 7 ? <^ 

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 K 1 1 1 1 1 1 1 1 ^ 1 1 1 1 1 \l 1 1 1 II 1 1 1 1 1 1 


oq 




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o in o m \ o o\ o o o o 
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I \\v\a% ;Cj9AiiBJB(Iuro9 

12; "^^/ sSaijcag Jiy ll'is JOj luiOjj 



Fig. 8. — Final temperatures and specific losses of bearings. 



18 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Locating the rubbing speed 12.7 on the A A scale of Fig. 8, and pro- 
ceeding as previously explained, noting that the rubbing speed is 
above 8.5 ft. per sec, we get the final bearing temperature 

^ = 146 deg. Fahr., nearly. 

This is by no means an excessive temperature. The tendency of 
machinery builders, however, is to limit the final bearing tempera- 
tures to approximately 150 deg. Fahr.; this is very conservative. 
According to Bearings and Their Lubrication, by L. P. Alford: 
"In practice it is necessary to design bearings to run at a much lower 
temperature than will cause damage, because of the requirements 
of the average customer. Such a maximum temperature is from 
140 to 160 deg. Fahr. It is probably true that the average bearing 
could just as well run at a temperature of 200 deg. Fahr 

The actual temperatures of large bearings was the subject of observa- 
tions by A. M. Mattice {Trans. A. S. M. E., Vol. 27), who states 
that examination of the temperatures of a large number of main 
bearings of engines of various makes showed more large engines 
running with bearings at temperatures over than under 135 deg. 
Fahr. Many bearings were running at over 150 deg., some consider- 
ably higher, and in one case a continuous temperature of i8c deg. 
was found, and in all of these cases the bearings were giving no 
trouble. 

H. M. Martin {The Design and Construction of Steam Turbines) 
says "turbines in which the bearing temperature is constantly about 
igS deg. Fahr. have given no trouble in practice, but a more usual 
limit of temperature is 165 deg. Fahr." 

The bearing oil loses its lubricating qualities at a temperature 
about 250 deg. Fahr., approximately. Returning to our example, we 
get the specific loss y, which in this case is read on the DD scale 

y=-4Si- 
Hence from {q) F=3iiiW = . 451X12. 7X9X4.5 = 232 ft. -lbs. per sec. 

Example 2. — Given a bearing running at a rubbing speed of 6.5 ft. 
per sec. and the conditions of a still-air bearing with small iron masses; 
Fig. 8 must now be used according to the rules for speeds below 8.5 ft. 
per sec. We get the final bearing temperature =138 deg. Fahr., 
and the specific loss on scale FF, y = .43. 

The maximum allowable final bearing temperature at a given room 
temperature determines the maximum speed at which the journal 
can be run without artificial cooling; thus, in the first example, if 
146 deg. Fahr. is considered as the maximum allowable bearing tem- 
perature, we cannot run this bearing at a higher speed than 12.7 
ft. per sec. without artificially cooling the bearing. 

In the following, we shall only consider cooling by means of 
water. 

If ZJ — the temperature difference, deg. Fahr., of the water before 
and after cooling (a practical, average value of D is 20 to 25 deg. 
Fahr.), 

Fi = actual rubbing speed of journal, ft. per sec, 

F2 = maximum speed in ft. per sec, at which bearing can be run at 

the maximum allowable temperature without water cooling, 
31 = specific loss in bearing, corresponding to the rubbing speed 

V2, and determined by the chart, 

then to keep the bearing at a temperature corresponding to F2, we 
must use 

y(Fi-F2W 



Q- 



{s) 



108 D 
gallons of water per min. 

Example 3. — Suppose, for instance, that we wish to run the bearing 
in Example i at a speed of 20 ft. per sec, but that the maximum allow- 
able temperature must be kept at 146 deg. Fahr., then we have 

Fi = 2o ft. per sec, 

F2 = i2.7 ft. per sec, 
corresponding to a temperature of 146 deg. Fahr. at 75 deg. Fahr. 



room temperature, ^ = .451, as previously determined (see Example 
No. i), then, using the value D = 2o deg. Fahr., we get from {s) 
.451(20-12.7) ^^ 

Q= 5T^ X 0X4.5, 

^ 108X20 y 't o> 

= .062 gal. of water per min. 

A very important use of Fig. 8 thus consists in the possibility of 
determining the limiting speed at which a bearing can be run without 
artificial cooling at a given maximum bearing temperature. If 
high final bearing temperatures are allowed, very high rubbing speeds 
may be used; in fact, the allowable speeds increase much faster than 
the corresponding temperatures; thus, at a final bearing temperature 
of, say, 195 deg. Fahr., and at a room temperature of 75 deg. Fahr., 
Fig. 8, determines a limiting rubbing speed of 40 ft. per sec. (against 
12.7 ft. per sec. at 146 deg. Fahr.), for a ventilated bearing and a 
limiting rubbing speed of 23 ft. per sec. for a still-air bearing. 

In determining these speeds, the chart is read in the opposite direc- 
tion, by starting at the final bearing temperature, then tracing parallel 
to the BB axis until the room-temperature curve is reached, thence 
horizontally to the left to the BB axis. The point reached on this 
axis is then connected with M ox N (as the case may be), and the con- 
necting line intersects the speed axis AA at points, which give the 
maximum allowable rubbing speeds at the given maximum bearing 
temperatures. 

Materials for Bearings 

Materials for hearings form an endless subject of discussion. The 
author is convinced that cast-iron is entitled to a far wider use than 
it has received. It has the well-known property of taking on a glazed 
surface which is practically proof against wear. As John Richards 
has put it, "there is no doubt that prejudice or mistrust prevents the 
use of iron bearings in many cases where they are best." Failures 
of cast-iron bearings are charged to the material, while failures of 
other bearings are charged to fate or luck. 

Those who oppose the use of cast-iron fail to recognize the numerous 
cases for which its use is so habitual that nothing else is thought of. 
Of these, the most striking are the unbalanced slide valves of common 
steam engines, which work under heavy loads and very indifferent 
lubrication. Eccentrics and eccentric straps, especially of locomo- 
tives, work under scarcely less favorable conditions of load and lubri- 
cation, with the additional condition of high speed. - The tables of 
planers and boring mills and all manner of sliding bearings in machine 
tools form additional illustrations. For steam engine cross-head 
shdes and cross heads nothing equals it. Finally, the line-shaft hang- 
ers made by Wm. Sellers & Co. since prior to 1850 have been made 
of this material. These bearings have run for thirty years without 
appreciable wear. 

A test of cast-iron and other materials for live spindle lathe bearings 
was made by the R. K. Le Blond Machine Tool Co. {Amer. Mack., 
Mar. 23, 1911). Four experimental i8-in. lathes were fitted with dif- 
ferent combinations of bearing materials, as follows: 

1. Hardened steel spindle with cast-iron boxes. 

2. Soft steel spindle with babbitt boxes. 

3. Hardened steel spindle with bronze boxes. 

4. Soft steel spindle with bronze boxes. 

The soft steel spindles were of 6o-carbon crucible steel; the bronze 
was made to the specifications of the Pennsylvania Railroad Company. 
After the end of some 8 years' service and treatment as far as possible 
identical for all four lathes, it was found that their rating as regards 
absence of wear and general satisfaction was in the order as given 
above; that is, the hardened-steel spindle with cast-iron boxes was 
the best combination. Both spindle and boxes were in as good con- 
dition as when placed in the lathe, and from aU appearances and 
tests showed absolutely no wear. 

Mr. Le Blond adds the following general observations: The 
question of bearing metals is a question of affinity. One metal has 



PLAIN OR SLIDING BEARINGS 



19 



an affinity for a certain other metal, and as very often illustrated in 
life, a soft spindle may be married to a bronze box when its affinity is 
babbitt. In other words, a successful bearing must be composed of 
two metals of entirely different degrees of hardness and disposition. 
The only exception to this rule is cast-iron and cast-iron. 

It is a matter of general knowledge that a soft-steel spindle and 
a soft-steel bearing will immediately cut and run together; in fact, it is 
practically impossibile to lubricate this combination so it will not 
cut. 

A soft-steel spindle and bronze bearing is probably the next worst 
combination, as the metals are very similar in hardness and under 
the very best conditions will scratch and cut. 

A soft-steel spindle and cast-iron bearing will give splendid wear 
if properly lubricated, but will not stand for the shghtest neglect. 

A soft-steel spindle and babbitt will give excellent service, stand for 
a great deal of abuse; in fact is as near a fool proof proposition as any. 

The hardened-steel spindle and cast-iron will stand as much 
neglect as any combination of metals, has a much longer life, will 
retain its accuracy for an indefinite period, withstand intermittent 
pressure or a series of blows which would peen out and loosen babbitt, 
and, from our experience, the most fool-proof bearing in the world 
to-day is the cast-iron and hard spindle. It has indefinite Hfe, re- 
quires absolutely no adjustment and will stand the maximum of 
abuse. 

The original patent of Isaac Babbitt (issued in 1839) was not for the 
alloy known by his name, but for the method of its apphcation. 
The exact formula used by the inventor is not known. Tin, copper 
and antimony were the ingredients, and from the best sources of 
information the original proportions in per cent, were as follows: 

Tin = 89.3 or 83.3 or 89. i. 

Copper = 3.6 or 8.3 or 3.7. 

Antimony = 7 . i or 8 . 3 or 7.4. 

This metal, when carefully prepared, is one of the best metals in 
use for lining boxes that are subjected to heavy weight and wear. 

A concise summary of modern practice with composition bearing 
alloys is given by John F. Buchanan {The Foundry, 1906) thus: 

To make the best grade of babbitt or anti-friction metal, proceed 
as follows: Select the purest metals that can be had, and the most 
suitable recipe for the duty of the alloy; make a preUminary mix of 
the refractories in a plumbago crucible, and pour it out for "harden- 
ing." Melt the metal which forms the basis of the alloy (it may be 
tin, lead, or zinc), and dissolve the hardening therein, at a gentle 
heat, using sawdust, tallow, or powdered sal-ammoniac for a flux. 
For making a large quantity in the ordinary brass furnace, make a 
cast-iron crucible 2 ins. smaller than the diameter of the furnace; 
lower it into the furnace and lute round. One word of caution is 
needed here. Zinc should not be melted in an iron pot, but if melted 
in a plumbago crucible it may be poured and mixed with the other 
components of the alloy already melted in the pot. 

The utility of babbitt metal is not to be gaged by its cost per 
pound. A cheap babbitt (lead or zinc base), well made, may give 
better service than a costly mixture which has been carelessly 
blended. Generally speaking, the commercial grade numbers of 
bearing metals are for: i, light loads and high speeds; 2, medium 
loads and moderate speeds; 3, heavy loads and slow or moderate 
speeds; and 4, heavy loads and high speeds. Such grading is reason- 
able, for the hardness of the alloys increases with the numbers, and 
price does not count. 

Babbitt metal, correctly speaking, is a tin alloy, but modern en- 
gineering practice and commercial usage favor the apphcation of 
the name to all metals capable of the same duty as babbitt. Hence 
we get three series of babbitt or anti-friction metals: ist, the tin 
series; 2d, the lead series; 3d, the zinc series. Tin is the most 
polishable of the soft metals, and it alloys readily with any of the 
useful metals employed for minimizing the friction of machinery; 



it has been made the basis of the best anti-friction alloys. Lead is 
undoubtedly the best anti-friction medium among metals, but it 
lacks stiffness to stand up to the work. Copper is the ideal bond 
for zinc alloys, and zinc is the most expansible and durable of metals. 
Zinc babbitts cast well, wear well, and fit snugly to the bearing. 
Owing to its highly crystaUine structure, antimony, the principal 
hardening element, should not exceed 20 per cent., as it is apt to 
separate and rub out of the alloy — 17 per cent, has been fixed as the 
limit by an eminent authority. 

The mutual relations of the metals determine the mechanical 
properties of the alloys. Zinc and antimony are too much alike to 
be used simultaneously, and tin alloys, without copper, are apt to 
spread under heavy loads. Due to its poor affinity for lead and tin 
and its low atomic volume, aluminum is not a suitable metal for anti- 
friction alloys. Bismuth, on the contrary, is a decided advantage 
up to about 1.5 per cent. This metal has been freely used in the 
production of some modern alloys, notably those with low fusibility, 
low contraction and high atomic volume. In Table 10 are given 
some special mixtures which have given complete satisfaction for 
the duty stated, and in Table 11 are given four grades of mixtures. 

In each case the metals represented by the figures 7, 17, and 6 
constitute the "hardening." These are copper-hardened alloys — 
the copper content being over 5 per cent. — and provide a series of 
cheap, serviceable, anti-friction metals. 

Table 10. — Miscellaneous Bearing Metals 



For Uning 


Tin 


Lead 


Zinc 


Anti- 
mony 


Copper 


Bis- 
muth 


Dynamos: high-speed. 

Marine engines 

Eccentrics 


88 

77 

S 

40 

34 

63 

42 

74.22 
78.84 

80 

80 

88 

6 

24 






8 

3 
IS 
10 
16 

2-S 

6.SS 
trace 

I 
10 

9 
16 


3-S 
3 
2 
2 
6 

2-5 

2 
3.60 
3-70 

8 

10 

8 

4 


-S 

•25 


17 
78 
48 
44 

13 SO 

14-75 

10 


30 

S6 

1,80 


Submerged bearings . . . 

Main bearings 

Slides, thrusts 


Railway trucks 

Axle-boxes (by analysis) 
Anti-acid metal (by 

analysis). 
Plastic metal 


I 


Genuine babbitt (hard) 




Genuine babbitt (No. 2) 








Universal bearing metal 
Anti-friction castings... 


78 


80 


•25 



Table ii. — A Series of 


Copper H 


ardened 


Alloys 


Grades 


I 


2 


3 


4 


Tin 


77 


77 

17 

7 


17 

77 

7 




Zinc 


77 


Lead 


17 
7 
6 


17 


Antimony 

Copper 


6 


6 


6 


7 



The composition of many common bearing metals, as determined in 
the laboratories of the Pennsylvania Railroad and published by Dr. 
Dudley {Journal of the Franklin Institute, Feb., 1892), is given in 
Tables 12 and 13. 

The bearing metal known as the standard of the Bureau of Steam 
Engineering of the United States Navy, also called anti-friction or 
anti-attrition metal, has this composition: 

Best refined copper 3.7 per cent. 

Banca tin 88 . 8 per cent. 

Regulus of antimony 7.5 per cent. 

The percentages are by weight. The mixture must be well fluxed 
with borax and rosin in mixing. 



20 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 12. — Composition of Bearing Metals (Per Cent.) 



Name of metal 


Copper 


Tin 


Lead 


Anti- 
mony 


Zinc 


Iron 


Camelia metal. 
Anti-friction 


70. 2 
1.6 


4-25 
98.13 


14-75 




10.2 


-55 
trace 


metal. 
White metal.... 


87.92 
84.87 
I-I5 
67-73 
— none 
80.69 

14-57 
-possibl 

12.4 

rus — tra 

5-1 

83-55 

iron, CO 
bismuth 

78.44 

-31 
15.06 
12.52 


12.08 
15-1 

16.73 
18.83 






Car brass lining. 


4.01 


trace 

9.91 

14.38 

Graphite 






Salgee metal.. . 
Graphite bear- 
ing metal. 
Antimonial lead 


85-57 


not de- 
termined 


Carbon bronze. 


75-47 

77-83 
92.39 


9.72 
Carbon- 

9.6 
Phospho 

2-37 






Cornish bronze. 
Delta metal... . 


e tra 
ce 


ce 
trace 


trace 
.007 


Magnolia metal 


i6-4S 
pper, zi 

19.6 




American anti- 




Traces of 
possibly 


nc and 

.98 
38.4 


-65 
.11 


friction metal. 
Tobin bronze . . 
Graney bronze. 


59 

75-8 

76.41 

90.52 

81.24 


2. 16 
9.2 
10.6 

958 
Mangan 
10.98 
Phospho 


Damascus 








bronze. 

Manganese 








bronze. 
Ajax metal. . . . 


ese — no 

7.27 
rus or 
arsenic 
88.32 


ne 






American anti- 


-37 
11-93 






friction metal. 
Harrington 


55-73 


-97 


42.67 
trace 


.68 


bronze. 
Carbox metal... 


84-33 
94-4 
9.61 

•94 
iS-O 
. 2 


14-38 
6.03 


.61 


Hard lead 








Phosphor 


79-17 
76.8 


10. 22 
Phos- 
phorus 
8 

Phos- 
phorus 






bronze. 
Ex. B. metal... 

















Table 13. — Composition of Bearing Metals (Per Cent) 





u 

1 


1-1 


a 


a 

B 

< 





3 
u 


Oh 




u 
N 


Plastic bronze 


64 

79-7 
64-75 

50 

2.25 
60.67 

3-7 

2.5 
7 

5-55 


30 

95 
30 

50 
-15 
32-97 

-25 


5 

ID 

5 




I 








Phosphor bronze. . . . 


.8 






Cyprus bronze 






-25 




JPlumbic bronze 










Parsons white brass. . . 


64-9 
4-6 

88.89 

58.38 
84 

88.33 










32-93 


Demo bronze 

Standard babbitt. 


7-41 


2. I 






Shonberg M. M. metal 








38.93 


Souther babbitt 


9 

II. II 




. 




German babbitt 



















The mixing of this anti-friction metal is a trick which must be 
learned. The best practice is to melt the copper, tin and antimony 
separately, adding the tin to the copper and the antimony to this 



mixture, fluxing it with borax with the proportion of about i 5 lbs. 
to 175 lbs. of the mixture; but satisfactory results are obtained by 
melting the copper first, dropping the cold tin into the melted copper 
and adding the antimony, which has been separately melted. This 
metal is carefully skimmed before pouring, and is poured into pigs 
and carried into stock as it stands. 

The journal bronze used on battleships of the United States Navy 
has this composition: Copper 82 to 84, tin 12.5 to 14.5, zinc 2.5 to 
4.5, iron (max.) 0.06, lead (max.) i.oo, all in per cent., with a normal 
of &3~^3i~3h It is used for bearings, bushings, sleeves, slides, 
guide gibs, wedges on watertight doors and all parts subject to 
considerable wear. 

Albert E. Guy gives the composition of the high-speed babbitt 
used in De Laval steam turbines as: copper 10, tin 80, and antimony 
10 per cent. For low speeds the metal used is: lead 77, tin 6 and 
antimony 17 per cent. 

The Mesta Machine Company, on rolling mill work, uses two 
grades of babbitt and a bronze. For the general run of work, a 
lead babbitt is satisfactory having this composition in per cent. : lead 
75, tin 12.5, antimony 12.5. For high rubbing speeds a mixture is 
made of i part of the above and 2 parts of genuine babbitt. This 
genuine babbitt, alone, is used on rolling mill engines and in bearings 
subjected to shock and pound. Its composition in per cent, is: tin 
82, copper 5.4, antimony 12.6. The bronze is a tough copper-tin- 
lead alloy very similar to Pennsylvania Railroad metal. 

The alloys division of the Standards Committee of the Society of 
Automobile Engineers in their report for June, 1911, specifies four 
bearing metals as follows: 

Babbit Metal, Specification No. 24 

Tin 84 per cent. 

Antimony 9 per cent. 

Copper 7 per cent. 

A variation of i per cent, either way will be permissible in the tin, 
and 0.5 per cent, either way will be permissible in the antimony and 
copper. The use of other than virgin metals is prohibited. No 
impurity will be permitted other than lead, and that not in excess of 
0.25 per cent. 

Note : This grade of babbitt is special, owing to the large amount of 
copper contained therein. It is used for the connecting-rod bearings 
of gasoline motor bearings, locomotive work, or for any service where 
machinery designers are confronted with severe operating conditions. 

White Brass, Specification No. 25 

Copper 3 . 00 to 6 . 00 per cent. 

• Tin, not less than 65 . 00 per cent. 

Zinc 28 . 00 to 30 . 00 per cent. 

Metal containing more than 0.25 per cent, impurities may be 
rejected. 

Note: This alloy gives good results in automobile engines, but 
provision should be made to have it generously lubricated. 

Phosphor Bronze Bearing Metal, Specification No. 26 

Copper 80.00 per cent. 

Tin 10. 00 per cent. 

Lead 10.00 per cent. 

Phosphorus 0.05 to o. 25 per cent. 

Impurities in excess of 0.25 per cent, will not be permitted. 

Note: This is a metal similar to that specified by many railroads 
for various purposes. It is an excellent composition where good anti- 
frictional qualities are desired, standing up exceedingly well under 
heavy loads and severe usage. It should be used only against hard- 
ened steel in automobile construction. 



4 



PLAIN OR SLIDING BEARINGS 



21 



Red Brass, Specification No. 27 

Copper 85 . 00 per cent. 

Tin S . 00 per cent. 

Lead S • 00 per cent. 

Zinc S ■ 00 per cent. 

A tolerance of i per cent, plus or minus will be allowed in the above 
percentages. Impurities in excess of .25 per cent, will not be 
permitted. 

Note: A high grade of composition metal, and an excellent bear- 
ing where speed and pressure are not excessive. Largely used for 
light castings, and possesses good machining qualities. 

The following particulars regarding Westinghouse practice with 
babbitted bearings are by Jesse L. Jones, metallurgist Westinghouse 
Electric & Mfg. Co. (Amer. Mack., Apr. 18, 1912). The company 
has adopted two principal babbitts — a tin-base babbitt that is very 
easy flowing and suited to pouring extremely thin linings. This 
babbitt is much tougher and but slightly softer than the original 
genuine babbitt formula which is often referred to as the U. S. 
Government Standard. 

The second is a lead-base babbitt that contains considerable tin, 
flows well and is much tougher and but slightly softer than the usual 
babbitts of the Magnolia class. 

Some use is also made of the lead-antimony, a hard genuine babbitt, 
and other special formulas that customers may specify. 

In order to insure the best results in bearings, only the very best 
grades of copper, lead, tin and antimony are used. The use of 
drossy lead, off grades of tin and antimonial lead results in inferior 
babbitt and unsatisfactory bearings, and is therefore most carefully 
guarded against. 

While the amount of copper in most babbitts is small, the use of the 
electrolytic grades is to be preferred, as some of the Lake brands are 
high in arsenic and this may cause poor adherence of the babbitt 
lining to bronze shells. 

Most of the brands of lead on the market are almost chemically 
pure but they contain varying amounts of dross and oxide and the 
only practical way of testing them is to run down 100 lbs. or more 
in a graphite crucible, boil up with green hickory wood, skim off the 
dross and weigh the clean lead. The same brand of lead may be very 
clean at one time and drossy at another, and the melting loss in 
making babbitt from it will vary accordingly, as will also the anti- 
frictional quahties. 

There is no real economy in using an off grade of tin running from 
93 to 98 per cent, of tin, instead of Straits, as it is necessary to pay 
for the tin content at the market price of tin, and the lead content 
at the market price of lead, so that all that is obtained gratis is a 
little iron, antimony, dross, etc., that will increase the melting loss 
and add nothing to the quality of the babbitt. 

The grade of antimony to be used has been the subject of very ex- 
tensive practical tests. It has been found that in some cases the 
better brands, having almost identical chemical analysis, give quite 
different results in the finished babbitt in regard to hardness. As 
antimony is used as a hardening agent, and as the total amount used 
in any babbitt is relatively small, the brand which has given the best 
practical results, although it is the highest priced antimony on the 
market, has been adopted. 

No adequate explanation has as yet been found to show why this 
particular brand gives better results than other brands of practically 
identical composition, but this fact has been checked so often that 
it is now accepted without question. 

Having secured the best materials obtainable they are melted to- 
gether in the proper proportions to produce the grade of babbitt 
desired. It is customary in making a genuine babbitt to combine 
the copper and antimony, or the copper, antimony and part of the 
tin to form a preliminary alloy or hardener. This is mixed with 
the rest of the tin, thus giving a more uniform product. 

A temperature of about 900 deg. Fahr. should be used in mixing 



a babbitt to secure satisfactory alloying, and the surface of the metal 
should be protected from oxidization by a layer of powdered char- 
coal. Dross is removed by boiling up with green hickory wood, and 
the babbitt may be deoxidized by means of vanadium, manganese, 
aluminum, magnesium, sodium, etc. When all new metals are used, 
deoxidization is, as a rule, unnecessary. 

Before pouring into ingots, the temperature of the babbitt should 
be lowered considerably, especially if water-cooled molds are not used, 
as a finer grain is thus secured. 

For pouring the ingots a bucket-shaped ladle with a bail and 
handle and a long, square-nosed pouring spout should be used. It 
gives a good surface as the metal is less agitated in the pouring than 
when the ordinary ladle is used. A few ounces of the babbitt should ^ 
first be poured into the mold, the stream interrupted for a second 
and then the pouring of the ingot completed. A cushion for the 
stream is thus formed and the surface is smoother as a result. Small 
air bubbles are removed by touching with a wooden pick before the 
metal solidifies. 

Taking so much pains to obtain ingots of good appearance may 
seem unnecessary when the babbitt is for one's own use, but it has 
been found that the nicer the appearance of the ingots, the better 
the bearings turned out, as the workman babbitting the bearings will 
take more pains with his work than when rough-looking ingots are 
given him. 

The Brinell hardness test has been found satisfactory as a shop 
test for securing uniformity in the babbitt. Tests are taken from 
the top, middle, and bottom of each kettle of the ingot metal and 
similar control tests are made daily on each of the various babbitt 
pots throughout the works where the bearings are filled. 




V2 Anchor 



H Anchor 




Baked Anchor Core 



Fig. 9. 



For all Split Bearings 
less than 9 Diameter 

Fig. io. 




For all Split Bearings 
9 Diameter or over 



Figs. 9 and 10. — Anchor core and standard anchors of Westing- 
house babbitt bearings. 

Bending, fluidity and peening tests are made daily on strips 1 2 X 
5 Xi in. Analysis, tensile, compression and specific-gravity tests are 
also made occasionally, while a babbitt inspector, who is a thoroughly 
practical man, has general supervision of all babbitt pots and the 
pouring of all bearings. 

Bearing shells for stationary apparatus are usually made from 
cast-iron, because of its rigidity and cheapness. Where mechanical 
strength, a certain amoimt of toughness and cheapness are desired, 
malleable iron is used. 

Shells of cast steel are made for some customers but they are not 
recommended eis they do not retain their shape. 

For street cars, etc., standard phosphor-bronze shells are used, 
because with such a bearing the return of a car to the barn is assured 
even if the babbitt melts and runs from the bearing. 

To prevent the babbitt lining from flowing, due to the revolution 
of the axle, all iron bearings are provided with cast anchor holes. 
These are made by adding to the green-sand core of the casting, 
baked anchor cores, secured with brads as shown in Fig. 9. 

There are two sizes of anchors used, | and f in. as shown in 
Fig. 10. Where bearings are bored before babbitting the cores are 
made of such length that the holes wil\ be standard after boring. To 
help the molder in setting the cores, the pattern maker spots the pat- 
tern so that it will leave small center marks on the green-sand core. 
Along the straight lips of each half bearing, the anchor holes should 
be very numerous and as close to the edge as is possible in casting. 



22 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



With bronze shells, undercut grooves or anchor holes, drilled in 
diagonally, may be added to prevent the lining loosening in case the 
bearing has been poorly tinned, but if properly tinned and babbitted, 
these are unnecessary. The greater the amount of babbitt in the 
anchor holes of a bronze bearing the greater will be the shrinkage 
and the more likely the lining will be to be loose and spongy. 

A bearing with large anchor holes seldom gives a clear, bell-like 
sound when struck with a hammer. But if the anchor holes are few 
and small, the bearing properly tinned and poured with a thin lin- 
ing, the babbitt becomes an integral part of the bearing, can only be 



Rough boring all bearings before babbitting is desirable, as it 
gives a lining of the babbitt of uniform thickness, a uniform grain 
and hence a uniform rate of wear. 

All iron shells are heated before babbitting to a temperature that 
will just admit handling them, say 350 deg. Fahr. This heating 
is done preferably in an oven, but it may be done over a coke or 
gas fire. In the latter case, especially with bronze bearings, the 
inner surface that is to be babbitted must be turned upward, 
otherwise a greasy deposit will form on the bearing that will prevent 
a good job of tinning, and hence the proper adherence of the babbitt. 



The Allowance given on this Sheet must eaiist after theBearing is in Elace. Normal Bearing Diameter to be Specified on Drawings 



Waste or Grease 
Lubrication 



Oil Ring Lubrication 



Solid Bearings 



Split Bearings 



Shaft Diameter 



Up to, Dot iDoluding 
3K" 



From. 3J^ up to, not 
Tncludins 4 " 



From 4 op to 7 
Inclusive 



Allowance 



.008 



.012 



.GIG 



.006 



.003 



The Workman will in 
all Cases work to the 
Minimum Bore Limit 
The Maximum Bore 
Limit is for the use of 
the Inspector to dete r 
mine -whether to pasrf 
or reject the Bearing 



.012 





Section U-V 




> 3 3>ia.rnclUBivi 



Bore H 




The Boring Operations 

let Allowance Bore F of complete 
Beeiriiig 

Slid Eccentric side Bore H obtained 
from Center at Dietance L above 
Center of Bearing, bringing Tool 
out uutU it touobes Bore F at 
Distance M below Center Line 

3rd Clcftrance Bore G of Top Half 
of Bearing 

^=^(X)These Dimension are for 

Reference In Drafting Room Only 



Ftg. II. — Standard bore finishes of Westinghouse babbit bearings. 



Nominal shaft diameter 



Horizontal 



Bore limits | 
-|- on 
bore 



Vertical 



Difference in size of 
shaft and bore 



Bore limits 
+ on 
bote 



Max, 



Min, 



Max 



Max, 



Mi: 



Max. 



Nominal 

shaft 
diameter 

Not 
Fron 



Bore 

A 
allow- 
ance 



Nominal 

shaft 
diameter 
Not 
From 



Allowance 



Bore 
F 



Bore 
G 



Bore 
H 
(.X) 



K 

iX) 



M 



Up to I inclusive. 



Above I up to I inclu.. . 
Above I up to i\ inclu . . 
Above i^ up to 2\ inclu. 
Above 2\ up to 4 inclu. . . 

Above 4 up to 4^ inclu. . . 
Above 4J up to 5 inclu . . . 
Above 5 up to Si inclu. . . 
Above 5 J up to 6 inclu. . . 
Above 6 up to 7 inclu. . . . 



Above 7 up to 8 inclu. . . . 
Above 8 up to 9 inclu. . . . 
Above 9 up to 10 inclu. 
Above 10 up to IS inclu. 



.0015 

.003 
.004 
.006 
,008 

.009 
010 
on 
012 
014 

015 
016 
017 
018 



.002 
.003 
.004 
.005 

.006 
.007 
.008 
.009 

-Oil 

.012 
.013 
.014 
.015 



.004 
.006 
.008 
.011 

.012 
• 013 
.014 
.015 
.017 

.018 
.019 
.020 
.021 



.003 
.COS 
.006 
.008 

.009 
.009 
.009 
.009 
.009 

.009 
.009 
.015 
.CIS 



.0015 
.002 
. 002 
.003 

.004 
.004 
.005 
.005 
.005 

.005 
.005 
.008 
.010 



.002 
.003 
.004 
.005 

.006 
.006 
.006 
.006 
.006 

.006 
.006 
.010 
.012 



li 



2h 



3 
3i 

4 
4} 

S 

ci- 
02 

6 

6i 

7l 



H- .002 

-f .002 
-H .002 
-f- .002 
+ .002 

-|- .002 
4-. 003 
+ .003 
-f.oo4 
4-. 004 

+ .005 
+ •005 
+ .006 
-I-.006 
-f007 



Not 
inc. 7i 



1-3 

I '-4' 

i'-6" 
i'-8" 
I'-io' 

2'-o" 



I'-I" 
l'-2" 

l'-3" 
l'-4" 
I '-6" 

l'-8" 
I'-IO" 
2'-0" 



+ .004 

-I-.004 
-I-.005 
+ •005 
+ .006 

+ .006 
+ .006 
+ .007 
+ .007 
+ .008 

+ .009 
+ .010 



+ .007 

+ .008 
+ .009 
+ .010 
+ .011 

+ .012 
+ .013 
+ .014 
+ .015 
+ .016 

+ .01 
+ .020 



+ .068 

+ .068 
+ .068 
+ . 100 
+ . 100 

+ . 100 
+ . 100 
+ . 102 
+ . 102 
+ . 102 



.032 

.032 
.032 
.047 
.047 

.047 
.047 
.047 
.047 
.047 

.064 
.064 



li 



1 A 



stripped off with great difficulty and leaves a white frost on the 
bronze. 

Iron bearings are cleaned in the tumbling barrel, or by the sand 
blast at the foundry. It is usually necessary to clean out the anchor 
holes by hand before babbitting, or even to pickle in hydrofluoric 
acid (especially on bearings provided with oil-ring lubrication), be- 
cause any adherent sand will be loosened by the hammering necessary 
in adjusting the mandrel, and this sand mingling with the babbitt 
when poured wiU ruin the bearing. 



The tinning of bronze shells is best done by immersing them in a 
pot of molten solder of half and half composition, using a saturated 
solution of zinc chloride as a flux, applied with a mop of clean woolen 
waste. Immediately after tinning, the bearing is placed on the man- 
drel and babbitted. Unless there is a clean film of molten solder over 
the entire surface to be babbitted there will not be a perfect adher- 
ence of the layer of babbitt. 

This will also be true if babbitt has been used for the tinning, as 
the babbitt has a much higher melting-point than the solder, and 



PLAIN OR SLIDING BEARINGS 



23 



maintaining a clear molten film with it is difficult. The presence of 
arsenic in the babbitt, due to the use of cheap antimony, or anti- 
monial lead, will result in loose linings also. 

In order to avoid blow-holes and imperfections in the babbitt lin- 
ing, it is very necessary to coat all mandrels with a very thin coating 
of clay wash. Put a pound or two of Jersey red clay in a pail of 
water and stir until suspended, then plunge the heated mandrel into 
it. The mandrel will soon dry and the molten babbitt will lie on it, 
giving a smooth surface, free from bubbles. 

This makes it possible to line a bearing with as little as ys in. of 
babbitt and the surface will be so smooth that only .008 to .010 in. 
need be machined out for the finish. Brass shells from 15 to 4! 



The babbitt is melted in cast-iron kettles holding about 500 lbs., 
and fired by gas. On first melting the new ingots, or in remelting 
the babbitt which has solidified after standing in the kettle, it will 
be found that the tin in the babbitt will commence to liquate at 
about 450 deg. Fahr.; hence it is necessary for satisfactory work to 
heat the babbitt to about 850 deg. Fahr. on starting up, and stir 
very thoroughly before pouring into the bearings, as otherwise the 
babbitt will not be of uniform composition. 

After once thoroughly alloyed in this manner, there is compara- 
tively little tendency for the tin to liquate, so long as the tempera- 
tures given as satisfactory pouring temperatures are maintained, al- 
though stirring of the babbitt during the pouring process is desirable. 



Horizontal Solid Bearings | 


Bearing Diameter 


DtUled 
Bole A 


B 


C 


D 


E 


H to 1 Inclusive 


% 


M 


v.„ 


%. 


%, 


Above 1 to 1)^ Incluaiye 


H 


■■ 


" 


•■ 




„ 154 „ljj 


% 


\ 


!f 


% 


H 


••m-^Vt 


'A 




" 


" 




•■ 2% " 3M 


1 


« 


% 


K 


" 


•• 3J4 .. 5 


IX 


.. 


" 


" 





One Kvne Type 




Vertical Bearings | 


Bearing Diameter 


A 


B 


C 


D 


Up to IJi Inclusive 


M 


Via 


'A 


'A. 


Above IJi to 2 J^ Inclusive 


V16 


Vm 


" 


V32 


.. 2M "5 


Y>. 


% 


^i 


Vk 


., 5 .. 10 


" 


,, 




H 











































Top of Bearing 



U^-— Length of Bearing -21 




Grease Bearing 
Compression Cup Fed Types 



Detail of Grooves 

Overflow ioi-^ ^ 

Automatic Feed Itl'^ 

'I ' 
-^ i5 L_ III 

' ^ < III 




Horizontal Split Bearings ] 


Bearing Diameter 


BriUed 
Qole^ 


B 


C 


D 


E 


IH to IH Inclusive 





Yt 


« 


V32 


V32 


Above 1% to 2}i Incluaive 


."^ 


" 


" 


% 


M 


,. 2>4 .. 2J^ 




■■ 


Vis 


Ma 


" 


.. '-H ■■ s 


^1 


•■ 


% 


v„ 


" 


.. 3 " 4>^ 


§1 




Vio 


M 


" 




a ra 












a^ 










■■ 4J< " -',i 




Yi 


'Via 


M 


Via 


" 1% "I'l" " 


CO 


■• 


" 


Via 


" 















Omit Groove on this End for Bracket Bearing 
\When less than H omit end Grooves 




Special - Used when necessary to ViewTjoth OH 
Rings from common Sight Hole 




Fig. 12. — Westinghouse practice for oil grooves in bearings. 



ins. in diameter are usually lined with -^ in. of babbitt and .014 to 
.016 in. machined out. Iron shells are hned with J in. of babbitt and 
^ in. machined out. 

The use of the clay is especially necessary where oil gets on the 
mandrels. The oil causes the babbitt to blister. Half an hour's 
babbitting will not sufiice to burn off the oil, but if the clay wash is 
used the oil is covered up and smooth bearings result. 

Cast-iron shells are rarely if ever tinned, as such tinning cannot be 
depended upon to hold the Uning in place. If the shells are made hot 
enough for the solder to alloy with the iron, the solder will oxidize 
and will not adhere. If kept cool enough not to burn the solder, the 
solder will fail to alloy with the iron, and hence will peel off when cool. 



The importance of the pouring operation may seem to be exagger- 
ated in this statement, but if it leads the manufacturer to employ a 
skilled workman for pouring bearings, instead of a laborer, the slight 
exaggeration will be justified, for the skilled workman will not only 
pour the bearing properly, but he will also see to it that the qual- 
ity of the babbitt, its temperature and the tinning are what they 
should be. 

The temperature at which the babbitt is poured is important. If 
much above 900 deg. Fahr. the shrinkage is very pronounced, and 
porous areas result, while the babbitt will be dirty and oxidized and 
its antifrictional qualities injured. Uniformity of temperature is 
desirable, and this is maintained by the use of a delicate thermostat. 



24 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



The thermostat is set for 860 deg. Fahr. and the gas is shut off when 
880 deg. Fahr. is reached, or if the temperature falls below 840 deg. 
Fahr. more gas is turned on. 

The shape of the lips of the ladle used for pouring bearings is very 
important. The lips should not be sharp but rounded, so that the 
stream will not strike either mandrel or shell, otherwise a burnt 
streak will result. A broad stream or an intermittent stream will 
produce porous areas or masses of blow-holes. A good pourer will 
keep both elbows close to his body, use a hand leather, so that he 
can grasp the handle of the ladle near the bowl, and hold his body 




X)=Diameter of Shaft 

.B=1.8Z)+K" 
C=1.45£>+'/4" 
E=1.SD+H" 

Fig. 1,3. — A plain bearing with formulas for dimensions. 



t = .05£)+J< 
d = .25D 
s = .35D „ 
e = .08Z)f H 




Fig. 14. — Heavy pedestal bearings with table of dimensions. 











Dimensions 


IN Inches 












A 


B 


c 


D 


E 


F 


G 


H 


I 


J 


K 


L 


M 


4 


6i 


si 


2 


S 


I 


1 


3i 


Si 


li 


i| 


2i 


li 


4i 


7 


6 


2j 


Si 


li 


i 


4 


6i 


II 


li 


2f 


If 


5 


8 


61 


2i 


6 


li 


i 


4i 


6f 


2 


li 


2i 


li 


6 


10 


8 


2i 


8 


li 


i 


6 


9 


li 


li 


3 


If 


7 


12 


9i 


3 


10 


li 


i 


8 


II 


li 


li 


3i 


li 


8 


13 


xoi 


3i 


lOi 


li 


i 


8i 


12 


2 


li 


3i 


li 


9 


IS 


Hi 


4 


I2i 


li 


i 


10 


14 


2 


li 


4 


2 


10 


16 


I2J 


4i 


I3i 


ij 


I 


loi 


IS 


2i 


li 


4i 


2i 


12 


20 


IS 


S 


17 


2 


li 


14 


19 


2i 


2 


S 


2i 



Remarks. — In the column F there are two bolts to shafts 7 ins. dia- 
meter; above that, four bolts in each bearing. The side brasses or cheeks are 
set up with screws K, but in a manner free from the common objection to this 
method. The screws are inserted from the inside, and have enlarged ends to 
give bearing enough to meet all requirements. The recesses to receive these en- 
larged ends can be cored in the main casting, and the cost of construction is no 
more than in the case of common set-screws, which should never be employed 
unless of very large size. The curves at A'' are developed to suit the height 
and area of base required. The bosses at O should be at least the depth of 
the plinth or base flange. Two are preferable for shafts larger than s ins. 
in diameter. When the caps become heavy the oil box can be made rectan- 
gular to remove useless metal, and is preferable in that form for bearings 
exceeding S ins. in diameter. 

When pedestal bearings of this kind are made without side brasses, or with 
a half shell on top, the transverse dimensions can be reduced, and should 
always be as small as possible. For mounting on masonry a sole plate should 
be used. This is generally required for the lateral adjustment of shafts. 



almost rigid while pouring, thus avoiding any surging of the metal 
in the ladle, or splashing. 

If the pourer is not very skillful, a sheet-iron bridge may be riveted 
to the lip of the ladle so that it will extend some distance below the 
surface of the metal. It can be adjusted so that it will give a stream 
of the diameter found best for the bearing being poured. This will 
not only regulate the stream out but keep the dross out of the 
bearing. 

Bearings are preferably poured in a vertical position. Some half 
bearings are poured with the convex side upward through holes cast 
in the shells for the purpose. Very large bearings are usually poured 
with the concave side upward. 

All solid bearings are broached on a broaching machine, which is 
also used for pushing out the mandrel. This operation heats the 
bearing, making it necessary to allow it to reach the room tempera- 
ture before making the finishing cut. 



Housing 



Conical Collar 

to return Oil 

Reservoir 




Oil 

Beserroir 



Fig. 15. — A ring-oiled bearing. 




Section A-B c ^- ^ -^ 
jy Section C-D 

B 




Fig. 17. — An improvement on the ring-oiled bearing. 

The necessary allowances for the bore finishes of bearings are 
shown in Fig. 11, which gives the results of many years of experi- 
ence. For additional information on this subject, see Press and 
Running Fits. 

Oil grooves are cut in the finished bearings by hand because, as a 
rule, the babbitt lining is too thin to permit their being cast. Stand- 
ard forms of grooving are shown in Fig. 12. 

The most important element in the production of a satisfactory 
bearing is the pouring. The quality of the babbitt is important, the 
use of a thermostat is important, the tinning is important, but more 
depends on the actual pouring of the lining than on any other one 
element. 

Regarding oil grooves, there is great diversity of opinion and prac- 
tice. With film lubrication their presence on the pressure side of a 
bearing would seem more likely to do harm than good, while with 
ordinary lubrication the reverse is true. Some advocate blind- 
ended grooves to avoid escape of oil, while others object to blind 
grooves because of their liability to become clogged and useless. 



PLAIN OR SLIDING BEARINGS 



25 



With film lubrications, open-ended grooves are obviously inadmis- 
sible. One point is settled — the edges should be well rounded to 
facihtate the entrance of oil to the bearing and the same is true of 
the meeting edges of split boxes. Sharp edges of grooves act as oil 
scrapers, not oil distributors. 

Bearing Design 

A drawing of a simple split bearing with formulas for leading 
dimensions is given in Fig. 13, by C. F. Blake {Amer. Mack., Nov. 28, 
igoi), while Fig. 14 and the accompanying table give dimensions of 
heavy four-part bearings by John Richards (A Manual of Machine 
Construction) . 

The self-aligning (ball and socket) construction was introduced in 
1849 by Wm. Sellers & Co., as a feature of line shaft hanger bearings. 
It has now come into extended use for large bearings of high-class 
machinery. In connection with the oil ring, first published by 
Professor Sweet (Engineering, Jan., 1868), it is shown in Figs. 15 
and 16, Fig. 15 being a typical section of a bearing fitted with both 



devices. Fig. 16, with the accompanying table of dimensions, 
gives the practice of the General Electric Co. Bearings up to and 
including 9 ins. diameter have two, and above that size four rings. 
Fig. 17 shows an improvement on the oil ring for high speeds, by 
the Builders' Iron Foundry (Amer. Mack., Feb. 10, 1898), and applied 
by them to grinding and polishing stands. The loose ring is re- 
placed by a collar, which is forced on the shaft and revolves with it. 
The collar dips into a capacious oil cellar below as usual, and a wide 
circumferential channel is cast in the box for the ring to revolve in, 





Fig. 18. — A water-jacketed bearing. 



Fig. 16. — Dimensions of ball and socket, ring-oiled bearings. 



Nominal 












dia. of 




Dimensions in Inches 








bearings 












A 


B 


C 


D 


E 


F 


H 


J 


K 


L 


*N 


P 


R 


5 


T 


*U 


7 


21 


i 


22 


145 


SJ 


1 


4i 


12J 


lOi 


i| 


5f 


6f 


2} 






8 


24 


i 


25 


i6i 


(•i 


iJ 


4l 


14 


12} 


li 


6i 


7i 


2| 






9 


27 


i 


28 


m 


7J 


li 


5\ 


IS? 


14 


2j 


7 


8| 


3 






10 


30 


a 


31^ 


20^ 


8i 


li 


5! 


I7f 


16 


2f 


6J 


8i 


2} 


3i 


sf 


II 


33 


I 


34^ 


22i 


81 


li 


6 


I9h 


17} 


3f 


7f 


9i 


2i 


4i 


6 


12 


36 


i 


375 


24i 


9i 


li 


(>\ 


2li 


19 


4 


8i 


lof 


3 


4i 


6 


14 


42 


i 


A3i 


2&i 


9i 


li 


6i 


25 


21} 


S 


10 


12} 


3i 


6i 


7 


IS 


45 


i 


46I 


30i 


9i 


li 


7 


2t\ 


23} 


7 


"i 


I3l 


4i 


7 


7 




Plug 



}il Inlei 

Section on Center Line Half Elevation ' Half Section on Center Line 

Fig. 19. — Water-cooled bearing with forced lubrication. 



i 



26 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



except that at the top this channel is obstructed by projections a, 
which provide only sufficient room for the collar to revolve freely. 
Their ofEce is to scrape the oil from the ring. This not only deposits 
it on the top of the box, but the force with which the oil strikes the 
projections causes it to shoot down the channels provided for it 
endwise of the bearing. Collecting grooves are provided at the end 
of the bearing, as well as free return channels to the oil cellar. The 
obvious result is a positive flooded circulation of oil throughout the 
bearing. 

Fig. 1 8 shows a water-cooled bearing, without self-alignment, for a 
large vertical engine, by the Union Iron Works {Amer. Mack., Oct. 12, 
1905). Four passages abed are cast in the lower half of the bearing, 
the ribs which separate the passages having openings at alternate 
ends to provide a continuous flow, as indicated by arrows in the 
plan. The ends of the outer passages have tapped holes at e f for 
the water-pipe connections. The caps of the bearings have no 
water connections. 




Total Allowance for End Plar 
under 2 dia. ot Bearing '/lo 
2"to 8" dia. of Bearing H"„ 
over 8"dia. of Bearing Vl6 
End Play should be talien up ■ 
in one Bearing whenever 
possible 



Fig. 20. — Standard oil-retaining grooves. 



Size of 


4 


B 


r 


n 


F 


F 


G 


Size of 


A 


B 


r 


D 


F. 


F 


G 


bearing 
















bearing 
















l^ 


2 


2^ 


A 


i 


J 


A 


A 


5i 


6i 


6ii 


i 


A 




A 


J 


li 


2i 


2-h 


A 


i 


J 


A 


A 


6 


7f 


7A 


i 


A 




A 


i 


2 


2? 


2ii 


A 


i 


i 


A 


A 


7 


8f 


sa 


i 


A 




A 


i 


2}- 


2| 


2H 


A 


i 


i 


A 


A 


8 


9l 


9H 


A 


1 




A 


A 


2^ 


3i 


3A 


A 


A 


i 


A 


A 


9 


lOf 


lOii 


A 


1 




A 


A 


2| 


3i 


3A 


A 


A 


i 


A 


A 


10 


12 


I2A 


A 


i 




A 


1 


3 


3i 


3U 


A 


1 


A 


A 


A 


II 


13 


I3A 


A 


i 




A 


i 


3^ 


4» 


4A 


A 


A 


_3_ 


A 


A 


12 


I4J 


I4A 


A 


J 




A 


i 


4 


4i 


4ii 


i 


A 


A 


A 


A 


13 


ISi 


ISA 


A 


I 




A 


i 


4i 


Si 


sA 


i 


A 


i 


A 


i 


14 


i6^ 


i6A 


A 


I 




A 


1 


S 


6J 


6A 


i 


A 


i 


A 


J 


IS 


ni 


I7A 


A 


li 


i 


A 


f 



The General Electric Co. find water cooling to be increasingly 
effective as the cooling surfaces are brought nearer to the actual 
bearing surfaces. Their preferred construction of water-cooled bear- 
ing, having also forced lubrication, is shown in Fig. 19. A grid of 
cooling pipe is laid in recesses in the bearing sheet in such manner 
as to be imbedded in the babbitt. Both pipe and babbitt anchors 
are exaggerated in size in the illustration. 

Standard oil-retaining grooves, as applied by the General Electric 
Co. to split bearings, are shown in Fig. 20 and the accompanying 
table. 

End Play of Shafts 

The well-known freedom of large shafts, when in motion, to move 
endwise under small forces is thus explained by Lucian E. Ficolet 
(Amer. Mach., Dec. 15, 1910). 

Fig. 21 represents a loosely fitted bearing resting upon a journal 
and carrying a load P. When the journal rotates, the bearing is 



maintained in position by a force / P, acting opposite to the direc- 
tion of rotation and equivalent to the tangential effort due to the 
friction, / being the friction coefficient. 

For the present purpose the journal and its bearing may be rep- 
resented as in Fig. 22, by a block supporting the load P and resting 
upon a flat surface. If the block slide uniformly under the force X, 
obviously X =/ P. Suppose now, another force F be applied perpen- 
dicular to X, as in Fig. 23. When the block is on the point of slid- 



I 




Fig. 21. Fig. 22. Fig. 23. 

Figs. 21 to 23. — End motion of a rotating journal. 




Fig. 24. — Revolving element of the Glocker- White turbine governor. 

ing, the resultant R must be / P, since the resistance to sliding is 
the same in aU directions. It at once follows that the application 
of the force F, however small, changes the direction of sliding from 
the direction of X to the direction of R and a gradual creeping takes 
place in the direction of F. It is clear that this end motion wiU 
occur, no matter how small F is, or how great the friction, because 
R will always be the diagonal of the rectangle formed by the forces 
X and F, and R will change its angle, though not its value, to suit 
the value of F. 



PLAIN OR SLIDING BEARINGS 



27 



Advantage may frequently be taken of the freedom of revolving 
shafts to move endwise in the construction of machines of which 
delicate adjustment is an essential feature. 

An example is the well-known dead load tester for pressure gages, 
in which turning the plunger completely frees it of endwise friction. 
Another example is found in the Glocker-White governor, applied 
by the I. P. Morris Co. to four 13,000-h.p. turbines at Niagara Falls 
{Amer. Mach., Aug. 6, 1908), and shown in Fig. 24. 

The fly balls, a, are of special construction to suit the peculiar re- 
quirements of turbine governing. Through links, h, they act on 
the sleeve, c, through which connection is made with the other 
mechanism of the governor. The driving shaft, d, is cut off at e, 
the spindle, /, being supported stationary at its upper end, sleeve, c, 
revolving upon it. The result is absolute freedom of sleeve, c, to 
respond to the forces exerted by the fly balls. 

Bearings in which end motion takes place are free from the ten- 
dency to streak, which characterizes bearings with closely fitted end 
flanges. 

Thrust Bearings 

Thrust bearings are much less favorably situated as regards lubrica- 
tion than Journal bearings, as, except in the Kingsbury bearing, 
which see below, the film-forming tendency is absent. The best 




Fig. 25. — Multiple washer thrust bearing. 

that can be expected of such bearings is oily surface lubrication. 
The discovery of the film-forming tendency of journal bearings by 
Beauchamp Tower explained the weU-known fact that thrust bear- 
ings must be subjected to smaller unit pressures than journal bearings. 

Much less information is available on thrust than on journal bear- 
ings. A few figures for unit loads will be found in Table i of Per- 
missible Loads on Bearings and in Table 9 of Products of Pressure 
and Velocity of Bearings. 

A construction of multiple washer thrust bearing having peculiar 
merit, applied by the Newton Machine Tool Works to worm drives, 
is shown in Fig. 25 {Amer. Mach., Jan. 20, 1898). 

When several loose washers are interposed between the shaft col- 
lar and the face of the shaft bearing, it is obvious that slipping may 
occur between any pair of faces, and that this slipping will take place 
between those surfaces which at the moment offer the least friction. 
Should these surfaces from any cause increase their resistance, the 
slipping will be at once transferred to another joint, the various sur- 
faces acting as mutual safety valves to one another, any surface 
which gets into the condition of incipient heating or cutting being at 
once relieved by another taking up the work. The holes in the 
washers are larger than the shaft on which they are placed. This 
construction introduces an irregular compound motion of the sur- 
faces upon one another, the advantages of which are well understood. 



It would seem to the author that these holes might well be \ in. 
larger than the shaft. The washers should be covered to avoid 
criticism by the unthinking. 

Washers of vulcanized fiber have been used with conspicuous 
success in thrust bearings. They are used by the G. A. Gray Co. 
in the thrust bearings of their spiral geared planers, each bearing 
consisting of two fiber and one hardened steel disk. Fiber washers 
are also common in drilling machine thrusts. S. P. Yeo reports a 
test of the material under severe conditions {Amer. Mack., Oct. 24, 
1907), as follows: 

Our experiment was on two disks 9 ins. diameter with a 4-in. 
hole I in. thick. We used regular commercial red fiber, bored and 
turned carefully with cut oil grooves in it. The conditions these 
washers worked under were as follows: number of hours per day 
running, 9; revolutions per minute, 15; pressure per sqoiare inch, 
350 lbs.; disks running in oil. 

We found upon examining these disks after one month's service 
that the fiber had worn to a glazy surface and showed very little 
wear. The life of such a pair of disks was 15 years; the same size 
in bronze lasted about three months. 

Fig. 26 shows the step bearing of large Curtis vertical steam tur- 
bines. The bearing plate, or lower block, is of cast-iron rigidly 




Fig. 26. — Step bearing of the vertical Curtis steam turbine. 

held by the frame. The block is guided at the sides and carried 
on a large screw, passing through a steel nut and coming in contact 
with a steel block set in the bearing plate. It is essential that this 
plate should be rigidly held, that its upper face should be a true plane 
set at right angle to the shaft axis, and that the clearance should be so 
small that the relative alinement of blocks and shaft cannot vary 
appreciably. The step plate is likewise of cast-iron and keyed to 
the lower end of the shaft. Both plates are recessed so that the 
surfaces of contact are collars. Directly above the step plate is a 
cylindrical guide bearing. The contact faces of the blocks must be 
truly parallel, and the contact surfaces of the end of the screw be- 
neath the lower block and its mate must be likewise true and free 
from convexity. Oil is introduced through the center of the screw, 
passes upward, enters the recess in the center of the plate, passes out 
between the contact surfaces, and ascends upward through the guide 
bearing, as indicated by the arrows in the illustration. In service 
the plates are actually separated by a lubricating film; some four or 
five times as much lubricant as is necessary is usually pumped as a 
safeguard. The greater the quantity the greater the separation be- 
tween the plates and the less the danger of cutting out. The actual 
separation of the plate is, of course, only a few thousandths of an 



28 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



inch. From this fact it can be seen that this type of bearing calls 
for the best of workmanship. 

True film lubrication takes place in the Kingsbury thrust bearing, 
Fig. 27 {Amer. Mach., Mar. 13, 1913), which shows a bearing installed 
as part of a i7,soo-h.p. hydraulic turbine generator of the Penn- 
sylvania Water & Power Co. at McCalls Ferry, Penn. The 
diameter of the bearing is 48 ins. and it carries a load of 410,000 lbs. 
at a speed of 94 r.p.m. The illustration shows the stationary ele- 
ment, the upper babbitted surfaces of the shoes D being the surfaces 
on which the revolving step bears. The shoes are segmental, as 
shown by the detached one at the left. A recess in the lower side 
of each shoe receives a block E one end of which is spherical and rests 
on the similar spherical end of a second block H. When in action, 
the segmental blocks tilt slightly and admit a film of oil between 
themselves and the revolving step, on which film the load is carried. 
Wedges F are for adjustment. A continuous circulation of oil is 
maintained, an inner retaining ring A and an outer ring B retaining 
the oil, of which holes, C, establish the level. 




Fig. 27. — The Kingsbury thrust bearing. 

These bearings carry very much higher unit pressures than the 
usual type of thrust bearing. Low-speed bearings on light oils 
regularly carry mean pressures up to 300 lbs. per sq. in. Higher 
speeds regularly carry loads up to 500 lbs. per sq. in. Heavier oils 
regularly carry 900 lbs. per sq. in. In tests with light oils the oil 
film was shown to persist with mean pressures as high as 7000 lbs. 
per sq. in., at which pressure the babbitt face on the shoes flowed. 

The wedge shaped oil film in the thrust bearings is two or three 
times as thick at one end as at the other. On large bearings the cal- 
culated thickness is but a few thousandths of an inch at the thin end 
at moderate speeds with a light oil and pressures of 300 lbs. per sq. in. 
On small bearings it is often less than one thousandth of an inch. 

A feature of these bearings is the fact that the shoe tilt can vary 
with the speed, so as to adjust itself properly. Hence, when the 
bearing is at rest, the surfaces are parallel and in close contact in 
the vertical bearings. The starting and stopping of vertical bear- 
ings, some of which are loaded about as heavily at rest as when 
running, is their severest treatment. But as the surfaces are in 
contact all over when at rest, the mean pressure can be quite high 
and depends on the kind of oil and the bearing materials. 

General rules that have become well established for shaft bearing 
practice apply equally well to Kingsbury thrust bearings; that is, 
for low speed and heavy loads a thick oil must be used, while for 
high speed and light loads thinner oil is required. It is also found 
that for high speeds it is preferable to have the shoes faced with 
babbitt metal, the collar being of cast iron or mild steel. For very 
low speeds, coupled with high unit pressures, hardened steel or 
chilled iron may be used for the collars, and brass or bronze for the 
shoes. 

The friction loss in a Kingsbury bearing is very low. An approxi- 
mate rule for vertical bearings having six eccentrically supported 
shoes with inside diameter one-half the outside diameter, and loaded 



to 350 lbs. per sq. in. of shoe area, using dynamo oil and keeping the 
temperature at about 40 deg. Cent., makes the mean coefficient of 
friction .00009 times the square root of the r.p.m., and varying in- 
versely as the square root of the unit pressure. For example, if 
the shaft runs at 100 r.p.m., at a pressure of 350 lbs. per sq. in., 
the mean coeflScient of friction is .0009, an extremely low value, 
the result being due to the nearly ideal conditions for automatic 
lubrication. 

The starting friction is in some cases an entirely different matter. 
When the shaft is at rest with the load on the thrust bearing, which 
is the case with vertical shafts, the continuous oil film is not present, 
and the coeflicient of friction between the metallic surfaces is high, 
averaging about .15. At the instant of starting there is some rubbing 
between the metals. In the bearings with babbitted shoes this 
rubbing is frequently evidenced by a sound like that of a hot bearing. 
The rubbing lasts only a very brief time (about a quarter turn of 
the shaft), the oU film beginning to form at the instant of starting, 
and increasing in extent and thickness as the speed increases. If 
the bearing is provided with clean oil the only wear that takes place 
is that due to the rubbing of the metals when starting and stopping. 
For conditions such as exist in horizontal steam turbines there is. 
very little load on the thrust bearing at starting, and hence no reason 
to expect much wear. Even with vertical hydroelectric units, and 
similar machines in which the entire weight of the rotating part is 
carried on the thrust bearing at all times, it is found that the wear 
of the bearing is practically negligible. 

Many of these bearings have been fitted in hydroelectric power 
plants and some of them carry enormous loads, the maximum at 
this writing being 560,000 lbs. at the plant of the Mississippi River 
Power Company, Keokuk, Iowa. 

The reason for the unsatisfactory wear of step bearings lies in the fact 
that the wear of any bearing is proportional, other things being 
equal, to the product of the pressure on and the velocity of the rub- 
bing surfaces; and in a new flat step bearing, while the pressure is 
uniformly distributed over the surface, the velocity is greater as 
the distance from the center increases. Consequently the bearing 
wears much faster at the outside edges than in the center, and its 
effective area is practically reduced by wear, throwing increased pres- 
sure on the remaining portion and further increasing the tendency 
of the outer portion of the remaining effective surface to wear. The 
whole bearing is thus rapidly worn away in detail, as it were. 

This action is reduced if the bearing surface is a ring instead of an 
entire disk, and it is for this reason that collar bearings, such as the 
thrust bearings of steamships, do much better than step bearings. 

Theory indicates and practice confirms that the Schiele curve bear- 
ing, shown in Fig. 28 as the bearing of a worm, overcomes this 




Fig. 28. — The Schiele bearing as a worm step. 

action. This construction is one which the author feels has been 
unduly neglected. It conforms to Professor Sweet's dictum that 
"things that do not tend to wear out of truth do not wear much." 
Its basic principle is uniform wear, because its form is such that the 
pressure at different diameters increases as the velocity decreases. 
The cost of its construction is doubtless the chief cause of its neg- 
lect, but in heavy, rough machinery, in which unbored babbitted 



PLAIN OR SLIDING BEARINGS 



29 



bearings are admissible, the extra cost is confined to the journal 
where it is not serious. 

The construction of the Schiele curve, together with an approxima- 
tion to it which is practically sufi&cient, is thus explained by J. E. 
Johnson, Jr. {Amer. Mach., Apr. 21, 1904), who has had ample ex- 
perience with it in rugged work, and who unqualifiedly endorses it, 
one of his bearings having run for seven years without appreciable 
wear. 

X X, Fig. 29, is the center line of the shaft, A B the maximum 
radius of the thrust bearing, and A the point of beginning of the 
curve. With A B as a. radius from any point C a short distance 
from B on the axis, strike a short arc c c, intersecting A B ate. Draw 
the line c, C or a small part of it next c, and from point D with the 




e^-yil. 




v^M'^'' 




C D E F G H J 
Fig. 29. — Exact and approximate methods of laying out the Schiele 

curve . 

same radius proceed similarly, obtaining the line d, D and successively 
with points EFG, etc., until the outer portion of last line drawn 
comes down to the minimum radius desired for the bearing or to the 
radius of the shaft passing through the thrust bearing. In practice 
there is little to be gained by making J K less than half oi A B. A 
smooth cujve tangent to all the short lines outside their intersecting 
arcs is readily drawn and is the curve desired. The whole operation 
can be done in less time than it takes to describe it. 

For all practical purposes the curve may be drawn as two arcs of 
circles, as shown in the sketch, first an arc struck from the outside 
line of the bearing, prolonged, with a radius of 5-16 R, then tangent 
to this a second arc of radius 15-16 R, with its center on a line diverg- 
ing from the outer line of the bearing at 30 deg. passing through the 
center of the first arc. 

The double cone bearing has approximately the properties of the 
Schiele bearing and has found wide use for the spindles of precision 
machine tools. It originated during the early days of American 
watch making and was developed from the Schiele construction in 
order to take advantage of the grinding machine, since Schiele 
curves cannot be made on those machines. 

Figs. 30, 31 and 32, by Jas. Dangerfield {Amer. Mack., Mar. 27, 
1913) show approved constructions. The angles commonly used are 
3 and 45 deg., though angles of 4 and 5 deg. have been used in place 
of 3 deg. 



Fig. 30 shows a typical journal with angles of 3 and 45 deg., the 
groove A being for clearance in grinding. Fig. 31 shows one form of 
construction. In this case the spindle B is soft, the front bearing 
C is hardened and shrunk on; the rear bearing is the sliding sleeve D, 
with a pin E to key it to the spindle, and adjusted by the split bind- 
ing nut F. The bearings C and D are ground in place on their spindle 
to fit the bushes G, which are usually of hardened steel, but bronze, 
cast-iron and babbitt metal have been used. 




Fig. 30 




Figs. 30 to 32. 



Fig. 32. 
-The double cone spindle bearing. 



Fig. 32 shows a construction by Mr. Dangerfield in which both 
end thrusts are on the front bearing. The rear bearing is straight. 
iff is a split taper sleeve closed by the nut I and keyed to its bush K 
by the pin /. In case of wear of the side bearing causing shake, the 
thrust bearing is ground off enough to bring the side bearing to a fit 

Knife Edge Bearings 

The knife edge bearings of scales and testing machines form a 
class by themselves. According to J. W. Bramwell {Eng. News, 
June 14, 1906) loads of 10,000 lbs. per linear inch may be imposed in 
extreme cases, 5000 lbs. being, however, usually ample. For loads 
up to 1000 lbs. per in. the edge may be perfectly sharp. For 
greater loads the edge is rubbed with an oil stone so that a smooth- 
ness is just visible. The pivots should be thoroughly supported 
against deflection and consequent concentration of load and should 
be of high carbon — 90 to 100 points — steel. The temper of the 
seats should be drawn to a very light straw, that of the pivots being 
slightly darker. The angle of the pivots is usually 90 deg. 



BALL AND ROLLER BEARINGS 



The relations of the leading dimensions of ball and roller hearings 
may be determined from the following formulas and table by Robert 
A. Bruce (Amer. Mach., June 22, 1899). 

d = D sin [^)\a) Z) = — 4— W 



m 



R=d 






-i\(c) S=d 



/i8< 



SoV 



;+i\W 



in which 

« = No. of balls (no clearance) , D = diam. of ball centers circle, ins. , 
d= diam. of balls, ins., 2? = diam. of inner race, ins., 

S = diam. of outer race, ins. 

For a total clearance c between balls, add - to d in formulas (a) 

n 

and (b). 

Leading Dimensions of Ball and Roller Bearings 





180° 
■ « 


. /l80N° 


/ 


. /i8on,<'"'"^ 


/ 


n 


. /i8o\° 


. /i8on ^ 


7 


25° <J2' 51.4" 


.43389 


2.3048 


3.3048 


1 -3048 


8 


22° 30' 0" 


.38268 


2.6131 


3.6131 


I .6131 


9 


20° 0' 0" 


.34202 


2.9238 


3 9238 


1.9238 


10 


18° 0' 0" 


.30902 


3.2360 


4.2360 


2 . 2360 


II 


16° 21' 48.3" 


.28173 


3.S49S 


4.S495 


2.549S 


12 


15° 0' 0" 


.25882 


3.8637 


4.8637 


2.8637 


13 


13° 50' 46.1" 


.23932 


4.1786 


5. 1786 


3.1786 


14 


12° si' 25.7" 


.22252 


4.4940 


5-4940 


' 3-4940 


15 


12° 0' 0" 


.20791 


4.8097 


S.8097 


3.8097 


16 


11° 15' 0" 


.19509 


5.1258 


6.1258 


4.1258 


17 


10° 3S' 17.6" 


.18375 


5. 4422 


6.4422 


4.4422 


18 


10° 0' 0" 


.1736s 


5.7S88 


6.7588 


4.7588 


IS) 


9° 28' 2S.2" 


.16459 


6.075S 


7.0755 


5. 0755 


20 


9° 0' 0" 


.15653 


6.3925 


7-3925 


S.392S 


21 


8° 34' 1 7.1" 


. 14904 - 


6.709s 


7.7095 


5.709s 


22 


8° 10' 54 -S" 


.14231 


7.0267 


8.0267 


6.0267 


23 


7° 49' 33-9" 


.13617 


7.3439 


8.3439 


6.3439 


24 


7° 30' 0" 


.13053 


7.6613 


8:6613 


6.6613 


25 


7° 12' 0" 


.12533 


7.9787 


8.9787 


6.9787 


26 


6° SS' 23" 


.12054 


8 . 2963 


9.2963 


7.2963 


27 


6° 40' 0" 


.11609 


8.6138 


9-6138 


7.6138 


28 


6° 25' 42.8" 


.11196 


8.9314 


9-9314 


7.9314 


29 


6° 12' 24.8" 


.10812 


9.2491 


10.2491 


8-2491 


30 


6° 0' 0" 


.10453 


9-5668 


10.5668 


8.5668 



The following information on this subject is taken largely from a 
paper read before the A. S. M. E. {Trans. Vol., 29) and articles in 
periodicals by Henry Hess, the data sheets published by the Hess- 
Bright Mfg. Co. and the exhaustive treatise. Bearings and Their 
Lubrication, by L. P. Alford. 

Successful performance of ball bearings depends upon the follow- 
ing factors: (a) A high degree of accuracy as regards spherity and 
uniformity of diameter of the balls. The tolerance in first-class bear- 
ings between the diameters of the balls in any one bearing is .0001 
in. (6) A high (very high) degree of surface finish, (c) High 
elastic limit of the materials, (d) Hardness, and especially uniform 
hardness, throughout each and all balls — case-hardened balls or 
races are inadmissible, (e) True rolling contact. 

Successful operation of ball bearings requires attention to the 
following points: 

Bearings must be lubricated. The oft-repeated statement that 
ball bearings can be run without lubricant is pernicious. 

Bearings must be kept free of grit, moisture and acid. This pro- 
hibits the use of lubricants that contain or develop free acids. 



The inner race must be firmly secured to the shaft. It is best to 
do this by a light drive fit, reinforced by binding between a sub- 
stantial shoulder and a nut. 

The outer race must be a slip fit in its seat. 

When thrust is taken in both directions it should be by the same 
bearing. This avoids all strains due to flexure of the shaft or of 
the housing or due to temperature variation and, while doing away 
with the considerable shop costs inseparable from correct lengthwise 
dimensioning, avoids the danger of excessive end loads from forcible 
assembly consequent on an inaccurate lengthwise location of parts. 

More than one bearing should never be dismembered at a time, 
in order to avoid the danger of mixing balls from different bearings; 
such balls from different bearings are apt to vary more than is per- 
missible for the individual bearing. 

Lubricants may range from the lightest of spindle oils at high speeds 
to fairly heavy greases at low speeds. The less frequent the atten- 
tion given, the heavier should be the lubricant. An excess of lubri- 
cant, enough to force out at the closures, should be employed when- 
ever the entrance of grit or moisture is to be feared. Lubricants 
containing or developing acid or containing free alkali must be 
avoided, as must those that become rancid. 

A ball bearing, like a plain bearing, must have a running clearance; 
but in the well-made ball bearing this clearance is much smaller than 
in a plain bearing; in all new bearings this freedom (radial freedom) 
is less than .001 in. The radial freedom is accompanied by an 
endwise (axial) freedom of one race with reference to the other; this 
will vary with the ball diameter and ranges from .0006 in. to .006 
in. for new bearings. 

A properly made and not overloaded ball bearing will not show 
wear in the ordinary sense of plain bearings, i.e., a reduction of the 
diameter and increase in bore. That and reduction in ball diameter 
can occur only when abrasive grit is admitted to the bearing. Grit 
will quickly grind down a bearing at a rate depending only upon the 
sharpness of the grit and the amount of time the bearing is exposed 
to it. 

An overloaded bearing will not be worn, but the surfaces of the 
balls and races will be destroyed; that will show first by minute pin 
holes and later by flaking. A large ball will take more than its share 
of the load and may therefore bring about all the appearances of an 
overloaded bearing; to avoid this, no ball must vary by more than 
.0001 in. from its fellows in the same bearing. It is on this account 
that bearings must never be dismembered, as otherwise balls are 
likely to be mixed; neither must repairs be made by adding balls to 
a set. Such repairs should be undertaken only by the maker, who has 
full sets of even-size balls available. The balls of commerce are 
never sold within the necessary limits. 

Rust is absolutely destructive to a ball bearing. It is very readily 
recognized; even in a bearing which has been cleaned so that no red 
rust is to be seen, the presence of more or less pronounced pits and ex- 
coriations, not only on the race surfaces, but also on the other parts 
of the bearings, is clear evidence. These pits are very distinct in 
appearance from those due to overload; aside from that, overload 
pits are necessarily confined to the balls and the ball tracks. 

Although the presence of acid or alkali in many lubricants is well 
known, its destructive effect is not generally conceded, but attrib- 
uted to rust and overload. Nevertheless, it is a very serious menace 
with some lubricants; its ravages are clearly enough distinguished 
from overload, because they are found elsewhere than on the balls 
and ball tracks; the marks are also quite distinct from rust marks; 
30 



BALL AND ROLLER BEARINGS 



31 



the acid marks often are pits, but always show also clearly-defined 
irregular etchings, similar to, though less pronounced than, those 
produced by acid etching of damascened gun-barrels. 

A very considerable rocking freedom of the one race with refer- 
ence to the other as the result of grit cutting will do no harm. An 
amount of rocking that at first seems alarming in the individual 
bearing may be due to a radial clearance of but few thousandths of 
an inch. It is the radial clearance which determines the further 
usefulness of the bearing; as two bearings are always used at some 
distance apart in the support of a shaft or wheel, the rock is governed 
by the radial freedom of the bearings and the distance between them, 
not by the angular freedom of the bearings individually. To deter- 
mine its true radial freedom, the bearing must be so held that the 
races are moved only crosswise without any lengthwise or tilting 
motion. 

Bearings in which the balls or ball tracks are pitted or roughed up 
from rust, acid or overload are usually beyond repair. 

Bearings that are ground down by grit so as to be loose, can be 
put in good order by refilling with a new set of larger balls. The 
same amount of care must be exercised to have all these balls within 
.0001 in. as with a new bearing; it will not do to put in a few new balls 
only, nor will it do to accept a dealer's belief that the balls in his 



between the baUs, have become the accepted design by the Hess- 
Bright Mfg. Co. Some other manufacturers prefer the cut race 
filled with balls and without separators. 

Running tests and accumulated experience have proven that this 
type of bearing with separators will also carry a thrust load far be- 
yond what would be expected or what calculations based on the 
wedging action would indicate. The safe thrust-carrying capacity 
of such bearings is iV of the radial-load capacity, though under 
special conditions more may be imposed, depending on the relation 
of ball diameter, race curvature and number of balls. 

Typical correct mountings of bearings for radial loads — combined 
in some cases with moderate thrust — are shown in Figs. 3-6. 

Fig. 3 shows a mounting for radial load without thrust. 

The inner race A should be a light drive fit on the shaft, and should 
further be securely clamped between a shoulder on the shaft and a 
nut, or their equivalents. 

The shoulder C should not be too small; about half the thickness 
of the inner race for small bearings, and about f the thickness of 
the inner race for large bearings, is good design. 

The nut, when well set up, should be firmly secured against jar- 
ring loose. An effective device consists of a split spring-wire ring 
with one end bent inward to pass through a hole drilled through the 




Fig. 2. 




.Ot^l- -^U-:Ol" 



Fig, 3. Fig. 4. 




Fig. 5. Fig. 6. 

Figs, i to 6. — Radial bearings for radial and thrust loads. 



bin are within the necessary limit. Unless this grinding action has 
lasted too long, it will be practicable to restore the bearing to some- 
where near its original condition; even though such refilled bearing 
should be somewhat looser than a new one, that does not justify the 
expense of a replacement. 

Radial Bearing Mountings 

Fig. I shows the principle of the modern form of radial bearing. 
The curvature of the race cross-section is an important factor in the 
carrying capacity of the bearing. The local groove. Fig. 2, to permit 
assembling the balls, if used, must be confined to the stationary race 
and be placed at the unloaded side of the bearing. This requires 
two designs, one with the cut in the outer race, if the shaft revolves, 
and one with it in the inner race, if the housing revolves, or else the 
load must be limited to that permissible with straight races. More- 
over, at high speeds, the rotation of the balls at the filling opening 
injures the balls and races. For these reasons uncut races with 
such a number of balls as may be then assembled, and separators 



nut into the shaft, the body of the ring lying in a circumferential 
groove turned in the nut. 

The outer race B should be a slip fit in the box; it should not be 
bound endwise. 

Failure to securely clamp the inner race on the shaft may produce 
trouble. It has been found that the hard race occasionally cuts 
into the relatively soft shaft, particidarly when loads are heavy and 
of a pounding or vibratory character. A reliance on a drive fit 
alone is not bafe. Such fit may be poorly made, it may be destroyed 
by occasional dismantling, or it may be destroyed by the load peen- 
ing down the shaft surface. 

Failure to mount the outer race with a slip fit may prevent it 
from taking up an unrestrained position with reference to the inner 
race and the balls and so produce an end thrust uncontemplated in 
the initial selection of the bearing that might soon prove destructive. 

Fig. 4 shows a mounting for combined radial and thrust loads. 
This arrangement difl!ers from that of Fig. 3 only in having the outer 
race B of the bearing secured endwise in the case with slight clear- 
ance each side. Any thrust parallel to the axis of the shaft or any 



32 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 





^^^^^^W 






I — ^^^ 



-i 





Lb 




42 



> 



O 




^^- * 



tendency of the shaft endwise will be taken 
up by the bearing acting as a thrust bearing. 

Fig. 5 is a combination of the elements 
shown in Figs. 3 and 4; the remarks already 
given apply here also. 

Whenever there are two or more bear- 
ings on a shaft the parts must be so ar- 
ranged that whatever end thrusts there 
may be will be taken in both directions on 
the same bearing. Frequently designers 
make the mistake of taking thrust in one 
direction on one bearing and the opposite 
thrust on the other bearing. In that case 
any inaccuracy in the machining of shoul- 
ders on the shaft or in the case will cause a 
destructive thrust to be set up through the 
balls and bearings as soon as the end nut on 
the shaft is set up. Similarly, deflections 
of shaft or housing, or temperature varia- 
tions will set up such end thrusts. 

Aside from the avoidance of these possi- 
ble sources of trouble, the construction 
recommended has the advantage of less 
shop cost, because avoiding otherwise 
necessarily very accurate work. 

The illustration also shows the use of a 
distance sleeve to permit of the endwise 
binding of the inner races of both bearings 
by one nut only. 

Fig. 6 shows a construction for taking 
radial and thrust loads on separate radial 
bearings. A radial bearing is mounted as 
near the load as may be and in the usual 
way to take only radial load. Beyond it a 
similar radial bearing is similarly mounted, 
but with its outer race clamped endwise to 
take thrust. To prevent the radial load 
being imposed on this bearing the seat is 
counterbored to free the bearing diameter. 

The use of the radial type to take thrust 
is preferable for speeds above 1500 r.p.m. 
as its carrying capacity is but little affected 
by speed. If the thrust direction alter- 
nates there will be a slight endwise play of 
the shaft, since the inner race has an axial 
or endwise freedom of .0006 in. for the 
small bearings to .oc6 in. for the larger ones; 
this win be increased by too heavy loading. 

Fig. 7 shows a mounting for loose pul- 
leys, conveyor rolls and the like, on hori- 
zontal shafts consisting of a standard inner 
hub on which a loose puUey of any desired 
diameter, and of not more than the speci- 
fied width, may be secured by means of 
a key. The inner races are a light drive 
fit on a sleeve which is held by set screws 
or keys on the shaft. The outer races are 
a slip fit in their housings and one is con- 
fined with a slight end clearance, while the 
other race is free endwise. 

Fig. 8 shows a mounting for a mule 
pulley on a vertical shaft. The shaft is 
stationary and the pulley rotates. Two 
objects are gained by the peculiar form of 
mounting shown. One is retention of oil, 
which would tend to throw out, owing to 
centrifugal force, if the outer races rotated 



BALL AND ROLLER BEARINGS 



33 



in the usual manner. The other object is distribution of wear 
around both races. With a stationary shaft it is evident that clamp- 
ing the inner race causes all the wear to take place at a single point 
of that race. Where the outer race is fixed and the shaft rotates, 
concentration of wear is prevented by the standard mounting, 
which allows the outer race to creep slowly around. In the 
mounting illustrated herewith the outer race is attached to the 
shaft and the inner race rotates with the pulley. Thus the wear on 
the inner race is distributed by rotation, and the outer race distrib- 
utes its own wear by creeping. The objection to allowing the inner 
race to creep when mounted directly on the shaft is that the contact 
surface between the inner race and the shaft is not large enough to 
sustain the load without peening or wear of the shaft, or both. 

It is evident that oil will be retained without splashing around both 
upper and lower races, even if the shaft be reversed end for end. A 
certain amount of tilting of the shaft is also permissible. 

Fig. 9 shows a typical mounting for a high-speed spindle and 
pulley. In machine-tool work, especially in grinding, it is highly 
important that the spindle be subjected to no unnecessary forces 
producing sidewise wear of the bearing. The pulley, therefore, should 
be supported by bearings independent of those in which the spindle 
runs. The spindle itself is mounted in ball bearings in the usual way. 
The back end of the spindle extends without contact through a sta- 
tionary tubular support on which is mounted a second pair of ball 
bearings, whose sole function is to support the pulley. Driving con- 
nection between the pulley and the spindle is provided by a pair of 
splines or feathers riveted into the hub of the pulley and fitting loosely 
in keyways in the spindle. These splines permit the pulley bearings 
to be out of line with the spindle bearings or to wear faster than the 
latter without impairing the true running of the spindle. 

The rear ball bearing of the spindle {i.e., the one in the center) has 
a special outer race without a groove. This construction is provided 
against the possibility of the spindle expanding from the heat devel- 
oped at the wheel when grinding. With the form of race shown, the 
inner race can move indefinitely lengthwise without the outer race 
being forced to follow it, which it might fail to do if it also expanded 
slightly. 

Figs. lo, II and 12 show mountings on shafts without shoulders, 
with provision for securely clamping the inner race of the ball bearing, 
while distributing the peening effect of the load on a suflScient length 
of shaft. Fig. 10 shows a radial bearing mounted in the usual manner 
on a sleeve, the inner race clamped by means of a nut against a sub- 
stantial shoulder on a sleeve. The sleeve is locked to the shaft by 
means of two set screws with fitted ends sunk into the shaft. A 
spring ring is then snapped into a circumferential groove in the sleeve 
and into the slots in the set screws, securely locking them against 
jarring loose. 

Fig. II includes a collar thrust bearing. Lighter thrust in the 
reverse direction is taken by the radial bearing. The set screws must 
be large enough to let their pilot ends take the end thrust without 
danger of shearing. The set-screw heads must, of course, be accessible 
from the end or through suitable holes. 

Fig. 12 shows an adapter bearing used chiefly on line shafts. The 
adapter consists of two steel sleeves or bushes; the outer bush is 
driven home to its shoulder with a light drive fit. This bush is bored 
out taper to suit the inner split bush. The inner bush should be 
driven home on the shaft when the bearing and outer sleeve are in 
place; that will firmly and truly bind the whole on the shaft. The 
taper is sufiScient for adaptation to the ordinary variation in shaft 
sizes. After the inner bush is driven home good and hard on the 
shaft, the split collar is brought against the bearing and clamped on 
the projecting bush end, thus safeguarding against the tendency of 
vibration to back out the taper bush. 

For shafts requiring great steadiness of movement, as in precision 
grinding machines, ball bearings alone have been found inadequate 
and Fig. 13 shows the application of a floating bush to a thickness- 
3 



grinding machine at the Underwood Typewriter Works, (W. M. 
Byorkman, Avier. Mach.^Feh. 9, 191 1). 

The bushes are located close to the ball bearings, and have inside 
and outside clearances of about rffw in. They have also end 
clearances sufficient to permit appreciable movement. The bushes 
carry no load, but the clearances fill with oil which acts like a dashpot 
to suppress the minute vibrations which would otherwise arise in 
the ball bearings. Not being loaded, the bushes do not wear, and 
their only resistance is that due to the viscosity of the oil. To pre- 
vent endwise movement of the spindle, the two thrust ball bearings 
near the right-hand end of the spindle are provided with an adjustable 
take-up, which allows metal to metal contact to be made without 
crushing. 

In Figs. 3-6 the groove and lip end closures shown are very effect- 
ive in excluding dust and grit and in retaining oil. They should be 
bored out no more than it in. larger than the diameter of the shaft. 
Figs. 14-17 show still more efi^ective arrangements. 

Fig. 14 is a modification useful where much very destructive 
fine grit is afloat or liquid is encountered under slight pressure. 
The second outer groove is filled with a semi-solid grease that makes 
a definite, frictionless packing. Its wearing away is compensated 
from a hand- or spring-operated grease cup; the latter type is prefer- 
able, but demands a proper balance between grease consistency under 
various temperatures and the spring pressure. The hand-operated 
cup is more definite for all conditions, provided it is occasionally set 
up. 

Fig. 15 is used where liquid under considerable pressure must be 
kept out. For occasional submersion the outer groove is simply 
drained outward with a free drain hole. For continuous submersion 
this drain hole must be connected to a pipe whose end is clear and 
low enough below the liquid to drain. 

Many constructors prefer to pin their faith to some positive felt 
packing for keeping lubricant in and foreign matter out. Felt wash- 
ers, as usually arranged, soon lose their contact with the shaft as they 
grow hard and are then worn away. 




Fig, 16, Fig. 17. 

Figs. 14 to 17. — Oil-retaining and dust-excluding devices. 

Fig. 16 shows an arrangement for sealing at the shaft. A spring- 
wire ring encircles the felt washer; its pressure will force the felt 
into sealing contact. If the washer is laid up from a strip with 
scarfed ends instead of stamped from a sheet the spring ring need not 
be so stiff. 

Fig. 17 shows a seal applied between a rotating hub and fixed box. 
The hub face and ring, being both beveled will permit of very con- 
siderable wear. 



34 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



The felt should be thoroughly soaked in a good cylinder oil. Mutton 
tallow and similar acid producers must not be employed. 

Collar Thrust-bearing Mountings 
In collar thrust bearings, aligning washers are necessary to secure 



Wherever shaft deflections are liable to occiur, the thrust bearing 
must have an imderlying washer, or a special aligning washer must 
be inserted under the ball-seated race. In either case the aligning 
washer must be free to shift laterally to accommodate itself to the shaft 
deflection. A ball-seated thrust bearing without the aligning washer 




FiG.'27, ^IG- 28. 

Figs. 18 to 28. — Principles and arrangements of collar thrust bearings. 



uniform loading of the balls. These washers frequently take the 
incorrect form shown in Fig. 18. One plate is convexed to rest in a 
concaved aligning seat. This form is wrong in that both plates fit 
the shaft and so cannot move relatively. Fig. 19 shows this corrected 
by freeing the lower plate from the shaft. This allows for certain 
errors, but not for all. Fig. 20 shows the full solution for every pos- 
sible error of machining or of deflection. An aligning washer is added 
to receive the lower plate; this washer also is free of the shaft and also 
free of the seat, so that it may move crosswise. In Fig. 21 a complete 
unit-handling ball bearing is shown, which consists of two plates, an 
interposed set of balls, the lower plate convexed to seat in a concaved 
imiversal aligning washer, and a cage to hold all together. 

These constructions are safeguards to provide for small unavoid- 
able errors. It is clear that large errors will be accompanied by large 
eccentric action of the aligning washers, which will be accompanied 
by friction. 

Typical correct mountings of collar thrust bearings are shown in 
Figs. 22-32. 

The collar type of bearing, at speeds below 1500 r.p.m., will take 
higher thrust loads than the radial type. 

The bore of the rotating plate A, Fig. 22, is ground to a definite size 
and is usually seated on the shaft. The bore of the fixed plate B is 
rather larger. 

In order to insure an even distribution of the load over the entire 
series of balls, the seating surface of the stationary plate is spherical. 
If the thrust load is apt to be relieved sufficiently to permit a separa- 
tion of the plates this must be prevented by a suitable arrangement, 
since otherwise the cage with the balls would drop slightly and the 
balls would be pinched as the pressure was again put on. 




Fig. 31. 



Figs. 29 to 32. — Foot step bearings. 

is good only for angular misalignment, not for deflections involving 
lateral shift of the shaft axis. 

By arranging two bearings to face in opposite directions, Fig. 23, 
thrust in opposite directions is taken up on ball bearings. Care must 



BALL AND ROLLER BEARINGS 



35 



be exercised that no undue amount of thrust is set up by the initial 
adjustment of the nut behind the bearing. Setting up must not be 
carried to the point of feehng the bearing resistance; ball-bearing 
friction is so low that it is perceptible to the touch only under loads 
that mean serious overload. 

When the thrust in one direction is light, a plain disk step may 
answer to replace the second ball thrust, as shown in Fig. 24. 

Combinations of radial and collar bearings — the former taking radial 
and the latter thrust loads — are shown in Figs. 25-28. 

In certain combinations of sizes the rotating plate of the collar 
bearing might be large enough to come into contact with the stationary 
outer race of the radial bearing. The insertion of a ^washer will 
prevent that (Fig. 25). The inner race at A should be a light press 
fit. 

Where the radial load causes heavy hammering and is large as com- 
pared with the thrust, it is well not to rely merely on the press fit of 
the radial-bearing inner race nor yet on the end-clamping due to the 
thrust load. By interposing, as in Fig. 26, a locked nut between the 
collar and the radial bearings, the latter will be securely clamped 
endwise. This is the preferred construction: it should be used wher- 
ever possible. 

The action of the spherical seat in compensating for deflections of 
shaft and housing and inaccuracies of alignment will be best when the 
center of the spherical seat lies as nearly under the center of the 
radial bearing as is possible. 

The arrangements shown in Figs. 27 and 28 have been advanta- 
geously employed in motor boats to take the thrust of the propeller 
shaft as well as its weight. Similar combinations are in use in steam 
yachts of 300 h.p. and in worm-driven elevators. 

The arrangement of Fig. 27 is preferable. If the center of the ra- 
dial bearing be also the exact center of the ball seats of both thrust 
bearings, the aligning washers may be dispensed with. Generally, 
however, it is more convenient to use them and shorten the length of 
the block, incidentally providing for small errors and deflections. 

Figs. 29 and 30 show suitable arrangements for the lower end journals 
of vertical shafts. 

Figs. 31 and 32 show mountings for end thrust on upper or interme- 
diate journals, which difier from the mountings for lower end journals 
in the provision made for oiling. A cup is carried upward along 
the shaft to a height that will ensure the balls being about half 
immersed. 

Fig. 3 1 takes thrust on collar bearing only. The oil cup A may be 
tightly spun or threaded in, or otherwise arranged to prevent the 
escape of oil in any way other than by overflow at its top. 

Fig. 32 takes thrust and radial load on collar and radial bearings. 
The thrust arrangement is practically that of Fig. 31. In order to 
provide an oil level for the radial bearing this is not mounted directly 
on the shaft, but on a collar which has an annular space next to the 
shaft into which the oil cup A projects upward to a sufficient height. 

The two oil spaces may be separately filled, or the lower one by 
overflow from the upper. 

Figs. 29 and 31 take thrust only. Figs. 30 and 32 take radial and 
thrust loads on separate bearings. The sleeve or bush between the 
two bearings will prevent any contact between the rotating plate of 
the collar bearing and the stationary outer race of the radial bearing. 

The action of the spherical seat in compensating for deflections of 
shaft and housing and inaccuracies of alignment will be best when 
the center of the spherical seat lies as nearly in the center of the radial 
bearing as is possible. 

The Load Capacity of Ball Bearings 

The load-carrying capacity of radial ball bearings, according to the 
practice of the Hess-Bright Mfg. Co. (based on the researches made 
for it by Professor Stribech), may be determined from the formula: 
P =knd'' 



P = 



in which P = load on bearing, 

n = number of balls required to fill the races, 
i = diameter of balls, 

k = z, coefficient depending on the type of bearing, the 
material and the speed. 
For Hess-Bright bearings, in which the radius of curvature of the 
outer race groove = t'ijfi, and that of the inner race groove = f|(f, 
separated balls being used and uniformly distributed load and 
uniform speed below 3000 r.p.m. being assumed, k = g (for load in 
lbs. and d in units of \ in.). For full type bearings with the filling 
opening in one race at the unloaded side, otherwise as above, k = $. 
For both ball tracks interrupted by filling openings, inelastic cage 
separators for the balls, or for full baU type; and speeds not over 
2000 r.p.m. with a uniform distributed load, ^ = 2.5. For thrust 
load on a radial bearing of the first type in this tabulation k = 0.9. 

In general, the larger the number of balls, the smaller the value of 
k. The radial load bearing is, within the limits stated, practically 
unaffected by the speed as to its carrying capacity. 

Collar thrust bearings are made of three general types. In the 
first both races are flat; in the second one race is flat the other grooved; 
in the third both races are grooved. The load-carrying equation 
given by Mr. Hess is: 

kind^ 

in which P = load on bearing, 
n = number of balls, 
d = diameter of balls, 
5 = r.p.m., not exceeding 3000, 

fti = a coefficient depending on the material and the shape of 
the ball races. " V 

For the materials used by the Hess-Bright Mfg. Co. and for races 
having grooves with a cross-sectional radius equal approximately to 
.82d, ^1 = 25 to 40 (for loads in lbs. and d in units of | in.). For 
unhardened steel, such as is occasionally used for very large races and 
where there is no hammering or sharp blows, ki = .5. When one or 
both races are flat ki should be reduced to one-fourth the above valufe. 

The Standard Roller Bearing Company give the following load- 
capacity formulas for ball thrust bearings having a groove in each 
washer, stating that the ratings obtained from their use are very 
conservative and give a condition of loading under which a bearing 
can be guaranteed if properly installed and lubricated. 

The formula for the light-type bearing with 17 balls is: 

nd^ 

P = 32,000—^ 

aVS 

in which P = load on bearing, lbs., 
d = ha.\l diameter, ins., 
M = number of baUs, 

a ^ pitch diameter of the ball grooves, ins., 
5 = speed of the shaft, r.p.m. 
The formula for the medium and heavy bearings having 11 and 
9 balls respectively is: 

nd^ 

The notation is the same as given above. 

Ball thrust bearings are commonly of the two-point type to which 
according to the S. R. B. Co., the preceding formulas apply. 

Variati'^ns in speed cut down the carrying capacity; sharp varia- 
tions of small amphtude, particularly at high speed, have the more 
marked effect. Their reducing action is similar to the battering 
efi!ect of sharp load variations. 

Load variations reduce carrying capacity, the effect increasing 
with the amount of the load change and the rapidity of such change. 

Accumulated experience with various classes of mechanisms is so 
far the only available guide for estimating the reductions in the con- 
stants k that must be made to take these influences into account. 



36 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



The fricHonal resistances of ball bearings have, by actual measure- 
ment, been found to vary from .ooii to .0095. These are the coeflS- 
cients of friction referred to the shaft diameter, thus permitting direct 
comparison with those of sliding friction. The higher values are 
due to conditions that cause a preponderance of sliding as compared 
with roUing friction. It must be remembered that there is no such 
thing as a bearing having only rolling friction; that might be possible 
were balls and races made originally with absolute truth of surfaces 
and were such truth then maintained by the absence of deformation 
under load. Ball bearings having a coefficient of friction materially 
above .0015 under the greatest allowable load are inadmissible 
because too short-lived. The high resistance indicates the presence 
of too large an element of sliding. 

A good ball bearing will have a coefficient of friction, independent 
of the speed within wide limits, and approximating .0015. This 
coefficient will rise to approximately .0030 under a reduction of the 
load to about one-tenth of the maximum. 

Dimensions of Ball Bearings 

The dimensions of ball bearings are well standardized. Oddly 
enough, baU diameters are universally expressed in inches and frac- 
tions thereof, while the dimensions of the races are given in either 
millimeters or English units. German builders use millimeters with 
a single exception where a &m has developed a series in English 
units, adapting them for the British trade, though that firm also uses 
chiefly ball bearings in millimeters. Even in England most ball 
bearings are made to millimeters as is also the more general practice 
of American manufacturers who have followed the German example. 
This general adoption of the millimeter dimensions is due to the fact 
that early German makers rehabilitated the ball bearing by the devel- 
opment of the principles and construction data of the modern type 
and secured a wide vogue for their products that made the sizes 
standard. As a rule each manufacturer makes a wide and narrow 
type of radial bearing, with three series for each; namely, light, 
medium, and heavy. In some cases a fourth series has been standard- 
ized, known as extra heavy. 



Table 2.— Dimensions of Radial Ball Bearings — Medium 
Series 



Table i. — 


Dimensions of Radiai 


. Ball Bearings — Light Series 


No. 

of 

bear- 


Bore 


Diameter 


Width 


Corner at 

bore of 

inner race 


Radial 
load, 
lbs. 


ing 


Mm. 


Ins. 


Mm. 


Ins. 


Mm. 


Ins. 


Mm. 


Ins. 


200 


10 


0-39370 


30 


I. 1811C 


9 


0.35433 


I 


0.04 


120 


201 


12 


0.47244 


32 


1.25984 


10 


0.39370 


I 


0.04 


140 


202 


IS 


0.59055 


35 


1-37795 


II 


0.43307 


I 


0.04 


160 


203 


17 


0.66929 


40 


1.57481 


12 


0.47244 


I 


0.04 


250 


204 


20 


0.78740 


47 


1.85040 


14 


0.55118 


I 


0.04 


320 


20s 


25 


0.98425 


52 


2.04725 


IS 


0-59055 


I 


0.04 


350 


206 


30 


1.18110 


62 


2.44095 


16 


0.62992 


I 


0.04 


550 


207 


35 


1-37795 


72 


2.83465 


17 


0.66929 


I 


0.04 


600 


208 


40 


I. 57481 


80 


3.14962 


18 


0.70866 


2 


0.08 


860 


209 


45 


I. 77166 


85 


3-34647 


19 


0.74803 


2 


0.08 


950 


210 


SO 


I. 96851 


90 


3-54332 


20 


0.78740 


2 


0.08 


1000 


211 


SS 


2.16536 


100 


3-93702 


21 


0.82677 


2 


0.08 


1160 


212 


60 


2.36221 


no 


4-33072 


22 


0.86614 


2 


0.08 


1550 


213 


6S 


2.55906 


120 


4-72443 


23 


0.90551 


2 


0.08 


1670 


214 


70 


2-75591 


125 


4.92128 


24 


0.94488 


2 


0.08 


1820 


215 


75 


2.95277 


130 


5-11813 


25 


0.98425 


2 


0.08 


2130 


216 


80 


3.14962 


140 


5-51183 


26 


I .02362 


3 


0.12 


2650 


217 


85 


3-34647 


150 


5-90554 


28 


I. 10236 


3 


0.12 


2850 


218 


90 


3-54332 


160 


6. 29924 


30 


I . 18110 


3 


0.12 


3400 


219 


95 


3-74017 


170 


6.69294 


32 


1.25984 


3 


0. 12 


3750 


220 


100 


3-93702 


180 


7.08664 


34 


1-33858 


3 


0. 12 


3950 


221 


105 


4.13387 


190 


7-48035 


36 


I. 41732 


3 


0.12 


4600 


222 


no 


4-33072 


200 


7-87405 


38 


1.49607 


3 


0.12 


5000 



No. 
of 


Bore 


Diameter 


Width 


Corner at 
bore of 


Radial 

load, 

lbs. 


bear- 








inner 


race 


ing 


Mm. 


Ins. 


Mm. 


Ins. 


Mm. 


Ins. 


Mm. 


Ins. 


300 


10 


0.39370 


35 


1-37795 


n 


0.43307 




0. 04 


200 


301 


12 


0.47244 


37 


1.45669 


12 


0.47244 




0.04 


240 


302 


15 


0.59055 


42 


1-65355 


13 


0.51181 




0.04 


280 


303 


17 


0.66929 


47 


1.85040 


14 


0.55118 




0.04 


370 


304 


20 


0.78740 


52 


2.04725 


IS 


0-59055 




0. 04 


440 


305 


25 


0.98425 


62 


2.44095 


17 


0. 66929 




0.04 


620 


306 


30 


1.18110 


72 


2.83465 


19 


. 74803 


2 


0.08 


860 


307 


35 


1-37795 


80 


3 • 14962 


21 


0.82677 


2 


0.08 


noo 


308 


40 


1.57481 


90 


3-54332 


23 


0.90551 


2 


0.08 


1450 


309 


45 


I. 77166 


100 


3.93702 


25 


0.98425 


2 


0.08 


1750 


310 


SO 


I. 96851 


no 


4-33072 


27 


1.06299 


2 


0.08 


2100 


311 


55 


2.16536 


120 


4-72443 


29 


1.14173 


2 


0.08 


2400 


312 


60 


2.36221 


130 


5-11813 


31 


1.22047 


2 


0.08 


2800 


313 


65 


2.55906 


140 


5-51183 


33 


I. 29921 


3 


0. 12 


3300 


314 


70 


2.75591 


150 


5-90554 


35 


1-37795 


3 


0. 12 


4000 


315 


75 


2-95277 


160 


6.29924 


37 


1.45669 


3 


0.12 


4400 


316 


80 


3.14962 


170 


6.69294 


39 


1-53544 


3 


0.12 


5000 


317 


85 


3-34647 


180 


7.08664 


41 


1.61418 


3 


0. 12 


5700 


318 


90 


3-54332 


190 


7.48035 


43 


1 .69292 


3 


0.12 


6400 


319 


95 


3-74017 


200 


7-87405 


45 


I. 77166 


3 


0. 12 


7000 


320 


100 


3-93702 


215 


8 . 46460 


47 


I . 85040 


3 


0. 12 


7700 


321 


105 


4-13387 


225 


8.85830 


49 


r. 92914 


3 


0. 12 


8400 


322 


no 


4-33072 


240 


9.44886 


SO 


I. 96851 


3 


0. 12 


1 0000 



Table 3. — Dimensions of Radial Ball Bearings — Heavy 
Series 



No. 
of 


Bore 


Diameter 


Width 


Corner at 
bore of 


Radial 
load, 
lbs. 


bear- 








inner 


race 


ing 


Mm. 


Ins. 


Mm. 


Ins. 


Mm. 


Ins. 


Mm. 


Ins. 


403 


17 


0.66929 


62 


2 . 44095 


17 


0.66929 


I 


0. 04 


850 


404 


20 


0. 78740 


72 


2.83465 


19 


0.74803 


2 


0.08 


1050 


405 


25 


0.98425 


80 


3.14962 


21 


0.82677 


2 


0.08 


1320 


406 


30 


1.18110 


90 


3-54332 


23 


0.90551 


2 


0.08 


1600 


407 


35 


1-37799 


100 


3.93702 


25 


0.98425 


2 


0.08 


1900 


408 


40 


I -57481 


no 


4-33072 


27 


1.06299 


2 


0.08 


2200 


409 


45 


I. 77166 


120 


4-72443 


29 


1-14173 


2 


0.08 


2500 


410 


so 


I. 96851 


130 


S-11813 


31 


1.22047 


2 


0.08 


3400 


411 


55 


2.16536 


140 


S-51183 


ii 


I. 29921 


3 


0. 12 


3900 


412 


60 


2.36221 


ISO 


5-90554 


35 


1-37795 


3 


0. 12 


4400 


413 


65 


2.55906 


160 


6.29924 


37 


1-45669 


3 


0. 12 


4900 


414 


70 


2.75591 


180 


7.08664 


42 


1-65355 


3 


0. 12 


6200 


415 


75 


2.95277 


190 


7-48035 


45 


I. 77166 


3 


0. 12 


6600 


416 


80 


3.14962 


200 


7-87405 


48 


1.88977 


3 


0. 12 


7300 


417 


85 


3-34647 


210 


8.2677s 


52 


2.04725 


3 


0. 12 


8580 


418 


90 


3 54332 


225 


8.85830 


54 


2.12599 


3 


O.X2 


1 0000 


419 


95 


3-74017 


250 


9-84256 


SS 


2.16536 


3 


0.12 


II 880 


420 


100 


3-93702 


265 


10.43311 


60 


2.36221 


3 


0.12 


14000 



Thrust bearings are usually made in three series, light, medium, and 
heavy. The heavier the bearing the larger the balls for a given 
strength. 

At the 191 1 spring meeting of the Society of Automobile Engineers, 
the standards committee rendered a report containing Tables i, 2, 
and 3, giving recommended standard dimensions for the light, 
medium, and heavy series, respectively, for radial ball bearings. 
These standards are referred to as Ball Bearings Standards A. The 
last column gives a radial load rating in pounds for each bearing. In 
this connection the committee reported: 



BALL AND ROLLER BEARINGS 



37 





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■ 1 




Fig. ,^9. 



38 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 





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t) D D t3 t3 & t3 t) t) & D t3 ID & ti Id Id D Id id D & 

Phhhhh t:cipq>i|f:q fijtxijiiptj pqpqpcipq ptjpcjpqpq PHi*! 




BALL AND ROLLER BEARINGS 



39 



"Attention is called to the fact that the capacities given in the 
tables are based upon ball bearings manufactured of suitable work- 
manship and of suitable material and running at uniform speed and 
uniform radial load, the speed not exceeding 500 r.p.m. 

"It is further suggested in explanation of the load standards that 
it cannot be expected that all conditions will be covered by the loads 
given. For conditions of shock, end thrust, and a combination of the 
two, greater factors of safety wiU have to be used." 

The dimensions and capacities of Hess-Bright thrust-collar bear- 
ings, hght and medium weight series, are given in Tables 4, 5 and 6. 

The center of R is on the center line; its location is not quite def- 
inite, but wiU be approximated by drawing the ball seat tangent to a 
comer iillet r at the base. 

The inch dimensions are the nearest equivalents to the even mm. 
in which the bearings are made. The loads cited are safe for steady 
speeds and constant loads. Consult the manufacturer regarding 
loads for speeds above 1500 r.p.m. 

The bearings shown in Fig. 40 are the same as those shown in Fig. 
39 but with balled seat washer and enclosure. For loads and speeds, 
also for dimensions of bearings themselves, consult the upper table. 

It is advisable to give the aligning washer some freedom of side- 
wise movement as shown, to permit it to assume its best position in 
case of shaft oscillation. With such freedom allowed, the shaft may 
be a snug fit in the upper race. 

If side freedom cannot be given the aligning washer, the shaft 
must be a loose fit in the upper race. 

Roller Bearings 

• The well-known roller bearings with hollow, cylindrical, helical, 
flexible rollers made by the Hyatt Roller Bearing Co. are used in 
machinery in general, on line shafting and in automobiles. The 
rollers are wound from flat strip steel into a closed helical coil. Thus 
they are flexible and can adapt themselves to slight irregularities in 
either journal or box without causing excessive pressure. The cylin- 
drical hollows in the rollers serve as storage spaces for lubricant and 
the helical interstices distribute the oil the entire length of the box. 
One half of the rollers in a box have a right-hand helix, the other half 
left hand. 

In the form of bushings, two types are made, known as the stand- 
ard type and the high-duty type. In the standard type the rollers 
are of carbon steel with an outer shell or lining of special analysis sheet 
steel. The rollers run in contact with the shaft or journal. Where 
the bearing surface is generous it is satisfactory to operate the rollers 
direct on soft-steel surfaces. 

In the second type the rollers are of special analysis chrome 
nickel heat treated steel. The lining is tubular and a tubular sleeve 
is provided to shp over the shaft or journal. Both of these parts are 
also heat treated. Several devices are used to cage the rollers, which 
are squared on the ends to thrust against the ends of the box. As 
the allowable unit-bearing pressures of the high-duty rollers are higher 
than for the standard rollers the high-duty bearings have the practical 
advantage of being shorter for the same load than the standard 
bearings. 




Fig. 38. 





MH 


00 

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ro 


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m 


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CO 


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« 


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ro 


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rf 


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vn 


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r^ 


r^ 


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tn 














































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w 


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00 


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in 




M 




t-l 






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rf 


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t^ 


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PO 


m 
























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w 


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r/i 


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pn|^ 


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a 




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ro 


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PO 


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in 


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r- 


r- 


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fe. 


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rrt« 


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n|« 


a 


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p^ 


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ro 


PO 


PO 


ro 


ro ■* 


■<t 


^ 


m 


^ 






r- 


r- 


r~ 





SO 


-0 


10 


in 


m 


■^ 


rt rf 


ro 


fO 


PO 


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NO 


NO 


NO in 


m 


■^ 


ro 




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<:f 


00 







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00 







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00 







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OC 


N 


NO 





NO 


H 


Kl 











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M 


H 


M 


M 


N 


N 


N (M 


N 


PO 


PO 


ro 


PO 


PO 


■^ -^ 


Tj- 


in 


in 





S 


<M 


r- 


N 


t^ 


N 


r^ 


M 


r^ 


N 


^ 


N r* 


N 


r^ 


^ 


00 


ro 


00 


PO CO 


00 


00 


ro 


M 










N 


PO 


ro 


Tt- 


'+ 


m 


m 





r^ 


r- 


X) 


w 


o\ 


Ch 









Tf 


rt 




h 






































M 


M 




1 




















































t^ 


H 


o> 


NO 


on 





vn 


a> 


0\ 


<i PO 


ro 


vO 


n 


NO 


NO 


r- 


i> 10 


^ 


r^ 


ro 


W 




1 





W 






p^ 


•* 


-t 


r* 


r* 





w 


w 


PO 


NO 


t^ 


t^ 


a 


Ov ro 




NO 





Q 


H 


M 


„ 


M 


M 


M 


M 


M 


M 


N 


W N 


M 


M 


r<i 


(Nq 


« 


M 


pq ro 


ro 


PO 


■^ 


M 




H 


W 


w 




^ 






■^ 


•'t 


■^ 


^ SO 


«o 










■^ 


■* 




PJ 


't 







b' 


t^ 


CO 





W 


M 


r^ 


r^ 


in 


m 


N 


N 


NO 





vO 








m 


m in 





ro 


IN 




e 


M 


<N 


ro 


fO 


ro 


ro 


ro 


** 


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m 


10 m 


m 








r- 


t^ 


t^ 


t^ 00 


Ov 


U 


M 


m 








00 


\n 




r- 


Cl 





m 


in 


»+ 


« « 


PO 


ro 


in 





N 


IH 


M 00 


r- 


Ov 


>-, 





































PO 


r* 


M 









'A 




n 















































(1 




M 


M 


H 


M 


N 


M 


N 


M 


PO 


ro 


PO -'t 


•«*■ 


■* 


4 


in 


in 


in 


m NO 


NO 


r* 


00 


















































6 







(S) 




in 


^ 


in 


n 





N M 


in 





x^ 


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in 


m 


r- 


r^ 





NO 










■^ 




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a 


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ro 


•^ 


in >n 


NO 


» 





w 




a 
























w 


H 


M 


H 


M 


H 


M W 


H 


M 


P4 


M 


















































00 


no 


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<0 


Ch 


4 


« 


p^ 


r- 


vD 


-^ r^ 





ro 


PO 


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M 


M 








t- 


P 




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l-l 


ro 





00 







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-rt- 


in 





PO 


in 


0\ 


PO 


m 


--I 




00 


00 


1 


cq 


M 


M 


M 


H 


N 


N 


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p^ 


PO 


PO 


ro PO 


-* 


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■^ 


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in 


m 


in NO 


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r* 
















































« 




e 










ro 


n 


>+ 


m 


f)n 


10 





ro 





in 


in 


in 





in 


m 


m 













■^ 


in 


MO 


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r^ 




ifi 


a 









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PO 


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in m 


NO 


r- 





5 




e 






















*^ 




M 


•"^ 


M 


*~* 


•"" 


M H 


*" 


M 


Pi 






00 


r- 


00 


NO 


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ro 


H, 


n 


00 


r- 


in Tt 


C4 


|_l 


0^ 


00 


NO 


in 


PO PI 





r^ 


'^ 






■0 


ro 







r- 


M- 




JO 


^ 








0^ 


in 




OS 


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ro 





Tt 






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<-^^ 




o> 


00 


X) 


» 


r^ 


r- 


r- 


NO 


NO 


in 






■* 


■^ 


^ M- 




ro 




H 








ro 


w 


10 


t^ 


o\ 




PO 


m 


r- 


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ro 


in 


r- 


Ov 




ro 


in r^ 


'-' 


ro 


t^ 




^ 






















w 


M 


M 


M 


M N 


N 


N 


N 


CNl 


PO 


ro 


ro PO 


Tj- 


^ 


"t 




s 


W) 


n 


ro 


V) 


n 


in 


n 


10 





in 


in 





m 





in 





in 


m 


m 
























ro 


ro 


^ 


■* 


in m 


■0 







r^ 


00 


» 


C> Ov 













s 






































■^ 


'"' 


t-i 






D 


N 


ro 


■^ 


in 


vO 


f^ 


DO 


0\ 


r> 


„ 


N r^ 


■+ 


in 





r^ 


00 





^ 


ro 


m 


00 












































« 


« 






^ 


















M 


i_i 


y^ l-l 


i_i 


M 


M 


i_i 


M 


M 


•-< M 


l-l 


M 


M 




W 




(N 


N 


w 


<M 


<N 


N 


N 


w 


N 


N 


« (S 


« 


N 


« 


M 


N 


N 


N N 


Pi 


«N 


« 



40 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



The shafting bearing is of the standard type with a horizontally 
split box so that it can be put on a shaft anywhere and does not have 
to be shpped on over the end. 

The allowable pressures for the standard or commercial type are 
fixed by the quahty of the shaft or journal against which the rollers 
bear. For low speeds up to 50 r.p.m. the maximum limit lies be- 
tween 400 to 500 lbs. per sq. in. of projected journal area. The 
method of housing, particularly in its relation to the distribution of 
load, and quality of lubrication have an influence in determining 
these limits. 

With an increase of speed the allowable pressure decreases. For 
line-shaft bearings up to 3f| ins. dia. running up to 600 r.p.m., 30 
lbs. per sq. in. is considered good practice. For larger shafts the 
same factor is allowable up to speeds of 400 r.p.m. 

The high-duty bearings carry much greater unit loads. A rating 
of 750 lbs. per sq. in. at icoo r.p.m. is conservative. 

A limiting maximum speed for the standard or commercial bear- 
ings is about 1500 r.p.m.; for the high-duty bearings, 3000 r.p.m. 

Table 7 gives dimensions of the high duty type of Hyatt bearings 
with load carrying capacities at 1000 r.p.m. At 1500 r.p.m. the loads 
should be reduced by one-half and at 500 r.p.m. or less they may be 
increased in like proportion. The data are the results of extended 
tests and are very conservative. When the inner race is omitted 
these loads require the use of a heat treated or hardened shaft. 
When the inner race is used the shaft may be of soft steel. When 
the rollers bear directly on the shaft, cold rolled material affords a 
better operating surface than the same material machined, and 
should be used whenever possible. A high carbon steel such as 
.40 to .50 is preferable to low carbon and can be obtained commerci- 
ally at very little excess in price over the latter. A high carbon or 
alloy steel properly heat treated is preferable to either of the above. 
A shaft properly carburized, hardened and ground, gives the best 
possible bearing surface. 

Overhung loads should be avoided as the deflection of the shafts 
leads to concentration of the loads on one end of the bearings. When 
unavoidable, a second bearing should be placed a reasonable dis- 
tance from the first. 

Roller bearings should always be lubricated but never with solid 
lubricant which clogs the rollers and prevents their free rotation. 



Table 7.— Dimensions and Capacities of Hyatt Roller Bear- 
ings, High Duty Type 
Without Inner Race 



Diameter 


Outside 

diameter 

over all, 

ins. 


Short series 


Long series 


of shaft or 
axle, 
ins. 


Length 

over all, 

ins. 


Safe load, 
lbs. 


Length 

over all, 

ins. 


Safe load, 
lbs. 


1 .000 

1.I2S 

1.250 
1.375 
1 .500 


2.249 
2.374 
2.749 
2.874 
3-374 


1. 000 
1. 000 

1 .125 
I -125 
I -250 


460 
500 
700 
750 
960 


2.000 
2 .000 
2.250 
2.250 
2.500 


1200 
1340 
1700 
1900 
2340 


1 .625 
1.750 
1.87s 
2.000 
2.125 


3-499 
3.624 
3-749 
4.124 
4.249 


1 .250 
1 .250 

I -250 
1.375 

1-375 


1040 
1125 
1200 
1470 
1550 


2.500 
2.500 
2.500 
2.750 
2.750 


2530 
2730 
2925 
3490 
3700 


2.250 
2.500 
2.750 
3.000 

3-250 


4-374 
4-749 
4-999 
5-374 
5.624 


1.375 
I -500 
1.500 
1-750 
1-750 


1650 
2060 
2270 
3030 
3400 


2.750 
3 000 
3.000 
3.500 
3. 500 


3925 
4820 
5300 
6890 
7600 







With Inner Race 






Inside 


Outside 
diameter 
over all. 


Short 


series 


Long 


series 


diameter 
of inner 


Length 
over all, 


Safe load, 
lbs. 


Length 
over all. 


Safe load, 
lbs. 


raceway, ins. 


ins. 


ins. 


ins. 


-750 


2.249 


1. 000 


460 


2.000 


1200 


.875 


2.374 


1. 000 


500 


2.000 


1340 


I. 000 


2.874 


1. 125 


750 


2.250 


1900 


I. 125 


3.374 


1.250 


960 


2.500 


2340 


1.250 


3.499 


1 .250 


1040 


2.500 


2530 


1. 375 


3.624 


1 .250 


II2S 


2.500 


2730 


1. 500 


3-749 


1 .250 


1200 


2.500 


2925 


1.62s 


4.124 


1.37S 


1470 


2.750 


3490 


1-750 


4.249 


1.375 


1550 


2-750 


3700 


I -87s 


4.374 


1.375 


1650 


2.750 


392s 


2.000 


4-749 


1 .500 


2060 


3 .000 


4825 


2.250 


4-999 


I -500 


2270 


3-000 


5300 


2.500 


5.374 


1-750 


3030 


3-Soo 


6890 


2.750 


5.624 


1-750 


3400 


3-500 


7600 



Table 8. — Dimensions and Capacities of Norma Roller Bearings 
Light Service Series 



A 
Inside diam. 


B 

Outside diam. 


c 

Width 


r 






Loac 


, lbs. at r 


p.m. 






Mm. 


Ins. 


Mm. 


Ins. 


Mm. 


Ins. 


Mm. 


Ins. 


10 


100 


300 


500 


1,000 


1,500 


2,000 


30 


1.1811 


62 


2.4409 


16 


0.63 


I 


0.04 


1.430 


1,320 


1,170 


1,030 


790 


660 


610 


35 


1.3780 


72 


2.8346 


17 


0.67 


I 


,04 


2,090 


1,910 


1,700 


1,500 


1,170 


990 


930 


40 


1.5748 


80 


3-1496 


18 


0.71 


2 


0.08 


2,375 


2,200 


1,940 


1,720 


1,320 


1,120 


1,050 


45 


1. 7717 


85 


3-3465 


19 


0.75 


2 


0.08 


2,530 


2,330 


2,100 


1,830 


1.410 


1,190 


1,120 


50 


1.9685 


90 


3-5433 


20 


0.79 


2 


0.08 


2,680 


2,460 


2,180 


1,940 


1,500 


1,280 


1,190 


55 


2-1654 


100 


3-9370 


21 


0.83 


2 


0.08 


3,850 


3.520 


3,080 


2,800 


2,150 


1,830 


1,710 


60 


2 .3622 


no 


4-3307 


22 


0.87 


2 


0.08 


4.510 


4,180 


3,630 


3.300 


2,530 


2,130 


1,980 


65 


2. 5591 


120 


4-7244 


23 


.90 


2 


0.08 


5. 280 


4,840 


4,270 


3.740 


2,950 


2,480 


2,310 


70 


2.7559 


125 


4-9213 


24 


0.94 


2 


0.08 


5,500 


5,060 


4,400 


3.960 


3.080 


2,640 


2,420 


75 


2.9528 


130 


5.I181 


25 


0.98 


2 


0.12 


5,830 


5,390 


4,730 


4.250 


3.300 


2,750 


2,530 


80 


3-1496 


140 


5-5118 


26 


1.02 


3 


. 12 


6,710 


6,160 


5,440 


4.840 


3.740 


3,190 


2,970 


85 


3-3465 


150 


S-9055 


28 


1 .10 


3 


0. 12 


8,030 


7,260 


6,600 


5,720 


4.510 


3,740 




90 


3-5433 


160 


6.2992 


30 


1. 18 


3 


0. 13 


9,680 


8,800 


7.810 


7,040 


5,390 


4.5 10 




95 


3-7402 


170 


6-6929 


32 


1.26 


3 


0.12 


11,440 


10,340 


9,240 


8,140 


6,380 


5.280 




100 


3-9370 


180 


7 -0866 


34 


1.34 


3 


. 12 


12,980 


11,880 


10,340 


9,240 


7,260 


5,940 




125 


4-9213 


225 


8-8583 


40 


1-57 


3 


0.12 


15,000 


12,700 


11,550 


10,400 


7,400 






130 


S-I181 


230 


9-0551 


40 


1-57 


3 


0.12 


15,800 


13,350 


12,150 


10,900 


7,780 






140 


S-5118 


250 


9.8425 


42 


1.65 


3 


0.12 


20,400 


17,200 


15,650 


14,100 


9.950 






155 


6- 1024 


280 


II .0236 


44 


1-73 


3 


0.12 


24,700 


20,900 


19,000 


17,100 


12,100 






175 


6-8898 


315 


12.401S 


50 


1.97 


3 


0. 12 


32,340 


27,390 


24,860 


22,385 


15.950 






180 


7.0866 


320 


12.5984 


50 


1-97 


3 


0.12 


33,700 


28,550 


26,000 


23,300 


16,600 






190 


7-4803 


340 


13.3858 


54 


2.13 


3 


0. 12 


36,400 


30,600 


28,000 


25,200 


17,800 






200 


7.8740 


360 


14.1732 


54 


2.13 


3 


. 12 


41,500 


35,000 


32,000 


28,700 


20,350 






215 


8 .4646 


385 


15.1574 


56 


2 .20 


3 


0. 12 


44,600 


37,900 


34,300 


31,000 


22,000 







BALL AND ROLLER BEARINGS 



41 



Table 8. — Dimensions and Capacities of Norma Roller Bearings (Continued) 

Medium Service Series 



A 
Inside diam. 


B 
Outside diam. 


c 

Width 


r 






Load, lbs. at r 


p.m. 






Mm. 


Ins. 


Mm. 


Ins. 


Mm. 


Ins. 


Mm. 


Ins. 


10 


100 


300 


500 


1,000 


1.500 


2,000 


2S 


0.9842 


62 


2.4410 


17 


0.67 


I 


0.04 


1,650 


1,520 


1,320 


1,190 


925 


770 


725 


30 


1.1811 


72 


2.8346 


19 


0.75 


2 


0.08 


2,150 


1,980 


1,760 


1.540 


1,210 


1,000 


950 


3S 


1-3779 


80 


3.1496 


21 


0.83 


2 


0.08 


2,750 


2,580 


2,270 


1,980 


1,540 


1.320 


1,100 


40 


1.5748 


90 


3-5433 


23 


0.91 


2 


0.08 


3,520 


3.190 


2,860 


2,530 


1,980 


1.650 


1.430 


45 


1. 7716 


100 


3.9370 


25 


0.98 


2 


0.08 


4,400 


4,020 


3.520 


3,190 


2,420 


1,980 


1,760 


50 


1.968s 


no 


43307 


27 


1 .06 


2 


0.08 


5,280 


4,840 


4,180 


3.740 


2,860 


2,420 


2,200 


55 


2.1653 


120 


4-7244 


29 


1. 14 


2 


0.08 


6,600 


6,160 


5,390 


4.840 


3.740 


3,080 


2,860 


6o 


2 .3622 


130 


S-I181 


31 


I .22 


2 


0.08 


7,700 


7,150 


6,270 


5.610 


4,400 


3.630 


3.300 


65 


2.5590 


140 


5-5II8 


33 


1.30 


3 


0. 12 


8.360 


7,700 


6,820 


5.940 


4,620 


3.850 


3.520 


70 


2.7559 


150 


5.90SS 


35 


1-38 


3 


0.12 


10,120 


9,240 


8,340 


7,260 


5,610 


4,620 


4,180 


75 


2.9527 


160 


6.2992 


37 


I -46 


3 


0. 12 


11,880 


11,000 


9,680 


8,580 


6,600 


5.500 


4,840 


8o 


3 . 1496 


170 


6-6929 


39 


1-54 


3 


0. 12 


12,980 


11,880 


10,560 


9,460 


7,260 


5.800 




85 


3.3464 


180 


7- 0866 


41 


1. 61 


3 


0.12 


14,080 


12,980 


11,440 


10,120 


7.700 


6,160 




90 


3.5433 


190 


7-4803 


43 


I .69 


3 


. 12 


16,280 


14,960 


13.200 


11,660 


9,020 






95 


3-7401 


200 


7-8740 


45 


1-77 


3 


0.12 


17,380 


16,060 


14,080 


12,540 


9,680 






100 


3-9370 


21S 


8 -464s 


47 


1-85 


3 


0.12 


21,120 


19,360 


16,940 


14,960 


11,440 






130 


S.1181 


270 


10.6299 


55 


2.17 


3 


0.12 


29,800 


25,300 


22,900 


20,600 


14,600 






140 


5.5118 


290 


11-4173 


57 


2.24 


3 


0.12 


33,100 


28,000 


25,500 


23,000 


16,300 






150 


5.9055 


310 


12 -2047 


60 


2.36 


3 


0.12 


37,800 


32,000 


29,000 


26,200 


18,600 






155 


6. 1024 


325 


12-7953 


60 


2.36 


3 


0. 12 


40,500 


34.200 


31,000 


28,000 


19.950 






16s 


6.4960 


330 


12.9921 


60 


2.36 


3 


0. 12 


42,900 


36,300 


33.000 


29,700 


21,120 






170 


6.6929 


350 


13. 7795 


62 


2.44 


3 


0. 12 


47,300 


40,040 


36,300 


32,670 


23,320 






180 


7.0866 


370 


14.5669 


62 


2.44 


3 


0. 12 


50,200 


42,400 


38,600 


34.700 


24,650 






190 


7-4803 


390 


15.3543 


66 


2.60 


3 


0.12 


54,600 


46,200 


41,900 


37.800 


26,900 






200 


7-8740 


410 


16. 1417 


70 


2.76 


3 


0. 12 


57,500 


48,700 


44.300 


39.900 


28,300 






215 


8 . 4646 


450 


17. 716s 


75 


2.95 


3 


0. 12 


62,300 


54.600 


49.600 


44,600 


31.700 







Heavy Service Series 



A 
Inside diam. 


B 

Outside diam. 


c 

Width 


r 






Load, lbs. at r 


.p.m. 






Mm. 


Ins. 


Mm. 


Ins. 


Mm. 


Ins. 


Mm. 


Ins. 


10 


100 


.300 


500 


1,000 


1,500 


2,000 


25 


0.9842 


80 


3-1496 


21 


0.83 


2 


0.08 


3.410 


3.080 


2,750 


2,420 


1,910 


1,630 


1,540 


30 


1.1811 


90 


3 - 5433 


23 


0.91 


2 


0.08 


3.850 


3.S20 


3,080 


2,750 


2,200 


1,830 


1,710 


35 


r.3779 


100 


3.9370 


25 


0.98 


2 


0.08 


4,400 


3.960 


3,520 


3.080 


2,420 


2,050 


1,920 


40 


1.5748 


no 


4-3307 


27 


1.06 


2 


0.08 


5. 940 


S.SOO 


4,840 


4.400 


3,300 


2,860 


2,640 


45 


1. 7716 


120 


4-7244 


29 


1. 14 


2 


0.08 


7,260 


6,600 


5.940 


5. 280 


3.960 


3.300 


3.080 


50 


1.968s 


130 


5.1181 


31 


1 .22 


2 


0.08 


8,580 


7,920 


7.040 


6,160 


4,840 


3.960 


3.740 


S5 


2.1653 


140 


5.5118 


33 


1.30 


3 


0. 12 


9,460 


8,800 


7,700 


6,600 


5,280 


4.400 


4,180 


60 


2.3622 


ISO 


5 -9055 


35 


1.38 


3 


0. 12 


11,220 


10,340 


9,020 


7,920 


6,160 


5. 280 


4,840 


6S 


2.5S90 


160 


6.2992 


37 


1-45 


3 


0. 12 


12,100 


11,000 


9,680 


8,580 


6,600 


5.720 


S.280 


70 


2.7559 


180 


7.0866 


42 


1.65 


3 


. 12 


16,060 


14,740 


13,200 


n,66o 


9,020 


7.480 


7,000 


75 


2.9527 


190 


7.4803 


45 


1.77 


3 


0. 12 


18,040 


16,720 


14.520 


12,980 


10,010 


8,470 




80 


3 . 1496 


200 


7.874 


48 


1.88 


3 


0.12 


18,260 


16,940 


14.740 


13.200 


10,120 


8,580 




85 


3.3.164 


215 


8.464s 


SI 


2 .00 


3 


. 12 


19.140 


17,600 


15,400 


13.860 


10,780 


9,020 




90 


3-5433 


22s 


8.8582 


54 


2.12 


3 


0, 12 


23.320 


21,340 


18,700 


16,720 


12,980 






95 


3.7401 


24s 


9.6456 


57 


2.24 


3 


. 12 


27,280 


25,300 


22,440 


20,020 


15,840 






100 


3.9370 


265 


10.4330 


60 


2.36 


3 


. 12 


37,400 


34.5^0 


29,700 


26,840 


21,120 






130 


S.1181 


370 


14.5669 


75 


2.95 


3 


0. 12 


56,400 


47.800 


43.500 


39.100 


28,200 






140 


S.5118 


390 


15.3543 


80 


3.15 


3 


. 12 


61,800 


52,400 


47,600 


42,900 


30,400 






155 


6.1024 


410 


16. 1417 


80 


3.15 


3 


0. 12 


67,000 


56,800 


51.600 


46,500 


33,000 






180 


7.0866 


440 


17.3228 


80 


3. IS 


3 


0. 12 


77,100 


65,600 


59.500 


53,500 


38,100 






190 


7.4803 . 


440 


17.3228 


80 


315 


3 


0. 12 


77,100 


65,600 


59,500 


53.500 


38,100 






200 


7.8740 


46s 


18 .3071 


80 


315 


3 


0. 12 


82,700 


70,000 


63.500 


57.300 


40,650 






215 


8.4646 


46s 


18.3071 


80 


3. IS 


3 


0. 12 


88,200 


74.500 


67,500 


60,800 


43.300 







The Norma roller bearing, Fig. 39, is intended for conditions in- 
volving a degree of shock, jar and vibration too severe for ball bear- 
ings. They are for radial loads only. The inner race is ground to 
a true cylindrical surface and the outer one to a slightly convex sur- 
face, the roller having point contact with the outer and line contact 
with the inner race — a construction that permits and compensates 
for slight angular displacements of the ball. Table 8 gives dimen- 
sions and loads of these bearings, the reference letters being those of 
Fig. 39. The inner and outer diameters and the widths will be seen 
to conform to those of Tables i, 2 and 3 while the loads are much 



heavier. The ratings are for steady loads and speeds, although 
50 per cent, temporary overload capacity may be added. 




1< ^^ ^ 

Fig. 39. — The Norma roller bearing. 



42 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



The formula for the capacity of solid roller journal bearings used 
by the Standard Roller Bearing Co. is as follows: 

P= i^o.ooo 





mm ' ■ 



Fig. 41. 
Figs. 40 and 41. — Conical roller bearings. 

in which P = load on bearing, lbs., 

<f = diameter of rollers, ins., 

« = number of rollers, 

/ = length of each roller, ins., 

s = circumferential speed of each roller, ft. per min. 
Conical roller journal bearings have had 
their most extensive use in automobile prac- 
tice. Figs. 40 and 41 are from designs of 
the Standard Roller Bearing Company. 
Fig. 40 shows two constructions for auto- 
mobile hubs having two single rows of rollers, 
and Fig. 41 shows in section a double taper 
roller bearing. 

The load-carrying capacity is given by 
the following formula, from the practice of 
the Standard Roller Bearing Co.: 

d-^nl 
P= 130,000 — 

n which P = load on bearing, lbs., 

d = mid-diameter of the roUers, ins., 
« = number of rollers, 
/ = contact length of a roller with 

its bearing washer, ins., 
s = mean circumferential speed of 
each roller, ft. per min. 
Roller bearings have been used with con- 
spicuous success as thrust bearings under 
enormous loads, a bearing of this type for 
a speed of 100 r.p.m. and a load of 2,000,000 
lbs. (subsequently increased to 2,250,000 
lbs.) by the Standard Roller Bearing Co., 
being shown in Fig. 42 {Amer. MacL, July 14, 1910). 

The rollers are cyHndrical and in short sections — a feature which 
reduces the differential sHding to an amount where it does no harm. 
The bearings consist of three elements: 



Treads, consisting of two heat-treated, tempered and accurately 
ground steel washers or plates. 

Roll cage, usually of bronze,complete with rolls of steel,heat-treated, 
tempered and accurately ground; retaining band, ball thrust, etc. 

Leveling device, consisting of two washers or plates, one face of 
each being convexed and concaved respectively, thus providing a ball- 
and-socket base for the bearing and insuring an equal distribution of 
the load, even though all parts adjacent be not in exact alignment. 

The general dimensions of the cage are 2 ft. 7 ins. inside diameter 
by 6 ft. 55 ins. outside diameter. Details of construction are given in 
Fig. 42. The bearings shown are inclosed in the casing of the ma- 
chine, all voids around bearings being packed with grease, and large 
compression grease cups provided to supply lubricant as consumed. 

The after, or right-hand tread washer, was required to be extra 
thick, to avoid deflection, as it is supported at its outer edge only. 
The forward, or left-hand tread washer, is amply supported over its 
entire face. The plates are made of high-grade, special alloy steel, 
forged into form as washers. They are bored, turned and faced to 
approximate dimensions, allowances being made for grinding. The 
washers are then subjected to heat treatment, are hardened, drawn, 
and then carefully ground to very close limits, the two faces being 
as nearly parallel as human ingenuity and highest-grade grinding 
machinery can produce, each face being in itself perfectly level and 
parallel. 

The thrust cages are of phosphor bronze, carefully machined, 
special care being exercised in the location of the slots which carry the 
rollers. The rollers are manufactured from a high-grade, special 
alloy steel, carefully heat treated, and all rollers are ground true 
cylinders, the error in parallelity for a given roll and diameter of all 
rolls in the bearing being held within .0002 in., plus or minus, of the 
nominal diameter. Heavy steel retaining bands are provided which 
encircle the bronze cage and retain the rollers in their respective slots, 
a steel ball being provided at the end of each roll slot to care for the 
end thrust of the rolls in the slot, due to centrifugal force and the 
force generated by reason of the rolls being guided by the cage in a 
circular path, instead of their natural tangential path. 



Oil Holes 



Drill for Cotter Pin 

when Ass'mbl'd 




Fig. 42. — Roller thrust bearing carrying a load of 2,225,000 lbs. at 100 r.p.m. 

Large numbers of such bearings are in use, other notable examples 
by the same constructors being at the Niagara Falls power house, 
where they sustain a normal load of 156,000 lbs. (extreme load 190,000 
lbs.) at a speed of 250 r.p.m. 



SHAFTS AND KEYS 



For shaft couplings, including the Hooke universal coupling, see 
Index. 

The strength of shafts subject to simple torsion may be determined 
from the formula: 

in which Af = twisting moment, Ib.-ins., 

rf = diameter of shaft, ins., • 

5 = fiber stress at outer fiber, lbs. per sq. in. 

Suitably transformed this becomes: 



-^ 



3210CO h.p. 



5 r.p.m. 

Experience shows that for simple torsion and for short counter- 

321000 
shafts a suitable value of — ^ — is 75, giving for this condition: 



4 



3 7SXh.p., 



r.p.m. 



More exact calculations of cases involving combined bending and 
torsion, when justified by the known data, may be made by the 
formula: 

Me =M6 + \/m7+m7 

in which Jlfe = equivalent torsion moment. 
Ml, = applied bending moment, 
Mt = appUed twisting moment. 

The results given by all the above formulas may be obtained 
graphically by the use of Fig. i, by Prof. J. H. Barr {Avter. 
Mack, June 11, igo8) which can be used without numerical compu- 
tations for determining the following factors: (i) The diameter of a 
shaft for given moments and intensity of fiber stress. (2) The 
intensity of the stress in a given shaft under known moments. (3) 
The moment in a given shaft corresponding to any intensity of stress. 

As plotted it covers all possible moments and shaft diameters, and 
all intensities of stress up to and including 15,000 lbs. per sq. in. 



-5 







] 




2 


3 


simple 
4 


Twisting Moment or Equivalent Twisting M&ment 
Scale A 
5 6 7 8 9 


10 


11 


12 


13 









^ 


^^ 




/ 
/ 

/ 


\\ 


















/ 






















1 




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/ 
























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^ 








V 


/ 
/ 
/ 






















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K^ 




N 


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c^ 


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21'ers 


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s 








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Yd 


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e 




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K 






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7Ki(5 

10 I 

11 ?i 



12 



13 



14 3 

15 5 



Fig. I. — Diameters of shafts under bending and twisting moments. 



which corresponds to a value of 4280 lbs. per sq. in. for 5. ' 

For the combined bending and torsion of line shafts with hangers 
about 8 ft. apart, a rough and ready allowance for the bending action 
is made by making the formula read: 



3 lico 

^\-r: 



Xh.p. 



p.m. 

Similarly, for the more concentrated bending action of first movers 
or jack shafts, the formula becomes: 



d 



=^ 



25 Xh.p. 



r.p.m. 



43 



(a) To find the required diameter of a shaft for a given twisting 
moment and a given intensity of stress: Scale A represents the twist- 
ing moment in pound inches. Each numbered division may repre- 
sent 100, 1000 or 10,000 Ib.-ins., as the nature of the problem demands. 
Scale B represents the intensity of the fiber stress, each numbered 
division representing 1000 lbs. per sq. in. Locate points on the scales 
A and B corresponding with the quantities given in the problem. 
From these points erect perpendiculars to the scg,les and find their 
point of intersection. If this point falls on one of the diagonals, the 
shaft diameter required will be the smallest quantity given on the 



44 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



diagonal if each numbered division on the ^ -scale has been taken as 
loo lb. -ins. of twisting moment. If each division has been taken to 
represent looo lb. -ins. of moment, the shaft diameter will be the 
intermediate number on the diagonal. Similarly, if each division 
has been taken to represent 10,000 lb-ins. of moment, the shaft diam- 
eter will be the largest quantity given on the diagonal. If the point 
of intersection of the perpendiculars does not fall on a diagonal 
the values for a diagonal through that point can be obtained by 
interpolation. 

{b) To find the intensity of stress for a shaft of given diameter 
and acted upon by a given twisting moment: Find the diagonal 
corresponding to the shaft diameter or interpolate for its position. 
Determine the point on the A -scale representing the twisting moment, 
letting each numbered division represent 100 Ib.-ins. of moment if 
the smallest quantity on the diagonal represents the diameter given, 
or 1000 Ib.-ins. if the diameter is represented by the intermediate 
quantity on the diagonal, or 10,000 Ib.-ins. if the diameter is repre- 
sented by the largest number on the diagonal. Trace a horizontal 
from this point on the j4 -scale until it intersects the diagonal repre- 
senting the diameter. From this point of intersection trace a vertical 
until it intersects the £-scale. This last point of intersection will 
represent the intensity of stress required. 

(c) To find the twisting moment which will produce a given inten- 
sity of stress in a shaft of a given diameter: Locate on scale B the 
point representing the intensity of stress which is given and drop a 
perpendicular until it intersects the diagonal representing the diam- 
eter of the shaft. If necessary interpolate for the position of this diag- 
onal. From this point of intersection trace a horizontal until it 
intersects the ^ -scale. The point thus found represents the twisting 
moment required. Each numbered division equals 100 Ib.-ins. if 
the smallest quantity on the diagonal represents the shaft diameter, 
or 1000 Ib.-ins. if the intermediate value on the diagonal represents 
the shaft diameter, or 10,000 Ib.-ins. if the largest value on the diag- 
onal represents the shaft diameter. 

(d) To find the required diameter of a shaft for a given bending 
moment and a given intensity of fiber stress: Multiply the given 
bending moment by 2 and proceed as under (a). 

(e) To find the intensity of stress in a shaft of given diameter 
acted upon by a given bending moment: Multiply the given bending 
moment by 2 and proceed as under (b). 

(/) To find the bending moment in a shaft of given diameter and 
under a given intensity of stress: Solve as under (c) and divide the 
moment thus found by 2; the quotient will be the bending moment 
required. 

(g) To find the diameter of a shaft required for a given combined 
twisting and bending moment and with a given intensity of fiber 
stress: Lay out the value of the bending moment on scale A as out- 
lined for the twisting moment under (o). Similarly, lay out the twist- 
ing moment on scale C, using the same value for each scale division as 
was used for each division of scale A . Set a pair of dividers as shown 
on the chart to the length between these two points. Add this length, 
to which the dividers have been set, to the length previously plotted 
on the yl -scale. The point thus found will represent the value of the 
combined twisting and bending moment or the equivalent twisting 
moment. Use this point in the same way as the point representing 
the twisting moment was used under (a) and solve for the shaft 
diameter. 

(h) To find the intensity of stress for a shaft of given diameter, 
acted upon by given combined twisting and bending moments: 
Find the equivalent twisting moment as outlined under (g), then solve 
for the intensity of stress as indicated under (&). 

(i) To find the equivalent twisting moment for a shaft acted upon 
by a combined twisting and bending moment, having given the diam- 
eter of the shaft and the intensity of the fiber stress: Use the method 
outUned under (c) for a simple twisting moment; the result will be 
the equivalent twisting moment. For a given equivalent twisting 
moment there will be an indefinite number of sets of values for the 



simple twisting and bending moments. If either bending or twisting 
moment is known, the solution can be made directly. If the ratio of 
these moments is known, their values can be found by trial with the 
dividers by using the chart. 

The range of the chart can be still further increased by giving larger 
values to each numbered scale division of scale A. If each scale 
division of scale A represents 100,000 Ib.-ins. of moment, the corre- 
sponding shaft diameter is 10 times the smallest quantity on the 
corresponding diagonal. If each scale division of scale A represents 
1,000,000 Ib.-ins. of moment, the corresponding shaft diameter is 
10 times the intermediate quantity on the corresponding diagonal, 
and so on. 



HoUow Shafts 

The diameters of hollow shafts may be obtained from the formula: 

di*-d;* 



d^ = - 



di 



in which d = diameter of solid shaft, 

iii= outside diameter of hollow shaft, 
^2 = inside diameter of hoUow shaft. 

the two shafts having the same strength. 

The diameters of hoUow shafts having given weight relations 
with solid shafts of the same strength may be determined from Fig. 2 
by Henry Hess {Amer. Mach, Sept. 3, 1896). The use of the chart 
is best shown by an example: 

Required, hollow shafts of the same strength as a solid shaft 9 ins. 
diameter. FoUow the curve, starting at 9 ins. at the left, to its 
intersection with, say, the lo-in. dia. line; then trace vertically down- 
ward to the internal diameter boundary hne, which starts from zero 
of the diameter scale, and find that it is intersected at yf ins. diameter; 
i.e., a solid shaft of 9 ins. dia. is equivalent in strength to a hollow one 
of 10 ins. with a 7|-in. hole. Similarly, a 12-in. with loj-in. hole, 
and a i6-in. shaft with a is^^-in hole, are found to be equivalents. 

Should it be desired to find the equivalent hollow shaft weighing, 
say, 50 per cent, of the 9-in. shaft, then the weight-percentage curve 
at the right, starting from 9 ins., is traced to the 50 per cent, vertical, 
which it is found to intersect at a diameter of lof -|-. Tracing the 
intersection of this diameter with the 9-in. curve at the left, down- 
ward to the internal diameter boundary, gives a hole of 8f ins.; 
i.e., a hollow shaft of lof ins. dia., with an 8f-in. hole, is the equiva- 
lent in strength of a soUd shaft of 9 ins. dia., but has only half its 
weight. 

The results given by the chart, while not always agreeing with 
those given by calculations, are in sufiSciently close accord for all 
practical purposes. 

As a matter of fact, the hoUow shaft wiU be stronger than the solid 
shaft, to which it is nominally equivalent, because the centers of 
solid shafts of any considerable size are defective and of little value, 
and it is to get rid of these defects that hollow forging process is 
resorted to. 



The Torsion of Shafts 

The torsion of shafts under a given stress may be determined from 
the formula 

.ig6 Gd* 

in which — angle of torsion in circular measure 
i=length of shaft, ins., 
ilf = torsion moment, Ib.-ins., 
G = modulus of transverse elasticity, 

= 11,000,000 for machinery steel, 
d = diameter of shaft, ins. 



SHAFTS AND KEYS 



45 



g 

c 

C 



26" n 








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50 60 70 80 90 100 

Weight Percentage 



Fig. 2. — Solid and hollow shafts of equal strength and their weight relations. 



For a torsion moment, M, of looo Ib.-ins. and a length, I, of i in., 
this becomes, when converted from circular measure to degrees, 
fl -05315 

in which Q = angle of torsion, deg. 

The results given by this formula may be obtained graphically 
from Fig. 3 by W. H. Raeburn {A mer. Mach, June 2 2 , 1 905) , in which 

,05315 
the quantity — -yi — has been calculated for diameters from i to 5 ins. 

and plotted in the chart, the use of which is best shown by an example : 
Required, the angle of torsion for the shaft shown in connection 
with the chart. The ordinate to the curve at 2J ins. diameter is 
.00207, which, multiplied by 12 — the turning moment in thousands 
of Ib.-ins. — and by 60 — the length of the shaft in ins. — gives 1.49 
deg., the angle through which the shaft will twist. 

The smaller chart for shafts between i and 2 ins. diameter is to 
a smaller scale than the one for larger shafts, but the larger chart 
can be used for shafts below 2 ins. diameter, if it be remembered 
that a shaft half the diameter of another will twist through 1 6 times 
as great as angle, 16 being the fourth power of 2. Thus these charts 
can be used for shafts either smaller or larger than those directly 
given. 

The torque capacity of solid and hollow shafts, the latter of seamless 
steel tube sizes, may be taken directly from Table i which is cal- 
culated for a unit fiber stress of 10,000 lbs. per sq. in. The capacity 
for other stresses is in direct proportion. 

The Critical Speed of Shafting 

The most careful possible balancing of rotating parts is not sufficient 
to secure quiet running at all speeds, for there exists a critical speed at 
which the remaining slight unbalance, however small, which there 
must always be, produces violent vibrations. This speed may easily 
be reached in steam turbines and other high-speed machines, and 
for such machines the shafts must be proportioned with respect to 
the critical speed, the diameter being such that the critical speed 
shall differ from the working speed by a suitable margin. As regards 
the margin, the General Electric Co. has found a value of 15 per cent, 
to give complete security from vibration in hundreds of cases. 

The working speed may be above or below the critical speed. In 
the former case there is, theoretically, danger of damage to the shaft 




Dia, 2 2)i 
Fig. 



2)4 2?i 3 "•3« ZVi i% i iH iy. iH 

3. — The torsion of machinery steel shafts. 



as the critical speed is passed, for, theoretically, it should break at 
this speed. Actually it does not do so, though for reasons that 
have not been explained. Such danger as there may be is minim- 
ized by balancing the piece as accurately as possible and by passing 



46 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table i. — Torque Capacity of Solid and Hollow Shafts, the Latter of Seamless Steel Tube Sizes 

By C. W. Spicer 

Sizes to the left of the left-hand zigzag line are diflficult to straighten except with special equipment. Sizes to the right of the right-hand zigzag line are 



more than half as heavy as are solid shafts of the same outside diameters. 
















Outside 


Tube thickness — B. W. G. and fractions (upper line); decimals of an inch (lower line) 


Solid 
shaft 


diam., 
ins. 


20 

.035 


18 

■ 049 


16 

.065 


13 

.09s 


II 

.120 


10 

.134 


.156 


^6 

.188 


H 

.250 


Me 

.313 


% 

• 375 


.500 


5i 

.625 


'A 


9.28 
IS. 2 
22.3 


JI.9 
19.7 
29-3 
41 .6 


14 -3 
24.1 

36.7 
52. 1 
70.2 
90.4 
114. 


17. 5 
30.5 


54-2 


Torque 

57.1 
84.3 


capacity, 

60.8 

90.9 

127.0 


lbs. ft. 

64.6 
97.9 
139.0 

187.5 
244,0 


154.0 
211 .0 
279.0 
356.0 

444 


300.0 
388.0 
489.0 
729.0 
1.020 


312.0 
408 .0 
518.0 
783.0 
1,112 
1,497 
1,948 


546.0 
850.0 
1,230 
1,687 
2,228 
2,853 

3,554 
4,336 


1,286 
1,792 
2,398 
3.106 
3,918 
4,820 

5,855 
6,930 

8,155 


20 .7 
40.3 


H 


47.3 

68.4 

93.5 

122 .0 

154. 8 

191 .2 
232 .0 

325.0 


69.7 


'A 


30.9 

41-3 
52.5 
66.3 

79.9 
96.2 


79.3 
109.0 

144.0 
184.0 
228.0 
280.0 
391.0 
525.0 
678.0 


IIO.O 




55.6 

71.2 

89.2 

108.6 

131 .0 


117 .0 

154.5 
198.0 

247.0 
301 .0 
427.0 
574.0 
743.0 
938.0 


165.0 




169.6 
219.0 

273.0 
336.0 
479.0 
645.0 
838.0 
1,062 

1,312 
1, 580 


235.0 
323.0 




139.5 
168. 5 
234.0 
312.0 
393.0 
492.0 
608.0 
722.0 

847. 5 
937.0 


309.0 
378.0 

545.0 
740.0 
970.0 
1,226 

1,515 
1,831 
2,180 

2,577 
2,952 


429.0 
558.0 


iM 


650.0 
897.0 
1,181 
1,516 

1,885 
2,296 

2,740 

3,233 

3,762 

4,345 

4,975 
5,610 
6,31s 
7.0S0 


885.0 


2 


435.0 
554.0 
689.0 

853.0 
1,022 

1,206 

1,406 


1,320 


2M 


1,356 

1,751 

2,195 
2,690 

3,250 
3,842 
4,46s 
5,163 
5,920 
6,740 
7,590 
8.460 


1.880 


21.2 


885 .0 

1,044 
1,257 
1,488 

1.734 


2,580 


2?i 

3 


1,151 
1,383 
1,630 
1,910 


2,458 
3,030 

3,659 
4,360 
5,095 
5,925 
6,790 
7,76s 
8,730 
9,810 


3.435 
4,460 


3M 


1,877 
2,202 


5, 680 


3'/2 

3M 


5,200 
6,140 
7,172 

8,275 
9,475 
10,76s 
14,040 


7.08s 
8.720 


4 


3,400 


10,580 


■4H 

4?4 
S 


9,450 
10,880 
12,390 
14,040 


12,660 
15,040 
17.700 
20.670 



through the critical speed as rapidly as possible, for the element of 
time seems to enter. It is also well to provide special arrangements 
to prevent undue springing of the shaft as the critical speed is passed. 

Below the critical speed the center of gravity revolves ivithout 
the bow of the shaft while above it it revolves within the bow, for 
which reason the small vibration due to the remaining unbalance is 
less above than below the critical speed. Again, above the critical 
speed the vibration is less with slender than with stout shafts. 
These principles are employed in the high-speed single stage 
De Laval steam turbines which run on long slender shafts and 
above the critical speed. 

Axel K. Pedersen, analytical expert the General Electric Co., 
has discussed this subject {Amer. Mack., May 7, 1914) as follows: 

While the importance of the critical speed is quite evident, and 
the definition easily understood, the nature and cause of this condi- 
tion and the statement that any shaft has a particular critical speed, 
no matter how carefully the balancing is done, deserves a more 
thorough explanation. 

An improper and careless balancing of the rotating parts would 
produce heavy vibrations at much lower speeds; maximum vibra- 
tions, however, first occur at the critical speed, and this has a con- 
stant-value dependent only on the shaft dimensions, and conditions 
at the supports and manner of loading, but not on the amount of 
lack of balance. 

As to the cause for the critical-speed condition, it is evident that 
even a careful and excellent static and dynamic balancing of the body 
is not mathematically perfect, and that, therefore, the center of 
gravity of the rotating body does not coincide with the center of 
rotation as fixed by the shaft, or in other words, that an unbalanced 
body, exists. 

Consequently, when running, the center of gravity rotates in a 
small circle around the center of rotation; this creates a centrifugal 
force, which, however small, produces an additional deflection of 
the shaft. The amount of this rotative deflection, which increases 
the radius of rotation, could easily be determined if the eccentricity 
of the center of gravity relative to the center of rotation were known. 



For increasing speeds, the centrifugal force, its direction rotating 
with the shaft and impressing vibrations on the bearings, increases 
in intensity. The smaller the intensity, the better the balance. 
The rotative deflection increases simultaneously, which is the same 
as an increase in the radius of rotation; hence, finally, a speed may be 
reached at which the rotative deflection becomes, theoretically, 
infinite. This is the critical-speed condition. 

It is evident that the rotative deflection is quite different from the 
initial static deflection of the shaft. A shaft with no static deflec- 
tion, as a vertical shaft, -produces a rotative deflection in exactly the 
same manner, and, consequently, critical-speed conditions are iden- 
tical for vertical and horizontal shafts. For the critical-speed 
condition, the static deflection is of no actual account. Where, 
nevertheless, the static deflection appears in critical-speed formulas, 
it is only due to transformation of the factors entering into the mathe- 
matical expressions, and merely indicates that the critical speed is 
dependent upon the stiffness of the shaft. 

While the critical speed is independent of the amount of the re- 
maining unbalance, it is not to be inferred that a low degree of balance 
is admissible. What is desired is the least possible vibration at the 
running speed and this is dependent on the remaining unbalance, 
being, as measured by the deflection of the shaft, directly propor- 
tional to the remaining eccentricity of the center of gravity of the 
revolving mass. 

When calculating the critical speed, allowance should be made for 
unavoidable inaccuracies in the assumption of the various conditions. 
Numerous factors tending to cause uncertainty exist. Thus, it is 
quite difficult to fix the point of application of the load. The condi- 
tions at the supports, whether the shaft is merely supported or partly 
fixed, inaccuracy of alignment, obliquity of the rotating parts, the 
increase in shaft-strength due to wheel hubs, are all factors of un- 
certainty which must be allowed for by assuming a proper factor of 
safety. 

Actually, if the balancing is done carefully, the shaft may be run 
at a speed very close to the critical value without any appreciable 
trouble being experienced. Under working conditions, however, 



SHAFTS AND KEYS 



47 



W 



\<--l 



I 



■^ 



Case No.l 
Overhanging Shaft 



m^- 



w 



w 



m 



■r-l- 



Case No.2 
Supported at Both Ends 

-I 



W^- 



-L- 



-^ 



Case No.3-a 



Case No.3-6 



Case No.4 




Fixed at One End Supported at The Other Fixed at Both Ends 



Wi 



U--27-— H 

1 I 



— 64.4-- 



Fig. 7 



^ 
.^ 



-75-- 



-60- 



-45 



--4w, 



^ 



W, 



w^ 



W,U --57- H 

I W,\< 39- 



■^ 



k 150 

Two Examples of the use of Chart 
Fig. 8 



->i 



190. 
200- 



250- = 



10,000 
J- 9,000 
■i,000 

r 7,000 

6,000 
5,000 



300 ^h 4,000 
350 J h 3,000 



400- 
450- 
500- 
550- 
600- 
650- 
700- 

800- 
_ 900- 

> 1,000 - 
1,100- 
1,200- 
1,300- 
1,400- 
1,500- 
1,600- 
1,700- 
1,800- 
1,900- 



p. 



-a 

ft 
m 

o 
+3 
'u 
U 

«H 

o 

m 

<i> 



2,000 



:M.500 « 

m 
fi 
O 

O 

1,000 -a 

900 § 

800 13 

700 I 

600 ^ 

• 500 g 

3 



400 ^ 
300 



200 
150 

- 100 



1,900- 
2,000- 



2,500- 

3,000- 
3,500-i 



-100 
I- 90 

80 
^ 70 

60 

50 

I- 40 

30 



4,000^ E 



'r- 20 



g 4,500-^ 

5* 5,000- 

-o" 5,500- 

|. 6,000- 

13 6,500- 

;| 7,000- 

'u 

O 

«H 8,000- 



,3 g 9,000^1 



> 10, 000- 
11,000- 
12,000- 
13,000- 
14,000- 
15,000- 
16,000- 
17,000- 
18,000- 
19,000- 



15 S 

OS 
01 

a 
o 

10 S 

9 ^ 

7 a 

o 

6 <^ 

<w 

■^ to 

cu 

4-^ 

> 



^ 3 

2 
1.5 



Fig. 



-Case numbers and critical speeds. 



lack of balance is likely to develop and troubles would surely follow ; 
therefore, it is advisable to allow quite a margin of safety, 20 per cent, 
above or below the critical value being considered satisfactory. 

The accompanying charts by Mr. Pedersen are based on an experi- 
mental theoretical formula by Professor Dunkerly which is suffi- 
ciently accurate for all practical purposes. The formula is as follows : 



I 
iV2 



Wo^ Wl^ «2^ fli^ 



+ 



in which 

iV = critical speed of a multiple loaded shaft, r.p.m., 
«o= critical speed of the unloaded shaft, 
«i, «2, W3= critical speeds of the shaft loaded with single loads 
Wi, Wi, Wi respectively, and not including the mass or 
weight of the shaft, the value na taking this into account. 

Introducing proper factors this formula may be made to read: 

N= ^^'77°_ 

in which ^0, ki, ki, etc. = critical-speed constants corresponding 
to speeds, Wo, Wi «2, etc. 

The constants hakikikz are determined by the charts, Figs. 5 and 6, 
and the final critical speed N by the chart. Fig. 4, using the sum 

^=^0-1-^1 -1-^2-1-^8-1- 

as combined constant. 



In the use of the charts the case number under which the problem 
in hand falls is first determined by consulting the diagrams in the 
upper part of Fig. 4. The critical-speed constant for the unloaded 
shaft is then determined from Fig. 5 followed by the determination 
of the constants for the various loads from Fig. 6. These constants 
are then added together and against their sum in the scale at the 
right of Fig. 4 the critical speed is read off. More in detail, the pro- 
cedure is as follows: 

1. Fmr an unloaded shaft, consult the upper diagrams of Fig. 4 for 
the case number; connect that number on the Mi axis of Fig. 5 with 
the length of the shaft between supports, thus locating a point on the 
M2 axis; connect this point with the diameter of the shaft and read the 
value of the constant k from its scale. Against this constant on the 
scale of Fig. 4 will be found the critical speed of the unloaded shaft. 

2. For a shaft with a single load, find first the constant for the un- 
loaded shaft as above. Divide the distance from the load to the 

/ 
nearest support, giving the ratio j of the upper diagrams of Fig. 4. 

Locate this value of j on the proper scale of Fig. 6; trace vertically 

to the curve for the case number and thence horizontally to the right. 
Connect the point found on the Mi axis with the load, thus locating 
a point on the if 2 axis; connect this point with the distance from 
the load to the nearest support, thus locating a point on the Mi axis; 
connect this point with the diameter and read the value of the con- 



48 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 





M^ 


M, 




i 


Case No.l- 






-300 


-I.OOO 

- 900 
E- 800 

— 700 




20- 






5-250 


— 600 

— 500 




19- 








=- 400 




18- 


. 










17- 






-200 


E. 300 




16- 






-190 


- 




15- 

14- 






-180 
-170 


^ 200 




13- 






-160 
-150 


~ 




12- 






-140 


I- 100 




11- 






-130 


- 90 

- 80 




10- 






-120 


- 70 

- 60 




9^ 






-110 


- 50 










^100 


E- 40 




8-] 
















f 90 


z. 30 




V-i 


^---.^ 




1- 80 g 


'- 20 




z 


^^' 




5- •-• 


^ 




6H 


"""'-^ 




§- 70 if 


I 










■*-> 




Case No.2- 


CO - 

C : 


^^--^ 


^/^ 


1- 60 1 


- 10 

- 9 

- 8 




"^^-~-, 


^,^'''''^ 


= 3 


— 7 j^ 




•3 1 




^"^ 


E- M 


- 6 ^ 




«H 4- 

o 


^/^"" 


^ 


nil III 

ween 


^ '1 

— 4 +j 

= 03 










— +j 


a 




u 


^^„,^ 


"*^ 


z.^ <u 


— 3 o 




o _ 


^^-^ 




- 40 ax-- 


E " 




S "; 


^^^"" 




- -S 


^ 2 




^^ - 






= s 


- 




^^-^'^25- 


• 




-30 


- 1.0 
r- .9 




^^--""'^ 






-. 


- .8 




,^-' 






- 


— .7 


Case No.3- 


^-^"^ 






I_ 


- .6 




2 J 






- 


^ .5 




1.9 r 






— 


E- •* 




1.8- 






_ 


r 




1.7- 






- 20 


I- -3 




1.6- 






— 


- 




1.5- 






— 


'- .2 




1.4- 






~ 


E 




1.3- 






- 15 


~ 




1.2- 






" 


- .10 










_ 


— .09 




1.1- 






V 


■— '08 










_ 


— .07 


Case No.4— 


1.0- 






— 


- .06 

'- .05 








_ 


=- .04 








- 10 










s- 


E. ,03 








h 9 


5 








i- 


1. .02 






^ tf 



Fig. 5. — Critical speed constants for unloaded shafts. 



stant k from its scale. For loads larger than the limit of the load 
scale — 20,000 lbs. — split the load into several parts. A load of 45,000 
lbs. may be split into three of 15,000 lbs. each, the constant for a 
load of 45,000 lbs. being equal to three times the chart constant 
for 15,000 lbs. This multiple value is then used as a combined 
value when finding the critical speed from the scale of Fig. 4. Note. 

For case No. i or the overhanging shaft, no ratio of y is found. 

For this case the point A on the M\ axis as shown on Fig. 6 is used 
directly as the starting point. 



3. For a shaft with multiple loads, find first the constant for the 
unloaded shaft as in paragraph (i). Next find the constant for 
each load separately as in paragraph (2). Finally take the sum of all 
the constants as the combined constant and find the critical speed 
from Fig. 4. 

4. Should the critical speed thus found he unsuitable, thus making 
it necessary to find a corrected diameter corresponding to a secoad 
more suitable critical speed, find the corrected diameter from the 
formula: . . IWi 

IWi 



di=di 



SHAFTS AND KEYS 



49 



Values of Ra1 
.50 .45 .40 .35 .30 


;io, f Jl^i ^2 
.25 .20 .15 .10 


20- 
19- 
18 — 
17- 
16- 
15— 
14— 
13 — 
12- 
11 — 
10 -| 

/7-i 
/J 

^- 5-E 

^ E 

W = 

•3 4^ 

t* - 

0) 

E l 

a _ 

3 3^ 

2-5-1 

2 — 
1.9 — 
1.8 — 
1.7 — 
1.6— 
1.5— 
1.4 — 
1.3 — 
1.2 — 
1.1 — 
1.0— 


/ 
/ 
/ 
/ 

/ 
/ 
/ 
/ 
/ 


—20,000 
^15,000 

^10,000 
i- 9.000 
=- 8,000 
=- 7,000 
=- 6,000 
^ 5,000 
=-4,000 

'— 3,000 
=-2,000 

I, 1,000 
t 900 
1^^800 ^ 
Sr, 700.^3 
i- ""600-^^:.-.. 
=- 500 1 

^ 400 'S 


\- 300 1 

r 4) 

E- 200 "S 
L in 

:- 100 

i- 90 
=- 80 
^ 70 
F- 60 
E" 50 
=- 40 

i- 30 

^ 20 
- 10 




r 6,000 
-5.000 
=-4,000 
L 3.000 

=-2,000 

r 500 

1- 400 
h 300 

j- 200 

L 100 

— 90Je 

-— 80 
r-^^70+; 

— 60 5- 

— 50i3 

r 40 a 
'- 306 

|- 20 

L 10 

— 9. 
E- 8. 
~ 7. 

— 6, 

— 5. 

r 4. 
r 3. 
i- 2. 

1 1.0 

^ ,9 

— .8 

— .7 
=" .6 
1" .5 

=- .3 

— .2 


-150 

:_i40 

-130 
-120 
-110 

rioo 

1-90 
\- 80 

[■ 70 
E- 60 

^ 50 „• 

E_ a 

: 

r 3 

: ^ 

-30 3 

- S 

- ^ 


^ 20 3 

- s 

I I 

-15 g 

^ a 
- 5 

r 5 

5-10 
r 9 

h 7 

h 6 

1" 5 
^4 












_:: :::::::: " "i ::: 




: — - ' 








"f — 












:: ::::__ _'_., : 




.. 1 




:.:. :::: : c :: 


^ 






t 4 




J I 




f L- _ 




..:. .;;:_: :--2 :. 








-I - .::::z;: :: : :: 




:.:_z. __ .. 




Z " : 




--'----f-- -_-- 








:::: ::z: -iz : : "": 








-I". .?::::;: _" : : 








_, i 




— z --(-- - -- 




2 - ji.::.: :;.:: 




: : _^:::::: ::::: 




.:':::::::: .:::: 




H / 




J .J 




/ 


\ I 




— 5&_ 


:::::::__:::_ :__: / 


L SS 


:Z::::::. :..::.: / 


_ _. ^.. ^y 


J.::::::::::;:::: / 


:::: :_:::: :;:;iz 


:::i : :::: / 






:::::::::::: ~.:rj'X-- - 


:::: : :: ::: :.z.: / 




Z - -_ / 


r z 


/_: i.-i / 


f 


f-- - -- - / 




z:::-! _ :__ :-.:: / 






f "z :::" "_2 


A For Case Noil 














:::::::::z:::::;3i,::: 


:::::::::::: ::::: / 


z iitz 


/ 




:::::::::::::::"':::: / 




•HI , ■ 


.- /--r -S>1 


---::. ''..<'_. / 


I — 1" — .«2" 


,'<! — / 


_ _ __ ;?' 


----;' --p^ — / 


: ; [:: a.. 


;::^r;-'f:;'_: / 






::! :::::. t:::^"::;:::: 


%'i :,'-"-''''.'- "- — :: / 


z 11 


r>S,'-: ,-^ -.: - / 






r ;; C/v 




L ^, 


:::::j':"";":" ~ " '-" / 


t -' 


— ^ ' - - - / 






Z ' ' 


;':::; : __ ::" 






c ^ -/^o^.^ 












/ >f „**if1 




^ X .> ^ 








"Z" ^ ^;"" "" 




















nl J-fl 




















:._ --Z-"::._ 








I"""; :::::: :.." 








^ 




i:?^ :::":: 





























Fig. 6. — Critical speed constants for single loads. 



in which (^1 = first diameter, ins., 

(^2 = corrected diameter, ins., 

iV^i = critical speed corresponding to diameter i\, r.p.m., 
iV2 = critical speed corresponding to diameter di, r.p.m. 

This procedure involves a slight error due to the neglect of the 
variation in the weight of the shaft. The error, however, is so 
small that it may be disregarded. It is to be understood also that 
a shaft of uniform constant diameter is to be assumed. This shaft 
should first be calculated for strength and its diameter be then 
checked for safety against the critical speed. 

The following examples illustrate the use of the charts on which 
they are solved by the dotted lines: 



Example i. — Fig. 7- — Case No. 3-a, Fig. 4. 

Diameter of shaft = 7 in. then from Fig. s yfeo = 

I 
Load W = 4000 lb., ratio y = 27 + 64.4 = 0.407 

and from Fig. 3 k\ = 

Hence combined constant k = 

and using Fig. 4, we get the critical speed A^i = 3150 r.p.m. 

Example No. 2. — Fig. 8. — Case No. 2. Fig. 4. 

Diameter of shaft di = 12 in., then from Fig. s i^o = 

4 



S6.o 



Lbs. 

I 
Load Wi = 2000 ratio ^ = .3 and from Fig. 6 ki= 32.0 

I 
Load W2 = 2SOO ratio ^ = .4 and from Fig. 6 ki= S3.0 

Load Wz = 2500 ratio 7 = • S and from Fig. 6 ^3= 60. o 

I 
Load W^ = 3000 ratio j = . 38 and from Fig. 6 ^4= 61.0 

I 
Load \Vi = 3500 ratio 7 = . 26 and from Fig. 6 ^5 = 48 . o 

Hence, combined constant k =310.0 

and the critical speed from Fig. 4 A'l = 1060 r.p.m 



Friction of Line Shafting 

The friction of line shafting fitted with plain bearings formed the 
subject of an extended series of observations comprising 188 separate 
tests by C. A. Graves, Chf. Engr. Edison Electric Illuminating Co. 
of Brooklyn {Amcr. Mach, June 5, 1913). 

The shafts were driven by electric motors and the power absorbed 
was obtained from the current readings. The hangers and loose 
pulleys on the line and counter-shafts were counted; the machines 
were stopped and the power input of the motor was measured. Next 



60 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMAN 



the belts were removed one at a time and readings were taken 
after each belt was removed until the belt connecting the motor 
and line shaft only remained. Finally, this was removed and 
the readings were taken with the motor running free. In many 
instances the belts were replaced in reverse order in order to check 
the results. 

The difference between the power observed with all belts on and 
with the motor running free gives the power absorbed by the bearings, 
subject to a slight correction due to the varying efficiency of the 
motor at various loads. The friction due to the increased load on 
the bearings when the machines are at work was not measured. It is 
believed to amount to about 20 per cent., but, in any case, it should 
be charged to the work done. 

The hangers and loose pulleys were treated in the same manner as 
equivalents, because the tests showed that the average loose pulley, 
either on a line or counter-shaft, absorbed nearly as much, and in 
some cases more, power than a hanger. 

Table 2 gives the results of the tests classified by industries, and 
in Table 3 the same data have been reclassified in accordance with 
diameters of the shafts. 

Mr. Graves also tested the shafts (eight in number) of a factory 
with a complete equipment of Hyatt roller bearings. The results of 
these tests are given in Table 4. The average of the table is .0286 h.p. 
per bearing. 

Table 2. — Friction Consumed by Line Shafting Fitted with 
Plain Bearings. Classified by Industries 



No. of 
installations 


Class of industry 


Horse-power per bearing 


tested 


Max. 


Min. 


Mean 


6 

7 

25 

5° 

100 


Stone working 

Wood working 

Clothing mfg 

Machine shops 

Various. 


•245 
.318 

• 037 

• 237 
.321 


.124 
•OIS 

.OIO 
.025 

•OIS 


.191 
.117 

.027 
.066 
.119 



Table 3. — Friction Consumed by Line Shafting Fitted with 
Plain Bearings. Classified by Size of Shaft 



No. of 


Size of 
shafts, ins. 


Average 
r.p.m. 


Horse- 


power per 


5 earing 


bearings 


Max. 


Min. 


Mean 


66 


H-i 


428 


.052 


.010 


.036 


706 


li 


382 


.079 


.016 




033 


37 


li 


42s 


.119 


.040 




062 


492 


I* 


392 


•193 


•03s 




089 


iSS 


i| 


218 


•113 


.029 




078 


409 


2 


242 


.300 


.028 




133 


21 


2\ 


264 


.321 


.124 




257 


83 


A 


243 


.300 


.085 




255 



The diameter 2 ins. is to be understood as including everything 
between lyf and 25 ins. and so for the other sizes. 



Table 4. — Horse-power Consumed by Line Shafting Fitted 
with Hyatt Roller Bearings 



Section of shaft ; . . 

Number of hangers 

Number of loose pulleys. . 
Total number of bearings. 
H.p. to drive shafting . . . . 

H.p. per bearing 

R.p.m. of shaft 

Diameter of shafting, ins. 



I 


2 


3 


4 


S 


6 


7 


22 


7 


II 


7 


7 


7 


7 


40 


24 


20 


19 


S 


8 


2 


62 


31 


31 


26 


12 


IS 


9 


I. 7 16 


.858 


•724 


.804 


.288 


• 549 


.281 


.027 


.028 


.026 


.031 


.024 


.036 


.031 


275 


300 


300 


200 


27S 


200 


240 


2 


2 


2 


lil 


lU 


lU 


lU 



22 

30 

938 

031 

180 

iH 



The power lost in friction of shafting formed the subject of an 
investigation by Prof. C. H. Benjamin {Trans. A. S. M. E., Vol. 
18). Sixteen factories were investigated, the results appearing in 
Table 5. 

Indicator cards were taken from the engine during the day, at 
intervals of about i hr., while the factory was in operation. During 
the noon hour, or after working hours at night, cards were taken from 
the same engine, when it was driving the line and countershafts only, 
no machines being in operation. Averages of these two sets of cards 
were assumed to show respectively the total horse-power and the 
friction horse-power. 

The figures in the column headed Horse-power to Drive Shafting 
include the power required to overcome the friction of the engine 
itself and that of all the shafting and coimters. If a deduction of 
10 be made from the percentages in the last column, they would 
show approximately the power required to drive shafting and counters 
alone. 

The friction of line shafting fitted with plain and ball bearings 
formed the subject of experiments by Dodge and Day, which were 
reported by Henry Hess {Trans. A. S. M. E., Vol. 31). As a result 
of the experiments, Mr. Hess concludes that: 

When the belts from line shaft to countershaft pull all in one 
direction and nearly horizontally the saving due to the substitution 
of ball bearings for plain bearings on the hne shaft may be safely 
taken as 35 per cent, of the bearing friction. 

When ball bearings are used also on the countershafts the savings 
will be correspondingly greater and may amount to 70 per cent, or 
more of the bearing friction. 

These percentages of savings are percentages of the friction work 
lost in the plain bearings; they are not percentages of the total power 
transmitted. The latter percentage will depend upon the ratio of 
the total power transmitted to that absorbed in the line and counter- 
shafts. 

The power consumed in the plain line and countershafts varies, 
as is well known, from 10 to 60 per cent, in diflerent industries and 
shops. The substitution of ball bearings for plain bearings on the 
line shaft only will thus result in savings of total power of 35X.io = 
3.5 per cent, to 3SX. 60 = 21 per cent. By using ball bearings on the 
countershafts also, the saving of total power will be from 70 X. 10 = 7 
per cent, to 70 X. 60 = 42 per cent. 

For additional information on the friction of line shafting see Index. 

Keys and Keyways 

The common driven key for securing a crank or gear to a shaft is 
commonly made with a width of one-fourth the diameter of the shaft 
up to about a 4-in. shaft, and above that somewhat narrower, say 
if ins. for a 6-in., if ins. for an 8-in., and 2j ins. for a lo-in. shaft. 
The depth should be from five-eighths to three-fourths the width. If 
the work is at all severe, the length should be not less than i| times 
the diameter of the shaft. The taper is commonly | in. per ft. 

This type of key is, however, a poorly designed thing at best; and 
under heavy duty, especially when the stresses alternate in direction, 
such keys are the source of much trouble. They seldom fail by shear- 
ing, but frequently fail from deformation due to the turning-over 
tendency of the forces to which they are subjected. Calculations of 
dimensions based on the shearing stress are, therefore, largely futile. 
There is no doubt that the success of the Woodruff key (which see 
below) is largely due to its better resistance to the forces which tend 
to turn it over. _ 

For the ends of shafts the Nordberg Mfg. Co. uses round keys, 
Figs. 9 and 10, which are much better than the customary form. The 
tendency toward deformation is absent; they are in true shear and 
are a driven fit in the direction of the shear, no one of these statements 
being true of the common construction. Moreover, with the taper 
reamer once provided, they are much cheaper than the square key. 
To overcome the tendency of the drill to crowd over into the cast-iron 



SHAFTS AND KEYS 



51 



Table 5. — Power Required to Drive Engine and Line and Countershafts Fitted with Plain Bearings 







(U 

IS 


en 




rt 


"0 


M 
« 




v 


M 


.S " 


bo • 




c 4:^ 


U-l *'"' 


CD 













03 03 


s 


g"- 


5 




/ 2|,3l 


II30 


I 4, 6 


580 


3, 3i 


1530 


2i,3 


1460 


{f^ 


II30 


3 


1065 


Ir 




I 4 


748 


f i|,if 




I 2, 3 


500 


2, 3 


900 


ii2i 


2490 


2, 6 




/2, 3 


1470 


)4 


1800 


/ 2, 2| 
\ 2^3 




674 


/ 1^2 

\3 


9881 


2i 


i6s 


3 


27s 


2 



a 

e 



a 
§ 
"0 




2^ 


3 

8 


tn 
g 


s 


a 
-3 


1-1 




:2; 


1 
3 
12; 




3 


<u 

S 
3 
I? 


<u 

E 

3 





170 
200 


"5 
68 


89 
28 


4 
6 


69 

27 






400.0 
74.0 


18 


78 


150 


46 


53 


5§ 


47 


43 


152 


38.6 


no 


142 


92 


4i 


79 


69 


80 


59-2 


190 


no 


141 


4 


96 


68 


300 


112. 


/ 180, 150 
\ 150 


114 


192 


4 


152 


123 


22s 


168.0 


/ 135, 135 
\ 135,150 


lOI 
lOI 


217 


3 


133 


250 


200 


40.4 


114 


58 


335 


3 


314 


313 


226 


74-3 


175,136 


102 


217 


3 


202 


258 


100 


47.2 


150 


274 


521 


3 


403 


454 


400 


190.0 


/ 160, 160 

\I25 


184 


484 


3 


435 


179 


350 


107 .0 


180 


180 


486 


3 


392 


428 


320 


241.0 


/ 175,160 

\ 175 


96 


131 


4 


89 


392 


140 


117. 


200 


74 


187 


3 


175 


184 


58 


91.6 


267 


19 


45 


6 


40 


53 


8 


39-2 


175 


37 


48 


4 


27 


30 




8-3 



so 

a 

"(3 


to 


0) 




2 


B 


^ 


cJ 


M 


fi^ 


157-0 


39-2 


57-0 


77.0 


25-3 


65.6 


47-9 


80.7 


64.0 


57-0 


91 .0 


54-2 


20.7 


Si-2 


40.0 


53.8 


24-5 


51.8 


108. 


56.9 


74-5 


69.7 


114. 


47-3 


17.0 


14-5 


45-7 


49.9 


28.6 


73 -o 


4.C 


48.6 






Wire drawing and polishing. . 

Steel stamping and polishing. . 
Boiler and machine work 

Bridge machinery 

Heavy machine work 

Heavy machine work 

Light machine work 

Manufacturers' small tools. . . . 
Manufacturers' small tools. . . . 
Sewing machines and bicycles 

Sewing machines 

Screw machines and screws. . . 

Steel wood screws 

Manufacturers' steel nails 

Planing mill 

Light machine work 



243-0 

17.0 
13-3 

II-3 
48.0 
77.0 

19.7 

34-3 
22. 7 
82.0 

32-5 
127.0 

100. o 

45-9 
10.6 

4-3 



One-half 

One- third 
Two-thirds 

Nearly full 

Full 

FuU 

FuU 

FuU 
FuU 
Full 

FuU 
FuU 

One-fourth 

FuU 
FuU 
One -half 



hub, a small pilot hole a, Fig. 10, is first drilled in the joint, after which 
a second hole b, as large as the proposed keyway wiU admit, is drilled 
in the shaft. Standard dimensions are given in Table 6. 

The Kennedy key, Fig. 1 1 , has found large use in the Pittsburg dis- 
trict in the most rugged roUing-miU work, for which it does better 
than any other. For such work the keys are made approximately one- 
quarter of the shaft diameter, and located in the gear so that diagonals 
through two corners of the keys intersect at the center of the bore. 
The taper of | in. per ft. should be on the top for a driving fit, the 
sides being a neat fit. The hub is first bored for a press fit, then 
rebored about ^ of the shaft diameter off the center, the keyways 
being cut in the eccentric side. That portion of the bore opposite 
Table 6. — Nordberg Standard Round Keys 



Dia. of 


Dia. of 


Cutting 


Dia. of 


Dia. of 


Cutting 


shaft, ins. 


reamer. 


length of 


shaft, ins. 


reamer, 


length of 




ins. 


reamer, ins. 




ins. 


reamer, ms. 


2tf-3 


3 
i 


4i 


13"] 






3 1^-3 i 


7 
8 


4i 


14 


21^ 


12 


si -4 


I 


4f 


IS -I 






4f -4i 


li 


5 


16 1 






5 


li 


4f 


17 


3i 


13 


Si 


if 


4I 


i8 J 






6 


li 


6f 


19] 






7 1 






20 |- 


3ii 




8 [ 


if 


6f and 8 


21 J 






9-' 






22 ] 






10 ] 






23 !■ 

24 J 


4^' 


i4i 


II ■ 


2 


lol 






12 1 













Reamer diameters are at the small end. 
ured on the diameter. 



Taper ^6 in. per ft., meas- 




FiG. II. Fig. 12. 

Figs, g to 12. — Improved forms of keys. 



52 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 




Table 7. — Woodruff Standard Keys 







>» 


^ 




U 




D 




^ 
^ 








i^2 


^ 


"o 

u 







Ui 




&• 


u 


in 




tn 


2 ,r 


M 
"o 










u 

r! 




"0 


s 


1) 
C 






■4-> 

a, 


a 




d 




M 






r* 


6 




.£1 




CIJ 


r^ 


Z 


« 


H 


« 


U 


il .2 ^ 


^ 


« 


H 


tt 


u 


ji .2 




a 


J 


c 




(i 




a 


6 


c 




d 


I 


i 


A 


A 




A 


B 




A 


A 




A 


2 


4 


A 


A 




A 


16 


_ 1 

••■ 8 


A 


A 




A 


3 


1 
2 


1 

8 


1 

16 




3 

64 


17 


li 


7 

32 


7 

64 




5 

64 


4 


1 


A 


A 




A 


18 


-rl 

•^ 8 


1 
4 


i 




A 


S 


5 

8 


1 

8 


1 

16 




A 


C 


^1 

Is 


A 


5 
32 




5 

64 


6 


f 


A 


A 




1 

16 


19 


-.1 

i 4 


A 


A 




A 


7 


3 
4 


1 

8 


1 

16 




A 


20 


li 


7 
32 


7 

64 




A 


8 


3 
i 


A 


5 
64 




A 


21 


-r 1 


i 


1 

8 




A 


9 


f 


1^ 


A 




A 


D 


-1 


A 


5 
32 




A 


10 


i 


A 


A 




A 


E 


li 


1 


A 




A 


II 


1 


A 


3 

32 




1 

16 


22 


if 


1 
4 


i 




A 


12 


7 
8 


A 


A 




A 


23 


if 


A 


A 




A 


A 


7 
8 


1 
4 


1 

8 




1 ' 

16 


i? 


-3 

'-a 


1 


3 

16 




A 


13 


I 


A 


A 




A 


24 


ih 


i 

4 


1 
8 




A 


14 


I 


A 


A 




A 


25 


-. 1 

I5 


A 


5 

32 




A 


IS 


I 


1 
4 


1 

8 




1 

16 


G 


li 


1 


3 
16 




A 




\ I 

.X ^ 

a I 

I 




Table 8. — Woodruff Special Keys 





>, 


&> 


>> 


M 


0) 






>> 




^ 




tn 


^ ^ 















en 


-S2 




<U 







^ s 

CJ 








u 


en 







Ic .- 







v-< 


!fl 




n 


.a 






,0 


0) 

s 






J3 


0) 







1) 


B 

d 


G 








^^ 


>, 





id 


0. 


n 


S B^ 


-o 


B 


Oh 


rl 


S 6 


OJ 

M 


X) 





S 


Q 


1) 
U 


zz 

i .2 ^ 


^ 


'^ 


Q 


ja 
H 


P 




^.2 


'0 


^ 




a 


J 


c 




(f 


e 




a 


b 


c 




d 




e 


26 


2| 


A 


A 




a 


A 


31 


3i 


7 

16 


7 
32 




13 

16 




A 


27 


2i 


1 
4 


1 
S 




a 


3 

32 


32 


3i 


i 


i 




H 




A 


28 


2i 


A 


A 




17 
32 


A 


33 


3i 


9 
16 


A 




If 




A 


29 


2i 


3 

8 


A 




M 


A 


34 


3^ 


1 


A 




If 




A 


30 


3l 


3 

8 


3 

16 




13 
16 


3 

16 



















Table 9. — Diameters of Shafts and Suitable Woodruff Keys 



Diam- 
eter of 
shaft 


Number 
of key 


Diameter 
of shaft 


Number 
of key 


Diameter 
of shaft 


Number 
of key 


A-f 
A-4 
A-f 

11_3 
16 4 


I 

2,4 

3,5 

3,5,7 

6,8 


f-A 

I 

lA-ii 

lA 

li-iA 


6,8,10 

9,11,13 

9,11,13,16 
11,13,16 
12,14,17,20 


if-iA 

il-if 

iH-il 

lH-2 
2A-2i 


14,17,20 

15,18,21,24 

18,21,24 

23,25 

25 



the keys remains as originally bored. This feature is not essential 
but is of obvious convenience in assembling and disassembling. 
For less severe duty this type of key may be made narrower, as in 




Fig. 13. 



Fig. 14. 



Fig. 15. 







Table 


10.— 


-Dimensions of Keys 


and Keyways 








X 
•0 


Taper key (| in. per foot) 


Straight key 


4J 

"o 
a) 

CO 


•a 
H 


"(3 

J3 

a 
Q 


•a 

6 


"o 

.s ■« 

0) 

Q 


Depth in shaft 

measured at 

center 


"o 
•« u 

OJ 

.S c 

■S " 

Q 


a 



4J 

a 

C "O 

Q 


Q 


X u 
S c 

•- (U 

ft 15 

Q 


Is 

.S c 

ft M 

Q 


A 


B 


C 


D 


jE 


F 


H 


C 


D 


E 


F 


H 


i 


A 


A 




A 


A 


.0195 


.0272 


A 


A 


A 


.0352 


.0273 


A 


A 


A 




A 


A 


.0384 


.0240 


A 


A 


A 


.0540 


.0397 


1 


A 


fj 




A 


A 


.0374 


.0252 


A 


A 


A 


.0528 


.0409 


i 


i 


A 




A 


A 


• 0S48 


.0389 


i 


A 


A 


.0704 


.0546 


1 


A 


i 




A 


A 


.0724 


.0525 


A 


A 


A 


.0880 


.0682 


i 


A 


A 




A 


A 


.0900 


.0662 


A 


A 


A 


. 1056 


.0819 


i 


A 


A 




A 


A 


. 1076 


• 0798 


A 


A 


A 


.1232 


.0955 


I 


i 


A 




A 


A 


. 1096 


• 0778 


i 


J 


i 


. 1408 


. 1092 


li 


A 


i 




i 


i 


.1448 


. losi 


_5_ 


A 


A 


. 1761 


.1364 


li 


f 


A 




A 


A 


. I§00 


• 1324 


i 


A 


A 


.2113 


■ 1637 


li 


A 


i 




A 


A 


• 2153 


.1596 


A 


A 


A 


.246s 


. 1910 


2 


i 


1 




A 


_3_ 


. 2192 


• 1557 


i 


i 


J 


.2817 


.2183 


2i 


A 


i 




i 


i 


.28S7 


.2142 


A 


A 


A 


.3169 


.2456 


2j 


t 


i 




i 


i 


.2897 


.2112 


f 


A 


A 


.3522 


.2728 


2i 


ii 


f 




A 


A 


.3561 


.2688 


a 


a 


M 


.3874 


.3000 


3 


3 


1 




A 


A 


.3601 


.2648 


i 


i 


1 


.4226 


.3274 


3h 


1 


a 




3, 


1 


.4306 


.3193 


i 


A 


A 


• 4931 


.3819 


4 


I 


i 




f 


i 


.4385 


• 3114 


I 


i 


1 


.563s 


.4365 



.22 
o .20 
"« .18 

CD 

^ .16 
® 14 

Q) 

4-* 

o 

e .12 






.10 
.08 
.06 
.04 
.02 



^ 






























































^ 


\ 


^^ 


























1 


^ 


^ 


s?. 


?> 


















\, 


C^^ 






■^ 




\ 


7 










^ 




/ 
/ 


^^ 


^ 


f^.^ 







^--'' 


-^ 


^ 












%c 


s. 






• 


*' 










-^ 






/ 




^^ 


■v^ 


•' 


>■ 




^ 


^^ 










/ 


^ 


K 


•' 


^ 

• 


\ 










"" 




^, 


^^ 


/ 




.,'' 


^ 









"" 


■\ 


^ 






^^ 




^ 




■*»-. 




^ 


\ 


■^ 








^^ 


^ 




y 


' 




'^^ 








^ 


:::^ 








^ 



.10 .20 .30 .40 .50 .60 

"Width of Key way, Diameter of Shaft =1.0 

Fig. 16. — The weakening effect of keyways on shafts. 

Fig. 12, and may be accepted as a sovereign cure for key troubles. 
In all cases the taper should be between the top and bottom faces. 
The Woodruff system of keys, because of its increased depth, obviates 



SHAFTS AND KEYS 



53 



the tendency of common square keys to turn over in their seats. 
It has come into large use, especially in machine-tool construction. 
Tables 7, 8 and 9 give the dimensions. 

The best method of dimensioning drawings of keyways in shafts 
which are to be cut on the milling machine is that shown in Fig. 13. 
By adjusting the cutter until it just marks the shaft and then sinking 
it for depth by reading the index dial, the correct depth is quickly 
obtained. The best method of dimensioning keyways in hubs is 
that shown in Fig. i 4, the convenience of which, to the workman, is 
obvious. The figures for these methods of dimensioning are easily 
obtained from Table 10, the reference letters of which correspond 
with those of Fig. 15. 

The weakening effect of keyways on shafts formed the subject of 
experiments by Dr. H. F. Moore {Bulletin University of Illinois 
Engineering Experiment Station, No. 42). The results of the experi- 
ments are given graphically in Fig. 16. Dr. Moore defines the 
efficiency of a shaft with keyway as the ratio of strength at the 
elastic limit of a shaft with keyway to the strength at the elastic 
limit of a similar shaft without keyway, and it is this efficiency which 
may be obtained from the chart. To use the chart locate a point 
defining the size of the ke5rway which will, in general, fall between 
two lines representing values of efficiency, and the efficiency of the 
shaft in question may then be estimated with sufficient accuracy. 
The space within the triangle OAB represents the range covered by 
the tests actually performed, ajid covers the proportions of keyways 
commonly used in practice. 




Table ii. — Brown & 



Sharp Mfg. Co.'s 
FOR Cutters 



Standard Keyways 



Diameter (Z?) of 


Width iW) of 


Depth (D) of 


Radius 


hole, ins. 


keyway, ins. 


keyway, ins. 


(R), ins. 


H to Me 


H2 


%4 


.020 


% to % 


H 


He 


.030 


15,^6 to iM 


H2 


%4 


■03s 


iKe toiM 


^6 


M2 


.040 * 


iKe toi^* 


H 


% 


.050 


x^He to 2 


He 


%2 


.060 


2>f 6 to 2}4 


% 


Ke 


.060 


2^6 to 3 


Ke 


Vxe 


.060 



BELTS AND PULLEYS 



The driving power of leather belts is summed up by Prof. W. 
W. Bird {Journal Worcester Polytechnic Institute, Jan., 1910) as 
follows: 

The h.p. that a belt will transmit depends upon the effective 
tension and the belt speed. The effective tensions depend upon the 
difference in the tensions of the two sides of the belt and on the sur- 
face friction, which depends upon the ratio of the tensions and the 
angle of wrap. 

Experiments and practice have shown that a belt of single thickness 
will stand a stress of 60 lbs. per in. of width and give good results, 
that is, it will only require an occasional taking up and will have a 
fairly long life. The corresponding values for double and triple 
belts are 105 and 150 lbs. per in. of width provided the pulleys are 
not too small. 

Experiments have shown that on small pulleys the ratio of the 
tensions should not exceed 2, on medium pulleys 2.5, and on large 
pulleys 3. The larger the pulley, the better the contact, the thinner 
the belt, and the better the contact for the same size of pulley. When 



The use of the tables is shown by the following examples: 

How much horse-power will a 4-in. si.ngle belt transmit at a speed 

of 4600 ft. per min., passing over a 12-in pulley? The factor is 920, 

therefore, 

4600X4 



920 



=20 h.p. 



How wide should a belt be in order to transmit 50 h.p. at 2000 ft. 
per min. on a 36-in. pulley? 
50X830 



W-- 



- = 2o.7-in. single belt. 



This gives a width of single belt which is beyond the usual limit, 
8 ins. being considered good practice for the maximum width of a 
single belt. 

W = =i3-in. double belt. 

2000 ^ 

How wide should a single belt be in order to transmit 2 h.p. at 
600 ft. per min. over a 4-in. pulley with 14a deg. wrap? 



Table i. — Constants for the Driving Power of Leather Belts 



Diameter of pulley 


Under 
8 ins. 


8-36 ins. 


Over 
3 ft. 


Under 
14 ins. 


14-60 ins. 


Over 

sft. 


Under 
21 ins. 


21-84 ins. 


Over 

7 ft. 


Thickness of belt 


Single 


Single Single 


Double 


Double 


Double 


Triple 


Triple 


Triple 


Factor 

Difference of tensions 


iioo.o 

30.0 

0.74 

2.0 

60.0 


920.0 
36.0 
0.89 
2.50 
60.0 


830.0 
40.0 
0.99 

30 
60.0 


630.0 

S2-S 

0.74 
2.0 
105.0 


520.0 
63.0 
0.89 
2.50 
105.0 


470.0 
70. 
0.99 

30 
105.0 


440.0 

7S-0 
0.74 
2.0 
150.0 


370.0 
90.0 
0.89 
2.50 

150.0 


330-0 
100. 


Per cent, of creep 

Ratio of tensions 


0.99 
1,. 


Tension on tight side 


150.0 







the pulley diameter in ft. is three times the thickness of the belt in 
ins. or in this proportion, we get equivalent results for different thick- 
nesses of belts. This gives a method of classifying pulleys. The 
belt has to adjust itself in passing over a pulley due to its own thick- 
ness. Some adjustment is also necessary on account of the crowning 
of the pulley. These adjustments account for the different ratios for 
the various pulley diameters. The effects of the crown and pulley 
diameters are not usually considered in belt rules, which is a grave 
mistake. The ratios are for 180 deg. wrap and decrease with 
less contact. 

The creep of the belt depends upon its elasticity and the load, and 
experiments have shown that this should not exceed i per cent, in 
good practice. In order to keep this creep below i per cent., it is 
necessary to limit the difference of tension per in. of width of single 
belt to 40 lbs. The ccrresponding values for double and triple belts 
are 70 and 100 lbs. per in. of width. These figures are based on an 
average value of 20,000 for the running modulus of elasticity of leather 
belting. 

Table i has been prepared on the basis of these limitations and 
gives a value for F in the equation 



Table 2. — Constants for the Driving Power of Leather Belts 



h.p.= — ^, — or W ■■ 



h.p.XF 



in which h.p. is the horse-power, V the belt velocity in ft. per min., 
and W the width in ins. 

Table 2 gives corrected values for F when the arc of contact or wrap 
is greater or less than 180 deg. On large pulleys the creep may exceed 
I per cent, if the wrap is over 180 deg., as the increased friction gives 
a greater difference of tensions. 



220° 


210° 


200° 



190 


180° 


170° 


160° 


150° 


140° 


130° 


120° 


980 


lOIO 


1040 


1070 


IIOO 


1 140 


II80 


1220 


1270 


1330 


1400 


810 


830 


860 


890 


920 


950 


990 


1040 


IIOO 


1 1 70 


1240 


730 


75° 


770 


800 


830 


860 


890 


930 


980 


1030 


IIOO 


560 


S70 


590 


610 


630 


650 


670 


700 


730 


760 


800 


460 


470 


480 


500 


520 


540 


S70 


600 


630 


660 


700 


420 


430 


440 


450 


470 


490 


Sio 


530 


560 


590 


630 


390 


400 


410 


420 


440 


460 


480 


500 


520 


540 


560 


320 


330 


340 


350 


370 


390 


410 


430 


450 


470 


490 


290 


300 


310 


320 


330 


340 


360 


380 


400 


420 


440 



Taking the factor iioo from Table i, in line with it in the 180 deg. 
column of Table 2 we find in the 140 deg. column the corrected 
value 1270. 

2X1270 



W=- 



600 



= 4 . 23-in. single belt. 



How wide a belt is required for 300 h.p. at 2000 ft. per min. 
over a lo-ft. pulley? 

W = ^°°^'^^° = 70 . s-in. double belt. 
2000 ' "^ 

This is too wide. Good practice calls for a change to triple at 48 
ns. unless for some special reason a narrower belt is necessary. 

^^ 300X330 ^ triple belt. 

2000 ^ '' ^ 

The belt speed is limited by centrifugal force, but below 5000 ft. 

per min. the loss on this account is largely compensated for by the 



54 



BELTS AND PULLEYS 



55 



increase of friction due to the decrease in the time element of the con- 
tact, caused by the increased velocities. 

The results given by these factors are well within working values 
and the belts will probably transmit 50 per cent, more power than 
these factors, but at the expense of the Ufe of the belt. A liberal 
allowance at the beginning means less annoyance, fewer delays in 
taking up the belts, longer life and less cost for renewals and repairs. 

The dimensions of belts in relation to the power transmitted and 
the effective pull may be obtained from Figs, i and 2. Fig. i con- 
forms to the usage of Wm. Sellers & Co. as deduced from the experi- 
ments made for them by Wilfred Lewis (Trans. A. S. M. E., Vols. 
7 and 20). Fig. 2 conforms to the recommendations of Carl G. 
Barth {Trans. A. S. M. E., Vol. 31) based on a re-analysis of the 



100 



90 



^ 80 



70 



g 60 



§ 50 

o 

a 
- 40 



30 



— ^ 1 — -- 


7 '■ 


■ >4-' ~ 'h ■ 




... ^^_ .. .. .- 


^2& 


-^± 


: ^^t5 - -- 


52 L ■« 


s^ 


i > 


..:. : . 5?^: _.. 


- - -SI'e 


^sk: 


Tjvi ::: ^is,7 " 


-- -3't^- - - --iX,>, — 


— -~9^,^ •^■ 


::^-5i:. . : :: ^?<a" 


^^I'TTjl ^« 


s?;^:" : _ :::: 


v^fK^ ^ : 


- - - T H.V ; - ^ r 




a^jtiti r-;x s&, s 


:::: :: S^ 'Sj^rfe:: ^S!'3t 5" 


flj^ T ,^ V^ > 


.. ... ^lP.it^~.-^\, V 


'CG^vS'i' — 'v 


e^g,^- ... - - 


iy,..^ L:_^^^_.^;; :::::::: :: 


'-'-'-'-^'-fm,'"'$iV ^^^^ 


= - =2 iiwr x^v 


^^■'<: ft ■<-r^ 


— s^./z SN — 


- -i-^i-j^..- _ ^«. ^ _ 


::: .s*c; , ::-^: 5 


■^l^ . < 


.:.; ::::.:.: .SiS: ::..i... 


. --- ^ftl- !>-- 


^■t . 


.... _ "^^ y 




_:"_:":. "" _ "'^s"~ 




s 









llillill II lllillllMM-Hfflitfll 


-_ _. ,' -- "s 


11:::::::+ + — -±i-:-:::::::::^i;i 


:::::::::::,?:::: ::::g 


10 ::.:::: ;:::.: :::-:-:2::- ::.:-::- 


10 ::::::::::ih::::: ::::::::::::::: 


H?fffe#HlH-^M 


„::::: ::::::::::Sz:::::::::::::::::::::: 


9 :: ::: ^ 


.- -- a -_::_: 


-^t 


8^^. . .■.|.yj 


?/ 


:::::::::: :r^|:;::-: 




7 ::""::"":::Sz"':":"::T(>r: ::-- ;:::-- 


|||||||||.l^|||||i^^j^fl:pjgrf|^ mm 


6 -LoV ^^.Mh^ i^ i.^ 




zz:-_::-_Sl--M'--^^------- ^ 


, -iSjt. S?^.,3£: ..i.:.. 


^---3=.^W--^7^ -T" 


^--=^^/ffi3?H ^--- 


.- - -L. ■v-UCi 4- - --- --- ^ 




^ fh-M-iMf-fira^ek \- 


l-'^/W\--^^-----------'W.V----\ 


, — ^y-s^x^Y s«;---5 


^--*7-7^^^ :::::: ::::^^:::: 


/y%^ Bf ^ ^ ^ ^ ^ ^ ^ ^ r^ 




2 "' 1Z~_~2 S 


P#^^^^^^^^^^^^^^^^ T^ 


M^^^^^^^^^^^^^^^^^^^^ W^ 


iV- - - - -- ' " 


|z.- -- - - .-- :: -- - - 


X. 1L +.:::::::::: 



same experiments combined with a study of the extended observa- 
tions of belts in service by F. W. Taylor {Trans. A. S. M. E., Vol. 
is). Mr. Earth's recommendations are intended to secure maximum 
economy of belts and of upkeep. He considers the proper working 
loads of belts when proportioned from this viewpoint to be so well 
within the capacity of any good joint that the kind of joint may be 
ignored. 

The author suggests that the Sellers chart be followed for main 
driving belts and that Mr. Earth's chart be used for machine belts. 

For arcs of contact other than 180 deg., the power transmitted 
and the effective pull, as given by the charts, are to be multiplied by 
the factors of the table below the charts. 

The charts should not be understood as giving, or as intended to 
give, the ultimate capacity of belts. As with every other machine 
member, the question regarding a belt is not how much it can be 
made to do, but how much it should be made to do. As a matter 
of fact, and as shown by the figures given below, belts will carry loads 
materially in excess of those imposed by either chart. 

An examination by the author of belt fly-wheels of 12 Corliss 
engines, ranging between 50 and 380 rated h.p. by three high- 
class builders (Amer. Mach., Oct. 28, 1897), gave the following 
values for the number of ft. per min. travel of i-in. belt for each h.p. 
of the rated capacity of the engines : 

Average of 1 2 480 

Maximum 750 

Minimum 375 

For additional information on main driving belts for steam engines, 
see Steam-engine Belts. 

As an example of heavy duty, Samuel Webber (Amer. Mach., 
Feb. 22, 1894) cites a main driving belt 30 ins. wide, f in. thick, 
running 3900 ft. per min. on a s-ft pulley and transmitting 556 h.p. 
foraperiodof 6 yrs., giving 210 ft. per min. travel of i-in. belt per h.p. 



50 






iS o 



lb 



^a. 









L — r L 


" 7 ""i*^ 


■-? : -'"="37-- 


'j ^> 


~t ~ ' - '^Sir. 


J - ^<S 


' "" : _^ij:>,_._ 


V 


^<& 


^•fe 


\ * 


air -^-- 


■--,' = -- = = :^rj^d ^^r25_ 


"v' '^Sd^- \ 


'^ ==^?; S 


I ' ^<t ^ 


._ .^4^!.,,. .S, - 


" " "sge/ V 


^"I s 


% s 




" . _:_ ^,.s.. 


>. \ 


: N\ 


\\ 


S v 


;:.::::':::..:::—:;:: s. 



1000 2000 3000 4000 SOOO 6000 7000 8000 
Velocity of Belt, Ft. per Min. 







,■' 5- 


^ "-^ - 


_,^: .^^.- . 


-t--? 5 


©• 7 . . s, 


W - \ 


TE^ S 


JoL'^. . .S^. _ - 


;yr r^_ __^ 


tft' r '^"^ 'l 


-Ci J^ ''' i^ ^ 


^r a^3S' ^s 


SZ-.^C . ^- -\ 


fSj ^^ ^ 


HJ-l^.4^ _ . \- I. 


^ft/vS^^: - % A 




1 y ^^l-- 


---'■-,' - 5^5 : 


i-^' i 


2.' : . ; _ . 5 


yS -S 


?^ - - - - ^ 



1000 2000 3000 4000 5000 600O 7000 
Velocity of Belt. Ft. per Min. 



Fig. I. — Sellers belt formula. 



Fig. 2. — Earth belt formula. 



Factors to be used for different arcs of contact. 



Arc of contact, deg 


90 


100 


no 


120 


130 


140 


ISO 


160 


170 


180 


190 


200 


210 


Factor 


.65 


.70 


.75 


.79 .83 


.87 


.91 


.94 


•97 


1. 00 


I -03 


1.05 


1.07 



56 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



The arc of contact of a belt on the smaller of two pulleys may be 
found by the following rule and Table 3, by Wm. Cox {Amer. 
Mach., July 20, 1905): 

Divide the difference between the diameters of the two puUeys 
in inches by the distance between their centers, also in inches. Let 
the quotient equal x. Now from the accompanying table find, in line 
with such ascertained values of x, the corresponding angle of the arc 
of contact of the belt on the smaller pulley. 

Example: Two pulleys of 80 and 30 ins. diameter are spaced 120 
ins. apart, center to center; what is the arc of contact on the smaller 
pulley? 

80-30 . 

= .4i6 = :ii; 

120 

Opposite .416 in the x column we find the arc of contact to be 156 deg. 

Table 3. — Arcs of Contact of Belts on Ptjlleys 



Table 4. — Comparative Power Transmitting Capacities of 
Pulleys of Various Materials 





Angle, 




Angle, 




Angle, 




degrees 




degrees 




degrees 


.000 


180 


•347 


160 


.684 


140 


.017 


179 


• 364 


159 


. 700 


139 


• 03s 


178 


.382 


158 


.717 


138 


.052 


177 


•399 


157 


•733 


137 


.070 


176 


.416 


156 


.749 


136 


.087 


175 


■433 


15s 


.765 


^35 


.IDS 


174 


• 450 


154 


.781 


134 


.122 


173 


.467 


153 


•797 


133 


•139 


172 


.484 


152 


• 813 


132 


•157 


171 


•501 


151 


.829 


131 


.174 


170 


.518 


150 


• 84s 


130 


.192 


169 


•534 


149 


.861 


129 


.209 


168 


•SSI 


148 


.877 


128 


.226 


167 


• 568 


147 


.892 


127 


.244 


166 


• 584 


146 


.908 


126 


.261 


165 


.601 


145 


•923 


125 


.278 


164 


.618 


144 


•939 


124 


.296 


163 


•635 


143 


•954 


123 


• 313 


162 


• 651 


142 


.970 


122 


■ 330 


161 


.668 


141 


•98s 


121 


• 347 


160 


.684 


140 


1. 000 


120 



The comparative transmitting capacities of pulleys made of different 
materials formed the subject of tests by Prof. W. M. Savtoon 
{Proc. Nat. Asso. of Cotton Mfrs., 1911). The results of these tests 
reduced to an arc of contact of 180 deg. and 250 lbs. per sq. in. tension 
on the tight side are given in Table 4. The tests were made at a belt 
speed of 2200 ft. per min. 

Idler Pulleys and Quarter Twist Belts 

The idler pulley may he made the source of great benefit, when properly 
laid out, although commonly looked upon as an unmixed evil. Used 
as a simple tightener, it is not to be recommended, but when so laid 
out to increase the arc of contact it may be made to reduce the 
tensions. 

Fig. 3 {Amer. Mach., May 26, 1910) shows the correct location of 
the idler pulley. Its obvious effect is to increase the arc of contact, 
especially on the smaller pulley where most needed. This, in turn, 
reduces the necessary tension on the slack side, increases the differ- 
ence of tensions, that is, the effective tension, and reduces the tension 
on the tight side for a given effective tension, these reduced tensions 
leading, in turn, to a corresponding decrease of pressure on the bear- 
ings. The idler should be on the slack side of the belt and near 
the smaller pulley. Either pulley may drive. Additional benefit 



Kind of pulley 



Cast iron 

Cast iron with corks proj. . 04 in. . 
Cast iron with corks proj. .015 in. 

Wood 

Wood with corks proj. .075 in. . . 



Wood with corks proj. .03 in 

Paper 

Paper with corks proj. . 087 in 

Paper with corks proj. (about) .015 in 



Comparative 

transmitting 

capacity at 

2 per cent. 

slip 



100. o 

107.0 
112. 1 
105.6 
104.8 

104.8 

i37^5 
122.0 
133-2 



may be obtained by mounting the idler on a weighted arm arranged 
to swivel about the center of the smaller pulley, as shown in Fig. 4. 
With this construction the tensions are independent of the elasticity 
of the belt and the objection to short belt transmissions disappears. 
Similarly, the weight of the belt in vertical transmissions no longer 
reduces the tensions on the lower pulley, and such transmissions 
become entirely practicable. Again, the effect of centrifugal force 
in causing the belt to leave the smaller pulley, reducing the arc of 
contact and carrying air between pulley and belt is overcome. 




Fig. 3. 
Figs. 3 and 4. — Correct arrangement of idler pulleys. 

The layout of quarter twist belts is shown in Fig. 5, the rule being 
that the central plane of each puUey must pass through the point 
of delivery of the other pulley. This construction should be used 
with narrow belts running on pulleys at a good distance apart only. 
Quarter twist belts will drive in one direction only. 

Guide pidleys shoidd be substituted for quarter twist belts whenever 
possible, and Figs. 6-14 show various arrangements of such pulleys. 
The rule is that the intersection of the central planes of consecutive 
pulleys shall be tangent to both pulleys. 

Thus in Fig. 7, in which pulleys A and B are of the same size, and 
either of which may drive in either direction and the shafts are at 
right angles, the intersection of the central planes of B and C is 
obviously tangent to both and so for the other pulleys. In Fig. 
8, A is larger than B and the same condition holds, as it does also in 
the increasingly complex arrangements of Figs. 9 and 10. 

In Figs. 1 1 and 12 A ov B may drive. In Fig. 11 C or Z> is loose 
on the shaft, while in Fig. 12 both C and D are loose. The loose 
pulleys should, if possible, be placed on the slack side of the belt. In 
Fig. 13 the guide pulleys revolve, nominally, at the same speed, but 
nevertheless one of them should be loose in order to provide for the 
differential action due to any slight difference in their diameters 
Fig. 14 shows a power distribution system through a 16-story building 
by means of vertical shafts, a single guide pulley only being used 
for each belt. Similar constructions distribute the power from the 
vertical shafts to line shafts on each floor. 

In all of the constructions shown, except that of Fig. 5, the belt 
will drive in either direction, the arrows being for assistance in tracing 
the motion. 

Holes through floors for quarter-twist vertical belts may be laid out by 



BELTS AND PULLEYS 



57 



i 



I 



the method shown in Figs. 15-18 by M. H. Ball (Mchy., Sept., 
191 2). The basic rule is that the center of the face of one of the 
pulleys at a point level with the center of its shaft must be in the same 
vertical line as the similar point on the other pulley, as indicated in the 
illustrations. The direction in which the pulleys are to turn deter- 
mines which of their sides must be in line, as it is always the sides 



Fig. 18 shows a method of laying out the floor holes for the drive 
indicated in Fig. 16. First draw an outline (plan view) of the two 
pulleys on the floor in full size, directly below and above the respec- 
tive pulleys to be connected by the belt. A starting-point for this 




Fig. S- — Quarter Figs. 6 to 10. — Substitutes for quarter 

twist belt. twist belts. 

from which the belt leaves the pulleys which should line up. Fig. 15 
shows how the pulleys should be set when the lower pulley turns to the 
left, as indicated by the arrow. Fig. 16 shows the setting when the 
lower pulley is driven in the opposite direction. The rules given 
apply to the aligning of pulleys at other angles as well, an example of 
which is shown in Fig. 1 7. 





Fig. II. Fig. 12. 

Figs, ii and 12. — Substitutes for quarter twist belts. 

layout can readily be found with a plumb bob. Then draw the center 
lines AB and CD through the faces of the pulleys, and divide the 
diameter of each pulley into eight parts, as shown, numbering the 
divisions i, 2, 3, etc. The numbers of the divisions must start from 




Fig. 13. 



Fig. 14. 
Figs. 13 and 14. — Substitutes for quarter twist belts. 



58 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



the sides of the pulleys which are opposite the arrow points shown in 
the plan view indicating the direction of rotation. Next, measure the 
distances from center to center of the shafts and from the center of 
the upper shaft to the floor. In the example shown the distance 
from center to center of the shafts is 96 ins., and the distance from 
the center of the upper shaft to the floor is 42 ins. As 96-^8 = 12, 




Fig. 18. 
Figs. 15 to 18. — ^Laying out holes in floors for quarter twist belts. 



on the top side of the floor, and as our measurements from the floor 
to the upper shaft determine how many spaces we are to set off, 
start numbering these divisions from the point E, the line EJ being 
parallel to the face of the upper pulley; then set off 35 spaces from 
E, thus determining point L, and draw line LO through /, making 
JO equal to LJ. This line indicates the position of the center of the 
belt at the floor line and a line of the same length parallel to it through 
K indicates the other center Une of the belt at the floor line. 

The layout for an angle of other than 90 deg., as indicated in Fig.- 
17, differs only in that the arc on the pulley outline extends only 
from the line EJ to the line GJ , Fig. 17, this latter line being parallel 
mth the face of the lower pulley. Any number of divisions more or 
less than eight may be used if preferred. 

Mnle-pulley stands may be laid out by the method shown in Figs. 
19 and 20 by Fred Howe {Woodcraft, June, 1912). 

Before the problem can be laid out as in Fig. 19 the diameters 
of the pulleys and the distances between the shafts must be known. 
Assume that shafts A and C are each horizontally 4 ft. 6 ins. from 
the mule pulleys, which are to be centered at E, and that the shafts 
are vertically 18 ins. apart. Draw two horizontal lines, 18 ins. 
apart, through the centers of A and C, and locate these pulleys upon 
their lines as shown. 

Pulley A is represented 4 ft. 6 ins. from pulley C, and the line H 
at the center of pulley A indicates the point where the turn in the 
belt is to be located. 

Having drawn a vertical line ff 4 ft. 6 ins. from C, measure off 
another 4 ft. 6 ins. from the last vertical line, and draw a third line 
at B. This line represents the location of shaft AB, were the belt 
stretched out in a straight line, without passing around the mule 
stand E. The inclined lines from pulley C to pulley 5 show the exact 
path or slope of the belt at every portion of its length from one pulley 
to the other. 

Let it now be assumed that the mule pulleys E are 10 ins. in diam- 
eter. Measure back s ins. from the middle vertical line H, and draw 
another vertical E, which will be the center of the mule stand; meas- 
ure from the horizontal lines through C and A to the lines G and F, 
and, scaling the drawing, shows about 2 ins. and 3 1 ins. or by calcu- 
lation 2g and 3f ins., respectively. 

If desired, the belt may be made to run with the mule pulleys level 
with one of the pulleys, either the upper or the lower, or they may 
be located anywhere between the two extremes, but the principle 
involved is the same, no matter where the mule pulleys may be 
located. The method here shown places the mules directly in line 




Fig. 19. 



Fig. 20. 
Figs. 19 and 20. — Laying out mule pulley stands. 



each division on the diameter of the pulleys is equivalent to 12 ins. 
Further, 42 -=- 1 2 = 35, which represents the number of spaces that the 
center of the belt will be from the center points of the sides of the upper 
pulley, as indicated at / and K in the engraving. Draw the line 
EJ through the point thus located in the rectangle representing the 
upper puUey. Then strike an arc with / as center and EJ as radius, 
as indicated, and divide it into eight equal parts. As we are working 



with puUeys C and A , so that the belt makes an even drop from one 
pulley to the other. 

Should it be proposed to locate the mule pulleys at any other point 
between C and A, the sum of the distances between the mules and 
the two pulleys is taken as above, and pulley B, laid down as 
described, no matter where between them line E may come. Should 
line E be moved toward or from shaft C, and the belt length 



BELTS AND PULLEYS 



59 



remain the same, there will be no change in the angle at which the 
mule shaft must be placed. And this angle is the same from the 
vertical as the angle of the belt is from the horizontal, viz., one in 
six, or the mule shaft must be suspended 2 ins. to the ft. of its 
length out of plumb. 

Fig. 20 shows the arrangement of the mules as found from Fig. ig. 
The mule puUeys are 10 ins. in diameter, and it wiU be assumed that 
all the pulleys are 6-in. face. Pulleys C and B being equal in diam- 
eter, the mules must be a distance apart equal to the diameter of the 
pulleys C and B, ov 12 ins. As the center of the face of each mule 
puUey must be placed with the middle of its face upon vertical line 
H, and as the shafts of the mules pitch i in 6, it is evident that the 
centers of the mule shafts will be 2 ins. apart on centers, and scaling 
the drawing, Fig. 20, shows this to be the case. 

The mule puUeys located as shown will run perfectly, keeping the 
belt square upon both driver and driven puUeys. In fact, in any belt 
transmission of similar character it is only necessary to look to two 
points. The first of these is that the upper mule pulley be so arranged 
that it receives the belt fair and square from drive pulley C. In 
fact, the upper mule pulley may be placed at any angle or at any dis- 
tance from C and the belt will track perfectly as long as the face of the 
mule puUey is fair to receive the belt squarely from puUey C. It 
makes no difference at what angle the belt leaves the mule puUey, 
except that this pulley must be so located that the belt leading away 
therefrom shall lead or track directly and fairly toward puUey B. 
That is all that is necessary for the upper mule pulley. 

The lower mule pulley may be so placed that it shall receive the 
lower fold of the belt fairand square from pulley A. Nothing else is 
so necessary as that the mule-pulley is located so that the belt guides 
fair toward the receiving side or face of pulley C. This means that 
the lower mule shall be moved bodily so as to guide the belt toward C 
and turned at the angle which may be necessary to receive the belt 
from puUey A. It makes no difference at what angle — within limits, 
of course — a belt leaves a pulley so long as the belt guides toward 
that pulley squarely at an angle of 90 deg. and on the center line of 
the face. 

In practice, it is usually necessary to locate a pulley so that it 
delivers the belt square to the next pulley, and then turn thepuUfey in 
or out, up or down, without moving it from its location, until it will 
receive the belt fairly from the last pulley over which the belt has 
passed. This applies alike to open belts, crossed belts, quarter-turn 
belts and mule belts, as in the present instance. 

Taking advantage of this fact, it is possible to move the mule 
pulleys a little so that both may be placed upon a single shaft. Refer- 
ring again to Fig. 20, it will be noted that the mule shafts are parallel 
and only 2 ins. out of line. But it should not be forgotten that the 
shafts are 2 ins. out of line in another direction, for, if the eye be placed 
to the right, so as to look along the direction of shaft AB, then the 
upper mule pulley will be found 2 ins. out of alignment with the lower 
pulley, and to bring both the mule shafts into alignment, the lower 
one must be moved 2 ins. to the left, while the upper one must be 
moved 2 ins. directly from the observer, toward the puUey on shaft 
AB. 

The reason why pulleys can be moved thus, and still allow the belts 
upon them to run properly, is due to the characteristic explained 
above. Take the case of the lower mule pulley: The belt, running in 
the direction of the arrow, leaves pulley B, Fig. 20, guided toward 
the lower mule pulley. Note what would happen were this pulley to 
be moved 2 ins. to the left. The belt as it left pulley B would be 
twisted slightly to the left but would still hit the mule pulley fairly. 
But as it is immaterial how a belt leaves a pulley, no harm is done in 
moving the lower mule pulley 2 ins. to the left. Its angle is not 
changed, therefore it receives the belt properly, and that is all that 
is necessary. 

Next, push the upper mule pulley horizontally backward 2 ins. 
This brings the two mule shafts in line as viewed from the right side, 
and causes the upper fold of the belt to twist a little as it leaves pulley 



C, but the angle of the upper mule pulley not having been disturbed, 
it still receives the belt fairly from C, and still delivers the belt 
squarely toward pulley B. Therefore, both mule puUeys may be 
placed upon a single shaft by making the slight changes described. 

When the pulleys upon shafts B and C are of unequal diameters, 
it will be necessary to use separate mule shafts and to adjust the 
shaft of each mule pulley square to the line drawn from one pulley to 
the other, as in Fig. 19. Otherwise, there is no change in the method 
of locating the mule puUeys and obtaining the angles of their shafts. 

Should it be found necessary to run the belt at two different angles, 
instead of using the same angle from pulley C to pulley B, lay down 
both angles in Fig. 19, adjust the mule shaft to right and left to be 




Fig. 21. — A triple drive in use eight years. 



65xlZ'Pulfey 
133 Rpm. 



60x/e Pulfey 
IZOHp. 




<■ za-0"- »k lo'-o'—— 

Fig. 22. — 170 h. p. transmitted by tandam drive. 

square with the line over face of puUeys from C to £ in Fig. 19, and 
then adjust the mule stand to and from the beholder, to be perpen- 
dicular to the belt line from E to B, Fig. 19. The lower mule shaft 
is to be adjusted in Mke manner, but to agree with the lower belt line 
in Fig. 19. Thus, when the puUeys are of unequal size, and the belts 
leave and reach puUeys C and B at unlike angles, there wiU be corre- 
spondingly different angles given to the mule shafts, to the right and 
left, and forward or backward. 

Tandem or riding belts, while frequently used as expedients are 
not usually looked upon with favor. There is, however, no good 



60 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



reason for this, as such drives have continued in use for years and 
with entire satisfaction. They may be used with great freedom, 
either to divide the power from a shaft or motor or to add the power 
of two motors. Fig. 21, by Chas. M. Young (Amer. Mach., Apr. i, 
1915), shows successful cases of the former and Fig. 22 of the latter 
kind. Power calculations need not differ from those applying to the 



ZOxe'Pullei; 



Line Shaft 
24")(.6" Pulley 
20 Up. 



Countershaft 
40 Kd' Pulley 
24 Up. 




Mainline 
AOla'Puiley 
'^ i6R.p.m. 



, 20/6 
, ,--rK-J'ulley 

Fig. 23. — Combined tandem and right angle drive. 

usual arrangement, the calculations being naturally based on the 
pulleys carrying one belt. In the case of Fig. 22 the pulley sizes 
must be carefully determined to give the proper speeds to the motors 
and avoid overloading one of them. 

Length of Belts 

The calculation of the length of a belt is occasionally necessary 
to meet cases where endless belts are to be carried over pulleys at 
considerable distances apart. Carl G. Barth {Amer. Mach., Mar. 
12, 1903) gives the following formulas of increasing accuracy in the 
order given: 



L = 



{D+d)7C 



x=i^±^+c. 



X=^^±^ + C. 



+CX+2C 

+2C 
+2C 



12- 
60- 



-X 
I3X 



2 ■ ' 60 — 18» 
in which Z, = length of belt (open) 

!? = diameter of larger pulley 
d = diameter of smaller puUey 
C = center distance 



(a) 
(&) 
(c) 



" m 



All dimensions are to be taken in the same units — feet or inches 
as preferred. 

Mr. Barth has tested these formulas by applying them to the 
limiting case (beyond what is possible in practice) in which d = o 

and C =— ' the correct value of L being Dtc. Under this test, formula 

(a), which is identical with Rankine's well-known formula, gives a 
result which is a little over 2 per cent, short, formula (b) a result 
which is about i per cent, short, and formula (c) a result which is less 
than four-tenths of i per cent, short. 

The length of a crossed belt is given by the exact formula 



£ = 2 '•yC^-D'^+D ( Ti^-cos 






in which the notation is as before with the addition that 

2 

Steel Belts 

Steel belts have been used to a considerable extent in Germany 
and with apparent success. The joint construction, shown in Fig. 
24 {Amer. Mach., Dec. 24, 1908), consists of two steel plates, an under 
and an upper, between which the ends of the belts are joined. These 
plates taper from a thickened section at the center to comparatively 
thin edges. The ends of the outer locking pieces are prolonged. It 
was discovered that when these extensions were not provided the 



belt would break near the inner pieces just after leaving the pulley, 
probably owing to the rapid straightening of the belt after its rapid 
motion over the puUey. In the size illustrated, the upper plate is 
made with a series of holes in order to lighten it. Both of these 
plates are shaped to a circular arc, whose radius is equal to the radius 
of the smallest puUey on which the joint is to be used. Thus, for a 
given joint there is a minimum limiting diameter of puUey on which 
it can run, but no similar maximum limiting diameter; for a given 
joint can be used on pulleys of any diameter larger than the one to 
which the plates are particularly fitted. 

The belt itself is made of a uniform quality of steel of an even 
thickness and is tempered. The ends are carefully brought together, 
fitted and soldered with a special solder that flows at a comparatively 
low temperature to avoid drawing the temper of the belt. This 
joining is then placed between the two plates already described, 
and these plates are fastened together by means of screws, as shown 
in the illustration. 

A number of interesting claims are made for these belts. Three 
of the most striking are: The small amount of slipping of the belt on 
the pulley, given in figures as less than -ro of i per cent., the narrow 
width of the belt compared with leather belts, the proportion being 
about as i to 5, and the great speed at which these belts can be run, ■ 
given as 100 m. per sec, or say 19,000 ft. per min. This latter 
figure is striking when compared with the limiting factor, usually 
given for leather belting as 4000 ft. per min. It is very common to 
run these steel belts at a speed of 50 m. per sec, or say, roughly, 
10,000 ft. per min. They have been used for driving belts in machine 




Fig. 24. — German steel belt joint. 

shops and other manufacturing estabhshments, installations of 250 
h.p. having been made. Table 5 gives some comparative data be- 
tween a rope drive, a leather-belt drive, and steel-belt drive for 100 
h.p., transmitted by pulleys 10 m. apart at a speed of 200 r.p.m. and 
a diameter of i m. The metric measurements are not translated into 
English measurements because the table is of comparative interest 
only. 



Table 5. 



-Comparison of Rope, Leather-belt and Steel- 
belt Drives 



Item 


Rope drive 


Leather-belt 
drive 


Steel-belt 
drive 


Breadth of belt space. . . . 

Breadth of pulley 

Weight of pulley 

Weight of rope or belt. . . 
Total weight of drive. . . . 


6 ropes 
45 mm. in diameter 
380 mm. 
1000 kg. 
240 kg. 
1240 kg. 
720 marks 
600 marks 
1340 marks 
13% 
13 h-P- 


500 mm. 

500 mm. 
520 kg. 
140 kg. 
660 kg. 
400 marks 
1300 marks 
1700 marks 
6% 
6 h.p. 


100 mm. 

no mm. 
270 kg. 
13 kg. 
283 kg. 
250 marks 


Cost of ropes or belts. . . . 
Total cost 


750 marks 
1000 marks 


Power lost in per cent. . . 
Power lost in horse-power 


.5% 
. 5 h. p. 



More recent information regarding several successful German 
installations of steel belts is given by R. K. Cronkhite {Amer. 
Mach., Nov. 21, 1912), the final conclusions being that: 

Steel belts from half to one-third, and in some cases one-quarter, 
the width of leather belts will do the same work as the leather belt 
without trouble. 



BELTS AND PUI.LEYS 



61 



Steel belts do not stretch or slip after being placed on the pulleys, 
and are not affected by variations in temperature to any perceptible 
extent, which makes them very reliable for use in damp places, such 
as laundries. They are especially adapted for use in paint and varnish 
works, as the accumulations of paint and other sticky substances can 
be washed off with gasoline and the belt kept in good condition. 

Being narrower than leather or other types of belting, they require 
puUeys of narrower face, which is an item in the equipment of a new 
plant or the installation of new drives in any factory. 

From the investigations made, it can be said that their first cost is 
considerably below that of leather or rubber belting. 

The experiments have demonstrated that steel belts are more sensi- 
tive than other types and that the shafts and puUeys on which they 
run must be in line and level or the belt will invariably run to the low 
side of the puUey, out of line, and will run off if the pulley is too much 
out of line. The use of canvas on the face of the pulleys is of decided 
advantage in connection with steel belting, as it forms a bed or cush- 
ion for the belt to run on, and at the same time greatly increases the 
pulling power of the belt. A special rubber covering, suggested by 
experiments, has proved satisfactory. 

In replacing a leather belt with a steel belt where the pulleys are 
crown faced, it is necessary to build the crown up to a flat face, as 
steel belts will not run on crown-faced pulleys. They will, of course, 
run on plain uncovered iron or steel, as well as on wood puUeys, but 
the use of the canvas or rubber covering is so beneficial that it seems 
almost necessary to good service. 

The following particulars regarding the practice of the Eloesser 
Steel Driving Belt Co., Ltd. of Manchester, England, are supplied by 
I. W. Chubb (Amer. Mack., May 14, 1914). 

At present the system is not regarded as suitable for comparatively 
small belts, as the cost of fitting and installation may considerably 
outbalance the mere material cost. In particular the length of belt 
has to be determined with considerable accuracy. For this purpose 
a small steel band of known section is mounted on the pulleys, driver 
and driven, and a tension frame is fitted to the ends of this measuring 
band. Using a calibrated nut and spring, the two ends of the frame 
are drawn together until the tension, as shown by a scale, is equal 
to that desired in the belt when running. One of the pulleys is then 
slowly rotated without driving the belt, the friction changing the 
tension indicated, while next the other pulley is rotated in the reverse 
direction, thus again changing the tension, and the mean of the two 
should correspond to the desired working tension. The band can 
then be cut to the exact length with ends meeting, and when removed 
from the tension apparatus will act as a template to the length of 
the driving belt itself. 

The material is stated to be charcoal steel, rough rolled at a red 
heat and then brought to from 2 mm. (.078 in.) to 9 mm. (.35 in.) 
thick by 12 mm. (.47 in.) to 200 mm. (7.87 in.) wide by cold working, 
the tensile strength claimed being about 212,800 lbs. to the sq. in. 

The sharp edge of the material is removed. The pulley should 
preferably be flat, any crowning not exceeding in height }i per cent. 
of the width of the steel belt. The rim of the puUey is covered with 
canvas and cork glued on in one length, a special cement being em- 
ployed for damp situations; the rim is first roughened by file or chisel 
cuts to avoid stripping. Above this covering a slight crown of cork, 
say H4 in- high, is glued, and with this the coefficient of friction be- 
tween belt and pulley is said to be equivalent to that between a 
leather belt and an iron pulley. If necessary, two or more belts are 
run in parallel, the ratio of width of belt to thickness being kept as 
high as possible. 

For jointing the belts steel plates are employed, milled to about 
the curvature of the pulleys, the belt ends being clamped and the 
belt itself brought by screws to the required tension, the ends being 
then threaded between the plates, which are secured by counter- 
sunk screws. The ends are, however, tinned first and solder is 
finally run into the joint by means of a blow lamp. A minimum 
efficiency of 99 per cent, is guaranteed and as the steel belt is about 



one-third the width of an ordinary leather belt, the pulleys may be 
narrower to that extent and consequently lighter. A steel belt, 
however, is not regarded as suitable for fast and loose pulleys, and 
where crossed belts are employed, the distance between the crossing 
point and either shaft must not be less than 70 times the width of 
the steel belt. For ordinary drives the puUeys may almost touch. 
Rope drives have been converted by fiUing the grooves with wood 
blocks. A cheaper method, however, is to cut a groove across the 
face of the pulley, into which is fitted a plate; to this is fastened a 
steel band shrunk round the pulley rim. It is stated that transmis- 
sions of more than 150,000 h.p. have been thus converted. 

Individual drives in use on this system range from 10 h.p. to 
3650 h.p., and some have been running for six years without showing 
stretch. In one case three belts are employed in connection with a 
roUing mill where the power varies from 600 to 1 200 h.p. in a second. 
In another case, in England, a couple of steel belts are employed for 
transmitting 450 h.p., showing a saving in power, as compared with 
the previous rope drive of 13 per cent. As to Germany, where 
Eloesser belts totaling some 200,000 h.p. are installed, in one case 
they displaced ropes transmitting 1650 h.p. from a single drum, 
while in another case 3660 h.p, was transmitted from one drum. 

Belt Shippers 

An improved belt shipper, which completely overcomes the common 
nuisance of the belt refusing to remain where.put, is shown in Fig. 25. 
The difiiculty is due to the weight of the shipper pole, which tends to 
bring the pole to the vertical position, with the belt half on each pul- 
ley. In this position the machine will not stand stiU if it has no work 
to do, and it will not drive if it has work to do. The arrangement 
shown gets rid of this effect of the gravity of the pole, with the result 
that the belt stays on either tight or loose puUey as desired. 

The sketch represents a shifter made of wood, the improvement 
consisting in having the pole play between pegs ab on the fork bar. 
The fork is shown in position to guide the belt to the loose pulley. 
The pole hangs in a vertical position against peg a. If the belt is to 
be shifted, the pole is pushed to the left as usual, and with the usual 
result, except that after the pole is dropped by the hand, it swings 
back by gravity to the vertical position again. The previous move- 
ment of the fork bar will, however, have moved the pegs to the posi- 



^^tP 



Fig. 25. — Improved belt shipper. • 

tions a'b', so that the pole will then be in contact with peg b in its new 
position b', ready to push the bar in the opposite direction whenever 
wanted, after which the pole will again return to the vertical position. 
The belt fork always stays where left, and, the pulleys being crowned, 
the belt also stays where put. The same result may be obtained by 
increasing the space between the forks — making this space span both 
pulleys instead of one, as usual. 



62 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Piilleys 

The dimensions of cast-iron pulleys may be obtained from the fol- 
lowing formulas and tables by J. W. See (Amer. Mach., July 23, 1881). 

B=AX.o62S + .s. 
C = ^X.o4 + .3i2S. 

Z)=^X02S+.2. 

£ = .4X.oi6+.i25. 

in which A = diameter of pulley, 

B = width of arms at center of pulley, 

C = width of arms at circumference of pulley, 

Z)= thickness of arms at center of pulley, 

£ = thickness of arms at circumference of pulley. 

All dimensions in inches. Change decimal results to the nearest 
sixteenths. 

Mr. See also supplies Table 6 of pulley dimensions, and the fol- 
lowing instructions: 

The pattern spiders should be of iron, parted, dowelled, the ends 
of the arms turned to size in the lathe, and the shallow recess H 
turned in the hub seat. The rims may be iron or wood, as policy 
suggests. The drawing shows how to shape straight and curved 
arms. The table gives dimension of arms where they would cross 
the rim and cross the center. The hub seat H is of such size as to 
receive quite a range of standard hub patterns, and make a nice, 
smooth job without sharp corners. The ends of the arms may be 
drilled to receive screws put through edge of rim to hold strings to- 
gether, if parted rim pattern is used. Some will prefer a single 
narrow rim to be drawn for any width. Some shops follow the 
vicious plan of casting all pulleys the full width of, say, a g-in. pattern, 
and then cutting to width in the lathe, using a special or drawing 
pattern for wider rims. 

The Rims. — Columns FG show the thickness at center and edge 
in the rough. The crown will be right for all widths. Pattern 
should be large enough to let casting finish to exact size — a matter 
very often neglected. All pulleys for general work should be 7 in. 
wider than belt. A good pulley trade calls for iron rim patterns of 
sundry widths to change on loose spider. 

The Spider. — The table gives size of arms at rim and center cross- 
ing, the diameter of the center web, radius of the fiUets, and diameter 
of the hub seat H, which is \ in. deep in all cases. The table makes 
the hub seat large enough to receive good-sized hubs, and still look 
right with small ones. 

To Draw Ctirved Arms. — Draw full size the diameter A; step off 
six points, a b c d c f; at each of these points strike circles C, of size 
given in table column C; strike circles at pulley center, sizes from 
columns B and I in table; with c for a center strike arcs h and i, the 
radius being to each side of circled; midway between these arcs and 
the points j and ^locate points I and m; with k for center and cl for 
radius strike arc n; with j as center and mc as radius strike arc 0; 
with kn for radius sweep inside of arm touching circle B, center being 
somewhere on arc w; with jo for radius sweep outside of arm, touching 
circle B, center being somewhere on arc o. 

For Straight Arms. — Draw lines touching circles B and C. Draw 
fillets p, touching edges of arms, and circle /. With one-half 
of /, minus F for radius, cut off the arms. Radius of q equals 
one-half of C. 

The edge view, or section of arms, as in Fig. 27, is made by circles 
E E and D from table, and side lines touching these circles. Radius 
of y equals E. Make these fillets nice, and thus avoid all sharp 
internal angles. 

Section of Arm, Fig. 29. — Draw circles r and s, representing width 
and thickness of arm; make / «« equal to v w; with u v for radius and 
« as center, draw sides of arm; put in circle x, touching the sides and 
the circle r. The fiUet p should have half circle section and present 
pure blended surface. 



The Hub Pattern, Fig. 30. — The intention is to have the hub patterns 
fit all pulley patterns within reason. Table gives diameter and 
lengths. The flanges should fit easily in the hub seats in the spider 
patterns. Radius of y is | in. in aU cases. FiUet is quarter circle. 
Hub patterns should be of wood. Core prints should be turned on 
the pattern solid. The prints are one size on all hubs. Make full 
set of straight core boxes i ft. long, and have in each two sliding 
ends to give shape of prints. By this means but few core boxes are 
needed, and the hubs and cores will interchange nicely for all common 
work. Taper both prints if desired. 

The above formulas and tables make no distinction between 
pulleys for single and double belts. For double-belt pulleys the 
author suggests the formulas: 

C =^X.o5 + .75, 

E =iC, 

F and G i more than for single-belt pulleys. 

The only suitable number of arms in a pulley, wheel or gear 
which is to be chucked by the arms is a multiple of 3, as such numbers 
permit strapping at three points without distortion. 

The above formulas and table are suitable for all ordinary cases 
of stock pulleys. For special cases and extra large pulleys, Fig. 
31 by S. E. Freeman (Amer. Mach., Dec. 3, 1896), which gives the 
practice of the Todd and Stanley Mill Furnishing Co. may be used. 
As will be seen, it is adapted for use in laying out rope sheaves as 
weU as belt pulleys. 

To use the chart, substitute the given dimensions in the proper 
formula; find the value pf the quantity under the cube root sign. 
Find this same quantity on the base line and trace upward to the 
various lines where read the required dimensions. Examples will be 
found below the chart. 

For the design of hollow pidley arms from their solid equivalents, 
see Arms of Spur Gears. 

The static strength of belt pulleys formed the subject of experiments 
by Prof. C. H. Benjamin {Amer. Mach., Sept. 22, 1898). The 
general conclusions arrived at are as follows: 

1. That the bending moments on pulley arms are not evenly 
distributed by the rim, but are greatest on the arm near the tight side 
of belt. 

2. That there are bending moments at both ends of the arm, 
that at the hub being much the greater, the ratio depending on the 
relative stiffness of rim and arms. An increase of the width of rim 
will undoubtedly help the arms. 

The rules deduced from the experiments for the rational design 
of cast-iron pulleys are as follows: 

1. Multiply the net pull of belt by a suitable factor of safety and 
by the length of arm in inches. Divide this product by one-half the 
number of arms and use the quotient for a bending moment. Design 
the hub end of arm by the usual rules to resist this moment. 

2. Make the rim ends of arms one-half as strong as the hub 
ends. 

The surphis of pulley face over belt width may be obtained from 
Fig. 32, by Carl G. Barth {Amer. Mach., Feb. 11, 1915) of which 
the lower line gives no surplus and is introduced for purposes of com- 
parison. The middle line gives about the surplus usually pro- 
vided, while the upper line gives the larger surplus which Mr. Barth 
uses and finds advantageous. The use of the chart is self-explanatory. 

The appropriate height of crown for belt pulleys according to Mr. 
Barth is given in Fig. 33. More usual proportions are given in 

Fig- 34- 

Parting split pulleys'hsli-vfz.yhetvieen the arms, Fig. 35, is a source of 
danger at high speed, as has been demonstrated by the experiments of 
Professor Benjamin (see Bursting Strength of Fly-wheels). This 
location of the joint is even worse in pulleys than in fly-wheels because 
the thinness of the rim provides less strength to resist the centrifugal 
bending stress. The construction shown is particularly bad because 



BELTS AND PULLEYS 



63 




the absence of a joint at the inner ends of the lugs aggravates the 
other bad conditions. Fig. 36, by Professor Sweet {Amer. 
Mach., Jan. 12, 1905), is a well considered design in which the weak- 
ness due to the parting is practically eliminated. 

Overhanging pulleys should be avoided, but when that is impossible 
the usual construction, Fig. 37, may be greatly improved by adopting 
the plan shown in Fig. 38. 

In Fig. 37 the pulley A is secured by the set screw C to the driving 
shaft B, which runs in the bushing E carried in the bracket D. The 




Fig. 28 



Fig. 30. 













Table 6 


— Dimensions 


OF 


Cast-iron Pulleys 
















a 
'0 
S 

S 


6 

M 


B 
- B 


1) 

.y s 


s 

tn 


1.4 


bO 


5- 


2 

u 

ni 

0) 


C 


S 

5 


s 

° c 

•5 S 
-a ^ 


S 

II 


+-> 

1) CO 

c « 

H 


=1 

•86 


.y B 




'0 


•73 
U 

(5 


A 


B 


C 


D 


£ 


F 


G 


F 


7 


.1 


5 


c 


D 


£ 


£ 


G 


H 


I 


6 


7 

8 


A 


5 

16 


1 
1 


A 


3 

16 


2f 


3 


SO 


3f 


2A 


lA 


1 


4 


f 


7 


9 


7 


15 
16 


9 

16 


1 


1 
4 


A 


A 


2i 


3 


52 


3f 


2f 


li 


15 

16 


1 
2 


3 

8 


7 


9 


8 


I 


f 


3 

8 


1 

4 


5 

16 


3 

16 


2f 


3 


54 


3i 


2A 


ll 


M 


A 


A 


7 


10 


9 


life 


1 


A 


1 
4 


A 


A 


3^ 


4 


56 


4 


2i 


lA 


I 


9 

T6 


7 

16 


7 


10 


10 


li 


tt 


7 
16 


1 

4 


A 


A 


3^ 


4 


58 


4i 


2| 


if 


I 


A 


A 


9 


II 


II 


lA 


3 

4 


A 


A 


A 


A 


3* 


4 


60 


4i 


2ii 


iH 


lA 


A 


A 


9 


II 


12 


li 


il 


1 
2 


5 

16 


5 

16 


A 


4i 


5 


62 


4! 


2f 


if 


lA 


f 


1 

2 


9 


II 


13 


lA 


13 

16 


4 


A 


A 


A 


4i 


S 


64 


4i 


2i 


iH 


li 


f 


i 


9 


12 


14 


if 


7 
S 


1 
2 


5 
16 


5 

16 


3 

16 


4i 


S 


66 


48 


,15. 
2l6 


lil 


li 


f 


1 
2 


9 


12 


15 


lA 


1 


A 


3 

8 


A 


A 


4i 


S 


68 


4i 


3 


If 


lA 


5 

8 


f 


9 


12 


16 


i| 


H 


A 


3 

8 


A 


A 


4i 


S 


70 


4l 


3 16 


iM 


lA 


5 
8 


1 
2 


9 


12 


17 


if 

li 

^^ 
2 


- 1 

T 3 
i 16 


5 

8 

JU. 
16 
11 
16 

3 

4 


f 
3 
8 

1 
2 


1 
3 

8 
1 
f 

f 
1 


1 
4 
1 
4 

1 
4 

1 
4 

1 
4 
1 
4 


4i 
4i 
4i 
4i 

5 
S 


5 

5 
S 
S 

6 
6 


72 


5 


3A 


2 


li 


f 


4 


9 


12 


18 


Hubs 


19 

20 


] 


diameter 


of shaft 




Diameter 
of hub 


Length 
of half 


Diameter 
of print 


22 


/ 


K 


L 


24 




I 


to li 






2i 


li 




i 


26^ 


2i 


if 


13 
16 


1 


1 


i 


5 


6 




lA 


to l| 






2f 


li 




If 


28 


2\ 


_ 7 
■•■ 16 


7 
8 


9 
16 


3 
8 


1 
4 


S 


6 




lA 


to if 






3i 


if 




li 


30 


2| 


i4 


a 


9 
16 


1 


i 


S 


6 




2A 


to 2 

to 2i 






3l 
4 


2 

2i 




i4 
If 


32 


2i 


lT6 




5 
8 


7 
16 


5 

16 


S 


7 




















34 


2| 


If 




f 


A 


A 


5 


7 




2A 


t0 2i 






4i 


2i 




2 


36 


2i 


T 3 
-■■4 


T-l 
■■■ 8 


ii 


7 

16 


5 

16 


S 


7 




2A 


t0 2f 






5 


2f 




2i 


38 


2i 


iH 


li 


H 


A 


A 


S 


7 




2H 


to 3 






5i 


3 




2i 


40 


3 


l| 


T-3- 

'- 16 


3 
4 


7 

16 


A 


7 


8 




3i 
3f 


to 3* 
to 4 






6f 
7i 


3i 
4 




2f 
3i 


42 


3i 


2 




3 
4 


7 
16 


5 

16 


7 


8 




















44 


3^ 


2 


T 5 
■l 16 


H 


7 
T6 


5 
16 


7 


8 




4i 


to 45 






8i 


4^ 


1 


3f 


46 


3l 


2i 


l| 


M 


A 


A 


7 


8 




45 


to 5 






9 


5 


4i 


48 


i\ 


2| 


■1 16 


7 
8 


1 
2 


3 

8 


7 


9 



















64 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Kind of Whee 








Formula for h 




h = Width of Arm at Center of Hub, Ins. 




3/ DW 1" 
J ia 4 


= Thickness of Arm at Center of Hub = .4 h. 
A'= Width of Arm at Rim = .67 h. 
6'= Thickness of Arm at Rim = .4 h^ 


.^(^"fV' 


Single Belt Pulleys 


In ^r' 






5^^-" 




3 / ZDW 1 " 
^ 8a "^ 4 


^ " ""' 






-^ 




Double Belt Pulleys 




^ 










^ — Diam, of Wheel, Ins. 


^ 




_ 




- 






3/ Td^^ni? 1" 
^ 10 a "*" 4 


Pr= Width of Belt. Ins. 

d = Diam. of Rope, Ins. 4 


^fflfl~ 




— 


~ 




- 




Manilla Rope Sheaves 


















l<ff ' 


















^ 
















- 


n == No. of Ropes or Groovea ^^flT 
















a = No. of Arms 


^^ 














v 


^ ,_„ __. 














'^\-- 


_^ 














^ -^ 






_ __ 














^-'^ 


3'4^ 


1^ 


















^-- 




















^ 


-' 




-^ 


' 


















. — ' 










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^ 




















. — ' 
















o^ 


r 






















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^^ 






















^ - -" 
















, 


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^^'' 
















yf 
























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2^2 


,-' 


•^ 


























^■- " 














"b 






^ 














































^ 


^ 
















































_^_-— -" 




^ 


^ 


















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-- — ^"^ 


2 




















,-^ 


•" 


























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— ■ 










y 


-^ 


















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.---■*■ 












■ 






^ 




















^ 


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^ 




















^ 






























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D 


iVz^ 


^ 
















r-^ 




































^ -- " 














„ — — — " 




f' 
















.— ' 




























^ 

























— — — - 


^ 












, ■ 


























^ 




r" 




































^ 


■^ 


























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-^ 








































-^ 


























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'" 
























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„^^-^ 




















. 






" 








1 








__ 




— 






























































— " 


"" 








































- 


^ 


















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1 ■" 








































_^^ 




■ 
















— ■ 




























































~ 
















































































































































































































^^^ 


mm^ 












































_ 












1 















5^2 

5 

4H 
4 

01 

» 

.£1 
3^2 I 

a 

i 

3 <j 



2H 



2 S 



1J^2 



70 80 90 100 



12S 



150 



175 200 



1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 30 40 50 60 

Number under the Cube Root Sign 

To find the dimensions of the arms of a 48 in. single belt pulley having 6 arms and for a 12 in. belt: substituting these factors in the quantity under the 
cube root sign for single belt pulleys gives '^ ^,V ^ 24. Locate 24 on the bare line, trace upward and read h=z\, &=ij, fe'=2|, and 6' = J, all in ids. 

Again for a rope sheave 8 ft. diameter having 6 arms and for 8, if in. ropes: substitute as before in the proper formula and obtain - 
Locate 140 on the bare line, trace upward and read h= 5^, h'= 3I, b= 2^ and b'= i| ins. 

Fig. 31. — Dimensions of arms of belt pulleys and rope sheaves. 



= 140. 




;3V2 

> 
1 

'2^2 



IVi 



7 




I I I 



U 



jHund 



123456789 

Width of Belt, Ins. = 5 
Fig. 32. — Surplus of pulley face over belt width. 



10 11 12 



4 



7 8 9 10 11 12 13 1415 16, 18 19 of an Inch, 

6" 3" 1" 6'.' 17 



5 6 

64 16 G4 i 

Crown of Pulley 
Fig. 33. — Height of crown of belt pulleys. 



BELTS AND PULLEYS 



65 



collar F, which is taper-pinned to the shaft prevents end play. This 
design is bad, as the bush E wears bell-mouthed at the pulley end and 
the bending effect on the shaft due to the pull of the belt on the pulley 



^^^m. 



m?^ 



CrowningL 



t: 



Crowning | Width of Pace 



fs 


Under 6" 


A 


6 to 12 


i 


12 " l8 


A 


iS " 24 


A 


24 " 30 


■h 


30 ■' 36 


i 


36 " 48 


A 


48 " 60 


f 


60 & Over 




Fig. 34.- 



-Crowning for belt 
pulleys. 



Fig. 35. 

Fig. 36. 

Figs. 35 and 36. — Correct and in- 
correct parting of split pulleys. 



increases as the wear on the bush increases. This gives combined 
bending and torsion on the shaft in transmitting the drive. 

In the improved design, Fig. 38, these difficulties are overcome. 




Fig. 37. 



Fig. 38. 



Fig 

39. 

Fig, 

40, 



5 



1 



Figs. 37 and 38. — Correct and incorrect design of overhung pulleys. 

The bush is prolonged and the pulley runs upon its periphery. The 
drive is transmitted through the collar G, which is secured to the 
pulley, and also taper-pinned to the shaft. The collar F is the 
same as in Fig. 37. Thus the shaft is sub- 
ject to torsion alone, or practically so. 

The correct arrangement oj tight and loose 
pulleys is shown in Fig. 39, by Professor 
Sweet (Anier. Mack., Jan. 12, 1905), the 
hub of the tight pulley being shortened and 
that of the loose pulley lengthened at both 
ends to make it central with the pulley 
face. Fig. 40 sacrifices length of bearing 
where it is most needed, and Fig. 41 is cer- 
tain to wear bell-mouthed. The chambered 
construction, Fig. 42 is appropriate on tight 
pulleys only. 

A superior construction of counter-shaft 
pulleys by Carl G. Barth {Amer. Mach., 
Feb. 18, 1915) together with standardized 
dimensions is shown in Fig. 43 and the ac- 
companying table. The loose pulley is 
smaller than the tight pulley in order to re- 
lieve the tension on the belt when it is doing 
no work, a beveled edge being provided to 
assist the shipping of the belt. The surplus 
of pulley width over the belt width is greater than is customary. A 
stationary sleeve or quill of cast iron is provided as a bearing for the 
5 



I 



Fig 

41 



42. 

Figs. 39 to 42. — Cor- 
rect and in correct ar- 
rangements of tight and 
loose pulleys. 



loose pulley. The effect of this is to confine the wear to one side 
of the bush which continues to fit the hole regardless of wear, which, 
however, is almost negligible. A grease cup feeds into an annular 
chamber around the shaft, from which three grooves run the length 
of the sleeve inside and passages connect with three similar grooves 
outside to feed the surfaces where the shaft and loose pulley run. 

When arranged as usual, loose pulleys are much more effectively 
lubricated with grease than with oil, the former remaining in place 
much better than the latter. For small pulleys, the grease cup may 
be tapped into the end of the shaft — a suitable hole lengthwise the 
shaft and another crosswise within the pulley hub carrying the grease 
to the bearing. 



HJ= 



,.,,,/J>V/A'/,J)Wy^,^^ 



-F 







=4h 



'III ' i : I . : u____i.' ' 



.,„„;MI/,i,„r7T7H7 



6t4t 



,^...^^^.....AYr' A.-'J^'VA'AJ 




2" jpi smof^ill-Zfi-f 



W^^^ , 






V-A 



Section X-X 



j^2pt!& 




So ^/efTVm Yieop ioZ^ 
■^ -A ^^ shaff, ht!' from ZM shaft up 



Fig. 43. — Dimensions of tight and loose pulleys. 



Size of 


Sizes relative to shaft, j 


Width 


Sizes relative to belt. 


shaft, 


ins. 




of belt, 






ins. 




ms. 






ins. 










S 


K 


i 


/ 


M 


N 


p 


c\ 


B 


F 


G 


L 


H 


E 


W 


i\h. 


3/4 


2% 


3% 


2],i 


I 


m 


H 


2 


234 


iH 


5K 


He 


Me 


aH 


1% 


3% 


3^ 


4'A 


2lii 


i\i 


Vi 


Vi 


2H 


3M6 


iH 


SMe 


He 


Me 


4Me 


2 


4 


3Vi 


4K2 


2% 


m 


Vi 


M 


2H 


3M6 


iH 


5'Me 


He 


Me 


A'Vii 


2H 


m 


3% 


4% 


3'Ai 


iH 


H 


Vi 


2M 


3% 


iMe 


(>Vi 


5^4 


H 


sMe 


2H 


aH 


4M 


sH 


3H 


1% 


Vie 


Vi 


3 


3' Me 


iMe 


6M6 


M64 


H 


5 A 


2M 


5^ 


4K2 


SH 


3' Me 


m 


Me 


H 


3« 


4Me 


iMe 


61 Me 


%i 


H 


sH 


3 


sVi 


4% 


6 


4 


m 


Me 


ii 


3H 


AM 


iMe 


l^A 


%i 


H 


6He 


3H 


sli 


5\i 


6% 


4M6 


iH 


Ms 


H 


3?4 


^'Me 


i?« 


7Me 


%2 


Me 


6H6 


3M 


dVi 


SYi 


6% 


A^A 


m 


?4 


H 


4 


5>/« 


m 


7% 


H2 


He 


6H 


4 


7 


6M 


Hi 


5 '4 


iH 


H 


'A 


AVi 
5 

6 


S^He 

6M6 

6% 
7H 


iMe 
i?ie 
iK' 


8Me 
pMe 
9% 
loW 


H2 

764 

A 


He 
'A 
A 
Me 


7Me 


K = 
M= I. 
in. J 


[.S5 

2S5 

= 1.5 


-Hi 
+-K 
5-1- 


in 
iH 


= 1.37 

m. 


sS-\ 
062. 


-% 
>5 + 


n. 


8 

8?l6 



F = iyi6B+% in. 


G = ii + 'M6 = 


HeB 


-l-iJ'^ in. L = F + 2G. 


H = A2F %. 


£ = 


HeB+Me in. 







A self-oiling loose pulley is shown in Fig. 44 by H. J. White 
{Amer. Mach., June 22, 1905). The bushing is of hard composi- 
tion and the oil holes are plugged with hard felt or rattan. Mr. 
White says that with j pt. of oil in the oil space these pulleys will run 
three months without attention. 

The htirsting strength of pulleys of various materials and construc- 
tions formed the subject of experimental tests by Prof. C. H. 
Benjamin {Journal A. S.M. E., June, 1910) similar to those on fly- 
wheels (see Bursting Strength of Fly-wheels). The results are given 
in Table 7. The cast-iron pulleys Nos. 11 and 12 were not fractured. 



66 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 




Table 7. — Results of Bursting Tests of Belt Pulleys 



Fig. 44. — Self-oiling loose pulley. 

F. P. Read {Power, Apr. 22, 1913) reports the repeated failure, 
at a rim speed of 593 7 ft. per min., of 84 X 1 2 ins. cast-iron split pulleys 
of the usual type with lugs and bolts half-way between the arms. 





Kind of 
material 

in 
pulleys 


Rim 


Weight 
pounds 


Bursting 
speed 


No. 

of 

test 


Style 


m 

u 

u 
u 

D 

s 

5 


'Si 



V. 

m 


Depth 
inches 


r.p.m. 


13 
u 

- . 
a 

c^ en 
(In 


. I 
2 
3 
4 
S 

6 

7 
8 

9 

10 

II 
12 
13 
14 

IS 

16 


wood 
wood 
wood 
wood 
wood 

wood 
wood 
wood 
wood 
wood 

cast-iron 

cast-iron 

paper 

paper 

steel 

steel 


solid 

solid 
2 sections 
2 sections 
2 sections 

2 sections 
2 sections 
2 sections 
2 sections 
2 sections 

solid 
solid 
solid 
solid 
2 sections 

2 sections 


24 

24 
24 

24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 


6.2s 
6.2s 

6.S 
6.S 
6-5 

6.5 
6.S 
6-5 

(>-s 

6.5 

6.0 
6.0 
6.0 
6.0 

6.75 

6.75 


1.62 
1.62 
1.78 
1.78 
1.78 

1.78 
1.78 
1.78 
1.78 
1.78 

0.406 
0.406 
I 75 
1-75 
0.0625 

0.0625 


29.37 
29.37 
29.67 
29.67 
28.81 

28.81 
28.81 
28.81 
28.81 
28.81 

70.44 
70.44 
77-37 
77.37 
41-75 

41-75 


2720 

2550 
2210 
2110 
2390 

2430 
2360 
2420 
2570 
2535 

3720 
3380 
2820 
2930 
2240 

2240 


284.7 
266.9 
231.8 
220.8 
251.0 

254-3 
247.0 

253-3 
258-5 
244.4 

389-4 
353-8 
295.2 
306.7 
234-5 

234. 5 



I 



FLY-WHEELS 



1000. 
,900. 


LO. 


11.5 


12.5 


14. 15. 16.5 18.5 20. 


22.5 


25. 




Badius, Ins. 

30. 35. 40, 


15. 50. 59. 60. 65. 70. 


80. 90. 10 


1 




L4. 1 \\\ 1 i Q=n^ 


U^ . 4r V— 


P 




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^ 


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>^ \^ \^ |\^ l^i^ 

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^_ V 54 s ^ 


Mi 


:::s;: 3 


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800. 


§ 

5 
^ 


i 


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i 


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i 




i— 

1 


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s — ^ ? 

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700. 






ffm irii|ii!ixii|ii|!|[i; 


650. 


-Si|g-ipi5'"ilt i^^i^ 






600. 








































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650. 





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500. 




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450. 


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17.5 20. 22.5 25. 



30. 35. 40. 45. 50. 55. 60. 65. 70. 

Radius, Ins. 



80. 90. 100. 



To find the rim tension in a cast iron -ftrheel 40 ins. radius running 350 r.p.m. : Find 40 on the base line and 350 on the vertical scale ; 
trace to their intersection and read 1450 lbs. per sq. in., rim tension. For a wheel of 400 ins. radius read as for 40 (that is, -3^) and 
multiply the stress by 100, and so for 4 ins. radius read as for 40 (that is, 4X 10) and divide the stress read by 100. 

Fig. I a. — Centrifugal tension in cast iron fly wheels. 



Stresses in Fly-wheels 
The stress in a ring revolving about an axis passing through its center 



For cast-iron having 10 = . 26 this becomes: 
5 = .o97»^ 



(6) 



due to centrifugal force is similar to that in a boiler shell due to inter- ^""^ ^^^^^ ^a.vmg w = .28 it becomes: 



nal pressure and is given, for any material, by the formula: 

in which 5 = stress on section, lbs. per sq. in. 
w = weight of material, lbs. per cu. in. 
V = velocity of center of gravity of rim, ft. per sec. 



5 = .1045^2 (c) 

For both iron and steel it becomes, with sufficient accuracy for fly- 
(d) wheel calculations: 



10 



{d) 



To find the total stress on the section for the calculation of the 



67 



68 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



dimensions of link and other joints, multiply the stress per sq. in. by 
the area of the section in sq. ins. 

The rim tension may be obtained without calculation from Fig. i 
by P. MuLLER (Amer. Mach., Nov. 28, 1901). The diagram for steel 
will also serve for wrought iron, which has practically the same 
specific gravity. The use of the charts is shown by an example 
below them. 



V = 443Ve (/) 

Before the experiments of Prof. C. H. Benjamin (summar- 
ized below) were published, the value of unity would have been sub- 
stituted in formula (/) for the efficiency of construction of a wheel 
cast in one piece. Those experiments show this procedure to be 
incorrect, such wheels giving way at velocities materially below those 
to be expected from the tensile strength of the material — the efficiency 



90. 100. 




15. 17.5 20. 22.5 25 



30. 35. 40. 

.Radius, Ins. 



60. 65. 70. 



To find the rim tension in a steel wheel 40 ins. radius running 350 r.p.m. : Find 40 on the base line and 350 on the vertical 
scale; trace to their intersection and read 1558 lbs. per sq. in., rim tension. For a wheel of 400 ins. radius read as for 40 (that 
is -V^) and multiply the stress by 100, and so for 4 ins. radius read as for 40 (that is, 4 X 10) and divide the stress read by 100. 

Fig. i&.— Centrifugal tension in steel fly wheels. 



The velocity of the rim at which bursting may he expected is given, for 
any material, by the formula: 

(e) 



te 



V=iM\- 
\w 

in which V = bursting velocity of rim, ft. per sec. 

t = tensile strength of material, lbs. per sq. in. 

w= weight of material, lbs. per cu. in. 

e = efficiency of construction, for values of which see Table i . 
For cast-iron, taking 19,000 lbs. per sq. in. as the tensile strength and 
.26 lb. per cu. in. as the weight, this becomes: 



of the construction being .85. Had deeper rim sections been used 
in the experiments, a larger value would probably have been found. 
The efficiencies to be substituted for other constructions are given in 
Table i. 

For steel, taking 60,000 lbs. per sq. in. as the tensile strength and 
.28 lbs. per cu. in. as the weight, the formula becomes: 



^=7S7\/e 



ig) 



No experiments have been made on steel wheels to determine 
their actual efficiencies of construction. 



FLY-WHEELS 



69 



The most essential J act disclosed by these formulas is that the stress 
increases with the square of the speed, doubling the speed multiply- 
ing the stress by four and neutralizing a factor of safety of four based 
on the stress. Much greater increases of stress are therefore pos- 
sible with fly-wheels than with steam boilers, and fly-wheels are 
correspondingly more dangerous than boilers. The Fidelity and Casu- 
alty Company, which insures both boilers and fly-wheels, finds the 
hazard on fly-wheels materially to exceed that on boilers. 

These formulas should he used with caidionwhen designing fly-wheels, 
as they are now known to have much less direct application than was 
formerly supposed. The condition of simple tension assumed, 
while true for an ideal revolving ring without arms, is seriously modi- 
fied by the action of the arms of actual wheels in restraining the free 
expansion of the rim. Because of this restraint, each rim section 
between adjacent arms is in the condition of a beam under a uniformly 
distributed load, the load being the centrifugal force of the material 
of the section and the stress due to this beam action is added to the 
simple tension stress, the resulting stress being always greater than 
that given by the above formulas. 




Fig. 2. — Method of failure of fly-wheels with flanged joints. 

The beam action is especially serious in the case of built-up wheels 
with joints located, as usual, half-way between the arms. A joint 
in this position is equivalent to a joint in the middle of a beam and 
not to a simple splice in a tension member. 

The beam action may be reduced by increasing the number of arms. 
Such increase reduces both the weight and length of the segments 
and the fiber stress due to the beam action — not the total fiber stress 
— is, hence, other things being equal, inversely as the square of the 
number of arms. 

Attention was first called to this beam action by J. B. Stan wood 
{Trans. A. S. M-. E., Vol. 14) and the truth of his analysis has been 
experimentally proven by Professor Benjamin {Trans. A.S.M. E., 
Vols. 20 and 23), who tested model fly-wheels to destruction by 
revolving them in a bomb-proof casing at increasing speeds until 
they gave way. Fig. 2, from a photograph of an actual case, shows 
the manner of failure of the common flanged and bolted joint located 
midway between the arms, and demonstrates not only the reality of 
this beam action but its preponderating importance in wheels of this 
construction. 

It is to be especially noted that, in repeated instances, wheels with 
this type of joint gave way through the solid rim and without failure 
of the bolts, as shown in Fig. 2, although the strength of the bolts 
was less than one-third that of the rim section, showing that the 
strength of this joint, as calculated in the usual way from the strength 
of the bolts, has nothing to do with the effective strength of the wheel. 

The results of these experiments have been corroborated by the experi- 
ence of the fly-wheel insurance department of the Fidelity and Casualty 
Company which has had several cases of failure of band fly-wheels 
with joints of the type shown in Fig. 2, in which the joint section 
went bodily out of the wheel and, in two cases, without affecting the 
remainder of the wheel or even bringing it to a stop. 

The beam action becomes an increasing factor as the radial dimen- 
sion of the rim decreases and is at its maximum in thin-rim belt 
pulleys. 

Wheels having j oints at the points of contrary flexure, that is, at one- 
fourth the distance from one arm to the next, have been repeatedly 
proposed as better adapted to meet the conditions of the beam action 
than those placed midway between the arms. Such wheels were 
tested by Professgr Benjamin and found not to be appreciably 
stronger than those of the midway joint construction. 



Professor Benjamin's experiments are summarized in Table i. 
The figures of the table are the averages of the experimental results, 
the number of wheels of each type tested ranging from two to four, 
except in the case of column 5 of which construction but one was 
tested. 

Regarding the wheel in column 3, Professor Benjamin considers 
that " if the tie rods had been more carefully designed and constructed, 
a greater speed could have been attained." 

For similar tests of belt pulleys see Bursting Strength of Belt 
Pulleys. 

W. H. Boehm, superintendent of the boiler and fly-wheel insurance 
departments of the Fidelity and Casualty Company, has calculated' 
the very useful Table 2 of safe speeds of cast-iron wheels of various 
types. The table is figured for a margin of safety, based on speed, of 
approximately three or a factor of safety, based on the stress, of nine. 
The table assumes the solid wheel to have an efficiency of construction 
of unity, which is not borne out by Professor Benjamin's tests and 
the table doubtless slightly overestimates the strength of the wheels. 
The Fidelity and Casualty Company accepts for insurance wheels 
having a factor of safety on stress of five, equivalent to a margin of 
safety on speed of 2.24. The company frequently insists on the 
addition of tie rods. Table i, column 3, to wheels with bolted flange 
joints. 

The fly-wheel cast in one piece is subject to uncertain initial strains 
due to shrinkage, but it is, nevertheless, by far the best of all common 
constructions. This is shown by Professor Benjamin's experiments 
and is, moreover, shown by common experience in which the failure 
of such wheels is the rarest of accidents. 

Constructioii of Fly-wheels 

In the design of wheels cast in one piece, the uncertainty of the shrink- 
age strains makes calculations regarding the strength of the arms of 
more than doubtful value. The author's empirical formulas for the 
arms of such wheels {Amer. Mach., April 23, 1896) have been used 
in the design of wheels from 33 ins. to 8 ft. diameter, and have been 
compared with wheels up to 20 ft. diameter with very satisfactory 
results. The formulas contain a factor for the diameter and another 
for the cross-section of the rim together with the usual constant. 
The author prefers a rectangular section having its greatest dimension 
radial, as it best resists the beam action, but the formulas provide 
for other sections by considering all sections of the same area as equiv- 
alents and taking the side of a square equal in area to the section 
as the base of the factor for the section. 

Referring to Fig. 3 for the notation, the formulas for the arm 
section at the outer end are: 

x = l in. -j-. 04^4-.! 53c 
J- y = hx 

all dimensions being in inches. The author's preference regarding 
the dimensions a and b is to make b = fa. 

The taper of the arms each side the center line should be from 
i to f in. per ft. in the side view and | to j^ in. per ft. in the edge 
view, depending on the size of the hub. The arm section is preferably 
that made by two circular arcs rounded over at the edges, as shown 
in Fig. 2, such section having a much more pleasing appearance 
than the more usual ellipse. The arms are usually six in number, 
but the same formulas may be used for a greater number of arms. 

For many cases in which a fly-wheel is desired but without definite 
requirements to permit calculations of the section for weight, satis- 
factory wheels will be obtained by making 
c = i in.-j-.o8i 

A superior fly-wheel by the Mesta Machine Company is shown in 
Fig. 4 {Amer. Mach., July 20, 1911). This wheel, which is of 17 ft. 
diameter, was designed for a rim speed of 10,000 ft. per min. The 
material is air-furnace iron having a tensile strength of 30,000 lbs. 
per sq. in. The wheel was divided as shown in order to reduce the 



70 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



spongy center of large sections, and the rim section is deep to reduce 
the beam action. The arms were cast with the rims but free at the 
hub ends, the hub being a separate casting of steel. Long sweeping 
curves connect the arms with the rim and hub ends. Complete cal- 
culations were made for the stresses at various sections, the extreme 
values being, for the arm section 3000, and for the rim section 2410 
lbs. per sq. in. 

Another superior fly-wheel (patented by G. M. Hinkley) is shown in 
Fig. 5 (Amer. Mach., May 17, 1900). It is used by the Allis Chalmers 
Company in their band-saw mills in which the rim speeds are regu- 



larly 10,000 ft. per min. — a figure that has, in some instances, been 
run up to 12,000 ft. At the date of publication about 300 of these 
wheels had been made, none of which had failed. The aim of the 
construction is to enable the wheel to relieve itself of shrinkage 
strains in cooling. The arms are arranged diagonally and pass from 
one side of the wheel rim to the opposite end of the hub, alternate 
arms being staggered with one another. As first cast, the hub is in 
two pieces, the central portion marked a being vacant. After the 
wheel has become entirely cold, this space is filled by pouring in mol- 
ten iron. As poured, the ends of the hub are separated by a core f in 



Table i. — Summary of Professor Benjamin's Experiments on the Strength of Fly-wheels 



Solid wheel, 
6 arms 




Wheel in 
halves, flange 
joint, 6 arms 




Wheel in 
halves, rein- 
forced joint, 
6 arms 




4 

Wheel in 

halves, link 

joint, 6 arms 




Segmental 
wheel, link 
joint, 8 seg- 
ments 





SoUd rim 

with separate 

spider, 

6 arms 




Solid rim 
with 24 tan- 
gent spokes 




Rim speed 



at failure 



feet per 
second 
I feet per 
[ minute 
Apparent rim tension at 
failure., lbs. per sq. in. 
by formula {d) 
Comparative rim speeds 
at failure 
Comparative rim tensions 
at failure 
EfiBciency of construction, 
e in formulas (e) and (/), 
assuming 19,000 lbs. per 
sq. in. tensile strength of 
cast-iron 



395 
23,700 
15.625 



•85 



194 

11,640 

3,764 

49 

24 

.19 



225 


305 


13.500 


18,300 


5,062 


9,302 


57 


77 


32§ 


60 


.26^ 


•49 



256 

15.360 

6,502 

6s 
42 

•34 



223 

13,380 

4,973 

56^ 
32 
.26 



393 

23,580 
15,445 



99 
.84 



424 

25,440 
17,978 

107 
"5 
• 94 



H-24- 




c='\lax b 



i 



"« 






i-f-4- 



1- 



Fig. 3. — Arms of fly-wheels cast 
in one piece. 




Fig. 4. — The Mesta fly-wheel. 



FLY-WHEELS 



71 



Table 2. — Safe Speeds of Cast-iron Fly-wheels. Margin 
OF Safety on Speed Approximately Three. Figures for 
Pad Joint do not Seem to be Justified by Professor 

Benjamin's Experiments 

Type of wheels and maximum obtainable efficiency of rim-joint 
No joint Flange joint Pad joint Link joint 

I. 00 .25 .50 .60 





Diam. in 


Rev. per 


Rev. per 


Rev. per 


Rev. per 


feet 


mm. 


mm. 


mm. 


mm. 


I 


1910 


955 


1350 


1480 


2 


95S 


478 


675 


740 


3 


637 


318 


450 


493 


4 


478 


239 


338 


370 


5 


382 


191 


270 


296 


6 


318 


159 


225 


247 


7 


273 


136 


193 


212 


8 


239 


119 


169 


i8s 


9 


212 


106 


150 


164 


10 


191 


96 


135 


148 


II 


174 


87 


123 


135 


12 


159 


80 


113 


124 


13 


147 


73 


104 


114 


14 


136 


68 


96 


106 


IS 


128 


64 


90 


99 


16 


120 


60 


84 


92 


17 


112 


56 


79 


87 


18 


106 


53 


75 


82 


19 


100 


50 


71 


78 


20 


95 


48 


68 


74 


21 


91 


46 


65 


70 


22 


87 


44 


62 


67 


23 


84 


42 


59 


64 


24 


80 


40 


56 


62 


25 


76 


38 


54 


59 


26 


74 


37 


52 


57 


27 


71 


35 


50 


55 


28 


68 


34 


48 


S3 


29 


66 


33 


47 


SI 


30 


64 


32 


45 


49 



If the revolutions given in the table be increased 20 per cent, the 
margin of safety on speed will be reduced to two atid one-half; if the 
revolutions be increased 50 per cent, the margin of safety will be 
reduced to two. 

thick, but the shrinkage of the rim compresses the arms and increases 
this space to about i| ins. After pouring the central portion, the 
ends are fastened together with bolts having the ends riveted over. 

These wheels are made of 8, 9 and 10 ft. diameter, the 8-ft. wheels 
having weights ranging between 5000 and 8000 lbs., the 9-ft. wheels 
between 6000 and 10,000 lbs., and the lo-ft. wheels between 10,000 
and 12,000 lbs. 

Another superior high-speed wheel by E. S. Newton {Avier. Mach., 
June 21, 1900) used without failure in band saw mills at speeds of 
10,000 ft. per min., is shown in Fig. 6. Wheels of 8 ft. diameter weigh 
about 6000 lbs. They have cast-iron rims and hubs with 16 wrought- 
iron, not steel, arms if ins. square. These arms are upset at each end 
and carefully tinned as far as they enter the cast-iron. They are also 
staggered. The rim is poured one day and the hub the next. The 
figure shows a wheel of 5 ft. diameter. 



Unusually large high-speed wheels have been called for in the con- 
struction of electric power-houses. Fig. 7 shows such a wheel by the 
AUis Chalmers Company, located in one of the power-houses of New 
York City (Amer. Mach., May 24, 1900). 

Except in its hub, the wheel is of steel throughout. The arms are 
hollow. The most striking feature lies in the reinforcing plates 
which are riveted to the sides of the rim casting. There are eight of 
these on each side, and the arrangement of the rivets will be seen to 
be such that the plates break joints with one another in such manner 
that there are fourteen effective plates in the weakest sections. The 
estimated weight of this wheel is 310,000 lbs. 

While this wheel has the joints half way between the arms the num- 
ber of arms is such as to greatly reduce the beam action. 




Bore 5%7~ 



Rim tension at 10,000 ft. per min., rim velocity by formula {d) 
2777 lbs. per sq. in., no failures. 
Fig. 5. — The Allis-Chalmers band saw mill fly-wheel. 



B Dia.,Tum ^^ 
-(L—o^lO Dia, 




Rim tension at 10,000 ft. per min., rim velocity by formula (d) 

2777 lbs. per sq. in., no failures. 

Fig. 6. — The Newton band saw mill fly-wheel. 



A wheel for a rim speed of 15,000 //. per min. is shown in Fig. 8 
{Amer. Mach., Jan. 27, 1913). The wheel is in use at the mills of the 
Illinois Steel Company at South Chicago and was made by the Wes- 
tinghouse Electric and Manufacturing Company. Its diameter is 
13 ft. 2 ins., its weight 100,000 lbs., and its normal speed 375 r.p.m. 

The assembled wheel shown is made with a cast-steel spider A, 
which has 12 arms made of a double 7|^X4-ins. square section, the 
corners being well rounded with a ij-in. radius. The arms have a 
liberal fillet at the hub and flange ends. The hub has a bearing of 26 
ins. on the shaft, which has a double-stepped fit and driving through 
a feather key. The rim is machined with 12 notches B 2% ins. deep, 
2x1 ins. at the outer periphery, having taper sides of 27 deg. 

The laminated slieets C, which occupy a width of i ft. 95 ins., are 
made from .0281 in. bessemer (not annealed) sheet steel, 12 being used 
for a circumference. Each sheet is made with two dovetails fitting 
into the notches machined in the spider rim. 



72 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 





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6 >^HOX-4v5-^ 



FLY-WHEELS 



73 



On each outer side of the laminated sheets is an end plate made of 
cast steel. Each end plate is accurately drilled, reamed and fitted 
with ten ij cold-rolled steel bolts D, the ends of which are fitted with 
hexagon nuts which set into counterbores in the plate. 

The laminated sheets are assembled with overlapping joints and 
when clamped together very little strain comes upon the bolts, as 
the thin sheet construction gives a very high slipping resistance with 
a comparatively light pressure. The bolts pass through the end 
plates and sheets. 

In the center of the group of laminated sheets is inserted a punch- 
ing of the same dimensions as the sheets, but fitted with six notches 
in each sheet, 3 ins. in width and i^ ins. deep. These notches are 
used for barring the engine, a special barring engine being attached 
to the equalizer set. 

Fly-wheel joints having an efficiency of 100 per cent, or more have 
been made by John Fritz {Trans. A. S. M. E., 1899) and by H. V. 



72 Holes equally spaced for 
Flywheel with Barring j 
arrangement 1 



I [<9»t^3re9# 




Fig. 8. — The Westinghouse fly-wheel for a rim speed of 15,000 ft. 
per min. 

Haight {Amer. Mach., Feb. 28, 1907). The two constructions are 
based on the same principle, Mr. Haight's wheel being shown in Fig. 9. 

The reason why ordinary joints are weaker than the parts which are 
joined is that those parts are cut away to provide room for the joining 
pieces. Mr. Haight's plan is to cut away the rim section throughout 
its circumference, carrying the section which is imposed by the joining 
pieces all the way around the rim, the result being that the rim is not 
weakened by making provision for the joining pieces. There is, 
in fact, no difficulty in making the joining pieces stronger than the 
rim, and hence this joint may have an efficiency exceeding 100 per 
cent. 

Moreover, the usual form of wheel-rim section involves a spongy 
center, which adds its due quota to the weight of the rim and to the 
strains to be carried by the rim section, while it adds very little to 
the strength of the section. Mr. Haight's construction involves a 
ribbed form of rim, by which this spongy center is largely eliminated, 
and hence the section of his castings should be stronger than that 



of the usual form; and, inasmuch as the link can then be made of the 
same strength as the casting, the conclusion would seem to be 
inevitable that the wheel as a whole should be stronger than a solid 
wheel having the usual section of rim. 

X>ividing the wheel with the joints at the arms neutralizes the beam 
action of the customary construction which Professor Benjamin's 
experiments have shown to be so injurious. 

The removal of the metal which would ordinarily occupy the chan- 
nels aa and its addition to the other parts of the section, where it 
acts to strengthen the section at the link as well as the remainder of 
the wheel, accomplishes the seemingly impossible. 

The bolt through the arm was placed there to provide for machining 
the rim. It is not needed to reinforce the link. 

In Mr. Fritz's design the same result is obtained by a cored section, 
Fig. 10, the action of the core being the same as that of the channels 




Section 

through 

Joint and 

Links 

Fig. 9. — The Haight 100 per cent, efficiency fly-wheel joint. 





Fig. 10. — The Fritz 100 per cent, efficiency fly-wheel joint. 

aa, Fig. 9, of Mr. Haight's construction. The joints are midway 
between the arms but the great number of arms (16) reduces the beam 
action to a probably negligible amount. The arms are hollow and 
join the rim segments by curves which avoid abrupt change of section. 
Four I links of unequal length are used at each joint, the object of the 
inequality being to distribute the stresses due to the links. Many 
of these wheels of 20 to 30 ft. diameter have been applied to the most 
severe rolling-mill duty and they have never failed. 

The design of hand fly-wheels is, as a rule, worse than that of plain 
fly-wheels. The thinness of the rim increases the stress due to the 
beam action and, with joints midway between the arms, such wheels 
are unsafe. 

In wheels of a size suitable for casting in halves, which includes the 
great majority, double arms should be placed at the joint, as in Fig. 
II by J. B. Stan WOOD {Amer. Mach., Apr. 4, 1907). This is a marked 
improvement over the midway joint, but it may be still further 



74 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 




improved by adapting the Haight principle, as suggested by Pro- 
fessor Benjamin {Amcr.Mach., April ii, 1907) and shown in Fig. 12. 

The ribs should be deep, both to resist the beam action and to 
bring the links more nearly to the neutral axis of the section. For 
ordinary cases the arms may be proportioned in accordance with the 
author's formulas for wheels in one piece. 

In segmental wheels the joints should be placed at the arms. 

Such a wheel of 22 ft. diameter, 96 ins. face and for three belts, by 
the Providence Steam Engine Co. {Amer. Mack., Nov. 11, 1895), is 
shown in Fig. 13. The space required for the pad for the arm joint 
makes the application of the Haight principle more diflScult in these 
wheels, but the more nearly it is adhered to the better the wheel will 
be. 



Section B-B 
Fig. II. — The Stanwood split band fly-wheel. 





Fig. 12. — The Haight principle applied to split band fly-wheels 
Table 3. — Dimensions of Shrink Links and Shrinkage Allowance. Practice of the General Electric Company 



Dimensions of standard keyways 


A 


J 


A 


J 


3 
4 

I 


4i 

5, 
6, 


S4, 6 
7h 9 




i| 




9, 


io|, 


12 





Dimensions 
of keyway 



B 



D 



Dimensions 
of key 



H 



Dimensions 
of keyway 



B C 



D 



Dimensions 
of key 



H 



Length of keyway and key 



/ 



K 



J 



J 



K 



I 
if 



3i 

32 
34 

4 
4i 



32 



3l6 

,ii 
O 16 

3l 
4i 



2$ 

2f 
2i 
3l 

3i 



T -S^ 

^ 32 

2l6 

9-2- 
2l6 

,11 

2 16 

,15 
2l6 

2-2- 
16 

3T6 

oil 

3 16 

3 16 

4A 
4A 



If 
2i 

3i 

3f 
4i 
4i 
4l 
Si 

Sl 
6f 
61 

7i 
7f 



2l6 

2-3- 
3l6 

2^ 
08 

2S 
38 

2I3 
3 16 



2fg 

2l6 

,11 
■= 16 
,15 
■=16 

3X6 



4f 

S 

Si 

Si 

54 

6 

61 

61 

61 

7 

7i 
7l 
7f 



98 

9f 
10 

io| 

II 

III 

iii 

I2f 
I3i 

i3f 
I4i 
145 



4l6 3 16 



4? 
. 9 

4T6 

4r6 
S 

si 

Sl 
cii 

16 
flS 
16 

6i 
61 

6f 
6H 
7^ 
7A 



3 16 
oil 
3l6 

4 

4T6 

48 

,1 5 

48 
44 
S 

si 

5 16 

3 16 

O 16 
rl5 
3 16 

6i 



4t6 

aXS. 

4t6 
■;-3- 

16 
ST6 
5 16 

1-15. 
16 

6A 
6A 
6H 
6M 

7i% 

71^ 
~ii 

7l6 

7I5 

7l6 



9 

9i 

9l 
lof 

io| 
iii 
III 
12I 

I2f 

I3I 
I3I 
14 
I4J 



4tt 

Ah 

44 

A lA 
4l6 

5 16 
ST6 

5f 

si 

61^ 
6A 

61% 

6f 

7 
1\ 



35 

38 
,13. 
3 16 

4 
■ 3 
4r6 

41^ 
. 9 
4t6 
A la 

4l6 

4ll 

Si 
sf 

Si 
5 4 

5 16 



4 
4i 

S 

si 

6 

6i 

7 

7i 



9 

9i 
10 
io| 
II 

III 



995 
49S 
995 
494 
994 

493 
993 
492 
992 
491 

991 
490 
990 
489 



12 

I2i 
13 

i3i 
14 

145 
IS 
isi 
16 

i6i 

17 

i7i 

18 

i8i 

19 

iQi 



ir.9 
12.487 
12 . g 
13 486 
13-986 

14.485 
14.985 

15-484 
IS -984 
16.483 

16.983 
17.482 
17. 982 
18.481 
18.981 

19.480 



20 

20§ 
21 
2li 
22 

22i 
23 

23i 
24 

24i 

25 

2Si 
26 

26i 

27 

27i 



19. 980 

20.479 

20.979 
21.478 
21.978 

22.477 

22.977 

23-476 
23-976 

24-475 

24-975 
25-474 
25-974 
26 . 473 
26.973 

27.472 



28 

28i 

29 

29i 

30 

3oi 
31 
3ii 
32 

32i 

33 

33i 

34 

34i 

3S 

3Si 



27.972 
28.471 
28. 971 
29,470 
29.970 

30.469 
30.969 
31-468 
31-968 
32.467 

32.967 
33 - 466 
33.966 
34-465 
34-965 

35-464 



FLY-WHEELS 



75 



The absence of shrinkage strains in segmental wheels makes feas- 
ible the application of the usual formulas for the strength of beams 
to their design. In large wheels the arms should be of I-beam or, 
better still, of oval hollow section. The total load should be taken as 
that due to the full pressure of the steam acting when the crank is 
at a right angle with the center line and a factor of safety of not less 
than lo should be included. 

For the design of hollow fly-wheel arms from their solid equivalents 
see Arms of Spur Gears. 

The Regulating Power of Fly-wheels 

Not much actual use is made of analytical methods in the determi- 
nation of the weight of steam-engine fly-wheels — resort being usually 
made to comparison with existing wheels. As will be seen below, 
the formula for the weight of wheels designed for a given fineness of 
regulation includes two coefficients — one for the steam distribution 
and piston speed and the other for the degree of regulation desired. 
The determination of the former for any given engine is so laborious 
that it is seldom made. Because of this the two coefficients have been 
frequently combined into one. The resulting formulas, while rational 
in form and sufficiently correct for the types of engine and the services 
from which they have been derived, are of limited application and, 
worse yet, their limits are unknown. 

The determination of the coefficient for the steam distribution 
and piston speed by analytical methods compels the resort to sim- 
plifying assumptions which vitiate, if they do not destroy, the value 
of the conclusions. The determination has, however, been made 
graphically with all necessary accuracy and for a wide range of 
conditions by Karl Mayer {Zeitschrift des Vereines Deutscher 
Ingenieure, 1893) and translated by Emil Theiss (Amer. Mach., Sept. 
7 and 14, 1893). Herr Mayer, with infinite care and patience, 
constructed a series of rotative effort diagrams from which a series 
of values of this coefficient was determined. 

In the operation of a fly-wheel under a varying impulse and a 
constant resistance, the velocity fluctuates between two limits which 
are expressed by the equation: 

-. — -. -. — -. — r— = a quantity called the coefficient 

greatest velocity— least velocity 

of steadiness, which is the reciprocal of the coefficient of fluctuation 

used by some writers. 

The value of the coefficient of steadiness having been selected to 
suit the character of the load, the weight of the wheel is then deter- 
mined to suit. Since an early cut-off and low piston speed will 
deliver more irregular impulses than a late cut-off and high piston 
speed, it is obvious that these factors also affect the weight of the 
wheel. 

The formula for the weight of the wheel is as follows : 

W=id .^ . — 
v^y<.r.p.m. 

in which TF = weight of wheel rim, lbs., 

i = coefficient for steam distribution and piston speed, 

d = coefficient of steadiness, 
«.A.^.= indicated horse-power, 

J) = mean velocity of wheel rim, ft. per sec, 
r.p.m. = revolutions per minute. 
Herr Mayer's determinations of the value of i are given in Table 4. 
Two assumptions run through the table: The length of the connect- 
ing rod is uniformly taken as five times the crank and the weight of 
the reciprocating parts is taken at an average value as given by a 
formula. The captions p, .jp and refer to the compression, 
which, in column p, is to the initial pressure; in column .jp, to seven- 
tenths of that pressure, while, in column 0, there is no compression. 
Herr Mayer's values of the permissible coefficient of steadiness, d, 
together with additional values, from Unwin's Elements of Machine 
Design, are given in Table 5. 



With the values of i determined, it is a comparatively simple 
matter for any engine builder to determine the values of d for his 
own wheels and thereafter to design others in a strictly rational 
manner. 

In doing this, and, indeed, in any application of this method, it 
should be noted that the value of i increases as the i.h.p. decreases — 
that is, as the cut-off is shortened. Values of the i.h.p. for the points 
of cut-off included in Table 3 should therefore be determined and the 
calculation of the weight of the wheel be made for the maximum value 
of the product of i and i.h.p. in order that the regulation may be 
satisfactory under the worst condition. 

While useful for purposes of comparison, the sections of Table 4 
for two- and three-cylinder engines have, probably, little real applica- 
tion. Wheels dimensioned in accordance with them would, no doubt, 
be so light as to be structurally too weak for use. 

In all that has been said, the weight of the arms and hub has been 
ignored. Their weight is so considerable while their effect is so small 
that, when applying the formula to existing wheels, their weight 
should be subtracted from the gross weight of the wheel. Calcula- 
tions of many large wheels have shown that the weight of arms and 
hub combined make up about 35 per cent, of the weight of the entire 
wheel. Their fly-wheel effect, on the other hand, adds but from 
7j to 10 per cent, to the value of the rim. 

Fly-wheels for Intermittent Work 

The design of fly-wheels for intermittent work, such as punching, 
shearing, etc., is based upon an entirely different procedure. The 
loss of energy being equal to the work done, the weight and velocity 
of the wheel must be such that the loss of energy does not involve an 
undue reduction of speed. The fundamental formulas are: 
W 



E = - 



2g 



w=- 



2gE 



(a) 
(&) 



in which £ = loss of energy of wheel = work to be done, ft.-lbs., 
W = weight of wheel,' lbs., 
j;i=normal or full velocity, ft. per sec, 
V2 = reduced velocity after work is done, ft. per sec. , 
g = acceleration of gravity = 3 2.2 
In equation (6) the reduced velocity may be expressed as a fraction 
of the normal velocity, that is, V2 = avi, giving 

2gE 



w=- 



Vl' 



-a'vi^ 

2gE 



,- ic) 

For belt-driven machines the limiting low velocity is that at which 
the belt runs off the pulley. According to Wilfred Lewis (Trans. 
A . S. M. E., Vol. 7) the experiments of Wm. Sellers & Co. showed that 
this would take place when the slip exceeded 20 per cent, of the belt 
speed, that is, a in equation (c) should not be less than .8. Intro- 
ducing this value and the numerical value of g gives for the limiting 
condition for belt driving 

TF=i8o^ 

Since in most cases the reduction of speed is momentary only, 
while in the experiments it was continued for some time, the limiting 
condition or one not far from it would seem to be admissible when 
other conditions do not prevent its use. 

Strict accuracy in calculations involving the energy of fly-wheels 
requires that the weight used shall be the weight of the entire wheel 
and that the velocity" be that at the center of gyration. The calcu- 
lation of the radius of gyration of such bodies as fly-wheels is laborious 
and is seldom made. The usual method is to make the calculations 
for the weight of the rim only and for the velocity at the center of 
gravity of the rim section. 



76 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 




t 
a 



o 
U 
u 
a 
•5b 
a 
W 



r2 






o 

1^ 



FLY-WHEELS 



77 



Table 4. — Values of the Coefficient i for Steam Distribution and Piston Speed for 

Substitution in Formula 
Single Cylinder Non-condensing Engines 





Cut-off i 1 i 1 i 1 J 


Piston 
speed 


Compression to 


P 


■7P 





P 


■yp 





p 


.7P 





P 


■7P 





200 
400 


272,690 
240,810 
194,670 
158,200 


241,530 
209,890 
165,450 
132,020 


218,580 
187,430 
145,400 
108,690 


242,010 
208,200 
168,590 
162,070 


220,280 
188,880 
151,440 
148,540 


209,170 
179,460 
136,460 
135,260 


220,760 
188,510 
165,210 


207,230 
176,080 
150,710 


201,920 
170,040 
146,610 


193,340 
174.630 


187,670 
167,860 


182,840 
167.860 


800 























Two Cylinder Engines with Cranks at 90 Deg. 





Cut-off J 1 i 1 i 1 i 


Piston 
speed 


Compression to 


P 





P 1 


P 





p 





200 
400 
600 


71,980 
70,160 
70,040 
70,040 . 


g5 


60,140 


59,420 ' 

57,000 
57,480 
60,140 , 


c 2 ] 

i^- 54,340 
R 10 


49,272 1 fj 2 
49,150 f '^. 

49,220 J ^ t 


f 50,000 


1 c 2 

37,920 1 rt j: 
35,500 J s^ 


} 36,950 



Single Cylinder Condensing Engines 





Cut-off A 1 i 1 i \ i 


Piston 
speed 


Compression to 


P 


■IP 





P 


■IP 


1 p 


■IP 





p 


■ TP 





200 
400 


292,730 

212,910 
141,900 


241,770 
171,970 


180,180 
117,380 


265,560 
194,550 
148,780 


226,310 
163,030 
143,710 


176,560 
117,870 
140,090 


234,160 
174,380 


206,030 
151,680 


173,660 
118,350 


217,980 
166,290 


195,400 
146,610 


171,000 
121,730 





















Single Cylinder Condensing Engines 





i 1 i 1 i 


Piston 
speed 


Compression to 


P \ -IP \ \ p \ ■IP \ \ p \ .Tp 





200 
400 


204,210 185,250 
164,720 148,780 


167,140 189,600 
133,080 174,630 


173,900 
164,970 


161,830 
151,680 


172,690 


165,930 


156,990 














* 



Three Cylinder Engines With Cranks at 120 Deg. 



Piston 
speed 


Cut-off i 


\ 1 ^ 




\ 




Compression to 


P 1 


P 


1 P 





P 





200 
800 


33,810 
30,190 


32,240 
31,570 


33,810 
35,140 


35.500 
33,810 


34,540 
36,470 


33,450 
32,850 


35,260 
33,810 


32,370 
32,370 



Table 5. — Values of the Coefficient of Steadiness 



For engines operating 

Hammering and crushing machinery 

Pumping and shearing machinery 

Weaving and paper-making machinery 

Flour milling machinery 

Spinning machinery 

Ordinary driving engines with belt transmission. 

Gear-wheel transmission 

Unwin's Elements of Machine design gives 
For engines operating 

Machine tools 

Textile machinery 

Spinning machinery. 

Electric machinery 

Electric machinery direct driven 



Values of d 



5 
20 to 30 
40 

5° 
50 to 100 

35 
SO 



35 

40 

SO to 100 

ISO 

300 



Determinations of fly-wheel effects using the entire weight of the wheel 
and the velocity of the center of gyration, have been much simplified 
by 0. S. Beyer {Amer. Mach., Oct. 17, 24, 1912). The simplification 
grows out of the fact that examination of a large number of fly-wheels 
for use on punching and similar presses has shown the quite constant 
relation that the weight of the rim is equal to 68.6 and of the arms and 



hub 31.4 per cent, of the weight of the entire wheel. Using these per- 
centages Mr. Beyer has calculated Table 6 for a variety of rim sections 
which embraces almost everything occurring in practice. With the 
aid of Figs. 14-17, this table will answer any question relating to the 
functions of ily-wheels used for intermittent work. 

The use of this table is as follows: To find the velocity in ft. per 
sec. of the center of gyration of a fly-wheel, select from the rim sec- 
tions shown in Table 6 and marked a, h, c, d, e, f and g, the one near- 
est, as to ratio of width to thickness, to that of the wheel, then locate 
in the first column the outer diameter of the wheel and trace over to 
the column headed by the same letter that identifies the selected rim 
section. 

The number found in this column gives the velocity in ft. per sec. 
of the center of gyration of the wheel when running at the rate of 
I r.p.m. Multiplying this number by the r.p.m. the wheel actually 
makes, gives the required velocity of the center of gyration. 

In the same manner the velocity in ft. per sec. of the outer circum- 
ference is found by tracing over from the outer diameter of the wheel 
in the first column, to the last column. The number there found is 
the velocity of the outer circumference of the wheel at i r. p. m., and 
multiplied by the actual r.p.m. of the wheel, gives the required 
velocity of the outer circumference. 

For sizes between those given in the table interpolation is necessary. 

The velocity at the center of gyration having been obtained from 



78 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 6, the energy for a wheel of given weight may be obtained 
from Figs. 14 and 15 (also by Mr. Beyer) which are identical, except 
that Fig. 14, for low velocities, is on a larger scale. 

To find the energy of a body of a given weight and moving with a 
given velocity, find on the scale of velocity the point that corresponds 
to the velocity in ft. per sec. at which the body is moving, and trace 
upward to the curve; then, from the point thus located on the curve, 
trace over to the scale of energy; the number identified on that scale, 
if multiplied with the weight of the body in lbs., will produce the 
required energy in ft.-lbs. 

The charts may be used with equal facility in the reverse direction 
to find the velocity at which a fly-wheel of known weight and diameter 
must be run in order to contain a given amount of energy. To do 



at each operation. The extreme value for belt-driven fly-wheels is 
20 per cent., at which the belt is liable to run oflE the pulley. This 
represents an abstraction of 2>(> per cent, of the energy. According 
to Mr. Beyer, for press work, the extent to which the diminution of 
the velocity of a fly-wheel is practicable depends upon the frequency, 
as compared with the velocity, with which the fly-wheel is drawn 
upon for energy. Thus: If an ordinary fly-wheel press, running at 
90 r.p.m., is tripped at regular intervals, say 15 times per min., the 
velocity may each time be diminished to the extent of 10 per cent, or 
even more. But if the press is run continuously, no greater diminu- 
tion than from 5 to 6 per cent, should be reckoned with. 

In the case of a heavily-geared drawing press, having an engine 
directly connected, or running under conditions otherwise favorable 



10 



In 
g5 



10 15 

Scale of Velocity, it. per Sec. 

Fig. 14. 

£= energy, ft. lbs. 
PF = weight of body, lbs. 
3 = velocity, ft. per sec. 

g = 32.2 



20 



25 



100 






















90 








































/ 


80 




















/ 


















/ 


/ 


70 


















/ 




















/ 




ieo 
















/ 


















y 


V 






^50 














/ 


















A 


/ 








° 40 

2 












/I 


















/ 


r 










30 








/ 


/ 


















/ 














20 






/ 




















^ 
















10 


^ 








































n 























25 30 35 



40 45 50 55 60 

Scale of Velocity, Ft. per Sec. 

Fig. 15. 



65 70 75 



E = - 



2 gE 

" W 



Figs. 14 and 15. Energy of i lb. at various velocities. 



this divide the given energy for which the velocity is required by 
the weight of the body; the quotient being the amount of energy con- 
tained in each lb. of the body's weight. Locate this quotient on the 
scale of energy, in Fig. 14 or Fig. 15, trace over to the energy curve, 
and down to the scale of velocity, where the required velocity in ft. 
per sec. may be read off. Then turn to the velocity table and find 
the velocity number corresponding to the outer diameter and type of 
rim section of the wheel, and divide by this number the velocity in 
ft. per sec. just found. The quotient is the r.p.m. the wheel must 
make in order to contain the given amount of energy. 

A leading question in connection with the design of fly-wheels for 
intermittent work relates to the permissible reduction of velocity 



to readily restoring the velocity to normal, the fly-wheel may be 
brought almost to a standstill. 

Obviously, other factors enter the problem, but they are as mani- 
fold as the kinds of work that may be done in the same press. 

Problems relating to the reduction of velocity and energy of fly- 
wheels may be solved by the use of Figs. 16 and 17, also by Mr. 
Beyer, which are almost self-explanatory. Thus, in Fig. 16, locate 
the permissible reduction of velocity on the vertical scale, trace 
horizontally to the curve and then down to the horizontal scale, where 
read the fractional part of the energy given out with the given reduce 
tion of velocity. Obviously the chart may be used in the reverse 
direction with equal facility. 

(.Continued on next page, second column) 



FLY-WHEELS 



79 



Table 6. — Velocities in Feet per Second of Center of Gyra- 
tion AND OF Outer Circumference for Different Cross- 
sections OF Rim, AND Different Diameters of Fly-wheels 
Running at i R.p.m. 




Velocity of center of gyration, in ft. per sec. for sections 



10 20 30 

100 100 100 

Scale of Energy Expanded 



Velocity 
of outer 
circum- 



Cross- 
















ference, 


section 


a 


b 


c 


d 


e 


/ 


& 


ft. per 


of rim 
















sec. 


12 


.0406 


.0411 


■ 0417 


.0423 


.0428 


• 0434 


.0440 


.0524 


l8 


.0609 


.0617 


.0625 


.0634 


.0642 


.0651 


.0660 


.078s 


24 


.0812 


.0822 


.0834 


.0845 


.0857 


.0868 


.0880 


.1047 


. 30 


.1014 


.1028 


.1042 


.1056 


.1071 


.1085 


. 1 100 


.1309 


c 36 


. 1217 


.1233 


.1250 


.1268 


.128s 


.1302 


•I319 


.1571 


■Z 42 


. 1420 


• 1439 


.1459 


.1479 


.1499 


■ 1519 


.1539 


.1833 


.C 48- 


.1623 


.1644 


.1667 


.1690 


.1713 


.1736 


.1759 


.2094 


* 54 


.1826 


.1850 


.1876 


.1901 


.1927 


■ I9S3 


• 1979 


.2356 


o 6o 


. 2029 


• 2055 


.2084 


.2113 


.2141 


.2170 


.2199 


.2618 


S3 66 


.2232 


.2261 


.2292 


.2334 


.2356 


.2387 


.2419 


.2880 


a 78 


.2435 


.2466 


.2501 


■253S 


.2570 


.2604 


.2639 


.3142 


.2638 


.2672 


.2709 


•2747 


.2784 


.2821 


.2859 


• 3403 


•0 84 


.2841 


.2877 


.2917 


.2958 


.2998 


.3038 


.3079 


.3665 


1 90 


• 3043 


.3083 


.3126 


.3169 


.3212 


-3255 


.3299 


.3927 


S 96 


• 3246 


.3288 


.3334 


.3380 


.3426 


•3473 


.3519 


.4189 


102 


■3449 


•3494 


.3543 


.3592 


.3641 


.3690 


.3739 


•4451 


108 


.3652 


• 3699 


• 37SI 


.3803 


.3855 


.3907 


.3958 


.4712 


114 


.3855 


.3905 


.3959 


.4014 


.4069 


.4124 


.4178 


■4974 


120 


.4058 


.4110 


.4168 


.4225 ■ 


.4283 


•4341 


.4398 


• 5236 



20 










































100 




































/ 








































/ 


' 






































/ 


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Si = normal velocity, ft. per sec 
92 = loss of velocity, ft. per sec. 
£1 



£1 = normal energy, ft. lbs. 
£2 = loss of energy, ft. lbs. 



Fig. 16. — Relation of energy expended and loss of velocity. 



7-5 



Similarly Fig. 17 gives the relation between loss of energy and of 
velocity. If, for example, a fly-wheel is to furnish 200 ft.-lbs. of 
energy during each cycle, but the working conditions of the press 
are such as to require the diminution of the velocity to be kept within 
the limit of 7.5 per cent, we turn to Fig. 17. 

Locating on the scale of velocity ratio — the one given; namely, 

, and tracing over to the curve, and down to the scale of energy 

. £1 
ratio ^, the number 6.957 will be found, and is the ratio of the total 

energy the wheel must have to the energy to be expended at a dimi- 
nution of the velocity not exceeding 7.5 per cent. 

In other words, the energy to be expended is to be multiplied with 
that number, to produce the total energy, or 200X6.957 = 1391.4 
ft.-lbs. From this total energy, the diameter being generally derived 
from surrounding conditions, the weight, velocity in ft. per sec. and 
the r.p.m. are readily settled with the aid of Figs. 14 and 15, and the 
velocity table. 

Another problem occurs when the amoimt of the expended energy 
is limited to a certain ratio to the total energy and this also may be 
solved by the aid of Fig. 17. 

Supposing, the total energy a fly-wheel requires to be 5 times the 
energy it may expend, the resulting velocity ratio is then found by 

locating the ratio 5 on the scale of energy ratio =^, and tracing up to 

■C.2 

the curve, and over to the scale of velocity ratio — , where the number 

100 

^— ^ will be found, which indicates that, if the total energy of the 

wheel to the energy to be expended is to be as 5 is to i, then the 
corresponding velocities must be as 100 is to 10.56. 



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Fig. 17. — Relation of total and expanded energy to loss of velocity. 



CONE PULLEYS AND BACK GEARS 



Graphical Solution 

The geometrical progression of speeds for driving machines by the 
cone pulley and back gear, or other means, is now generally accepted 
as correct. By this is meant that each speed should equal the one 
next below it, multiplied by a constant ratio. According to Carl G. 
Barth {Amer. Mack., Jan. n, 1912), the ideal value for this constant 
ratio in machine-tool practice is the fourth root of two or 1.189. 
The smallest ratio which Mr. Barth has found in the best speeded 
lathes of to-day is somewhat greater than this, being about 1.25. 
The ratios found in machine tools having the usual pattern of wide 
range cone pulley, range between 1.5 and 1.75, while ratios as high 
as 2 are occasionally found. Such ratios are too large to permit 
the selection of economical speeds for the work. 



Find the resulting ratio in the base line as at a. Trace upward to 
the curve for the desired number of speeds as at b. Trace to the left 
and read the required ratio of successive speeds as at c. 

To find the desired speeds construct a diagram as in Fig. 2, by 
Professor Sweet (Amer. Mach., Oct. 13, 1898). Lay off ab to any 
scale and call it unity. Lay off ac to the same scale and equal to the 
ratio found in Fig. i. The most convenient method of doing this is 
to take de and/c, Fig. i, for ab and be. Fig. 2, respectively. Draw 
the verticals through b and c, Fig. 2, and lay off bd to any scale to 
represent the lowest number of revolutions. Draw ad and extend 
it to e, through which draw the horizontal ef, when bf will represent 
the second speed to the same scale that bd represents the first speed. 
Proceed in this way as indicated in the diagram, finding points 
g, h, i, etc., for the various speeds. Should the diagram extend 



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20 80 40 

Ratio of Highest to Lowest Speed of Spindle 



50 



60 



Fig. I. — The geometrical speed ratio of cone pulleys. 



For feeds of machine tools the geometrical progression is also in 
general, though not universal, use. About the only type of machine 
for which this arrangement of feeds is still a matter of controversy 
is the drilling machine. 

The data given or assumed are usually the highest and lowest 
revolutions per minute of the machine spindle, the number of speeds 
and the diameter of the largest pulley, this last being determined by 
the available room. The assumed number of speeds should be re- 
garded as a trial number only and subject to correction, should the 
ratio, due to that number, be found too high or too low. 

There are three steps in the process: (i) finding the ratio; (2) 
fimding the speeds; (3) finding the diameters. 

To find the constant ratios consult the chart, Fig. i, by Prof. 
H. F. Moore {Amer. Mach., May 21, 1912) and proceed as follows: 
Divide the largest by the smallest r.p.m. of the machine spindle. 



beyond the limits of the paper, lay down the last value found on the 
paper to a reduced scale, and proceed as before. 

When finding the diameters of the steps three cases exist. 

Case I. Crossed belts. 

Case II, Open belts with pvilleys at sufiicient distance apart to 
make it unnecessary to compensate the tendency of the changing 
belt angle to alter the length of the belt. Machine tools driven from 
overhead countershafts are illustrations of this case. 

Case III. Open belts with puUeys so near together that the 
tendency of the changing belt angle to alter the length of the belt 
must be compensated. Foot-lathe drives and many speed cones 
are examples of this case. 

Since in Case I the belt length is constant, while in Case II it is so 
nearly constant that it may be regarded as such, the two cases may 
be treated as one. The distinguishing .feature of these cases is that 



80 



CONE PULLEYS AND BACK GEARS 



81 



the sum of the diameters of mating steps is constant, while in Case 
III this sum is not constant. 

To find the diameters of the steps for casfes I and II, the cones being 
alike, as is usual, and having an odd number of steps, proceed as in 
Fig. 3 : Draw a horizontal through a and from a lay down the speeds 
to the same scale as in Fig. 2 or to any convenient scale, giving ab, 
ac, ad, etc. Draw a vertical aO and make aO equal to the middle 
speed, ad. Draw hO, cO, dO, etc.; lay down Ri equal to the radius of 
the largest step and draw gA. Through h, the intersection of the 
highest speed line Oj with gh, draw hi at an angle of 45 deg. Now 
RiTi, RiTi, Rsrs, etc., are the radii of mating steps. 

The speed of the countershaft is ad. 

If the cones have an even number of steps there is no middle speed, 
point d is initially unknown and cannot be located at the start. 




Fig. 5-- 



—Ri — >1 

— Ri — ^ 

R^-J^ 
Drivers ' 

-Finding the diameters for Cases I and II. Unequal cones 
having either an odd or an even number of steps. 



In the case of unequal cones, two mating steps are naturally known 
and are used to locate O as in Fig. s : Makej^ equal to the radius of 
the largest driven step and bj equal to the radius of the smallest 
driving step (or, if those dimensions are given, make/i equal to the 
radius of the largest driving, and Im equal to the radius of the smallest 
driven, step). Draw and extendi^ (or draw and extend fm) given 
0. Locate i by laying down the largest driven step r^ (or locate h 
by laying down the largest driving step i?i), draw hi at an angle of 45 
deg. and proceed as before. 

The speed of the countershaft an is found by drawing On at right 
angles with ih. If the cones are very unlike On may fall without the 
field of the other constructions. 

The ratio of the back gear in all cases is the ratio of the highest (or 
lowest) direct to the highest (or lowest) back-gear speed. 




Fig. 3. — Finding the Diameters for Cases I and II. Equal cones 
having an odd number of steps. 




Fig. 4.— Finding the diameters for Cases I and II. Equal cones 
having an even number of steps. 



Figs. 2 to 5. — Graphical method of laying out cone pulleys. 



Proceed as in Fig. 4: Lay down the speeds as before and locate 0' 
at any convenient point on aO' . Draw O'b and O'f; lay down Rx, 
find h' and draw h'i' at 45 deg. as before. The coordinates of h'i' 
will give mating cones, but not equal cones. Find the center of 
h'i' and through it draw 0'(i, thus locating d. Now make aO=ad 
and find the mating steps as before. The speed of the countershaft 
is again ad. 

This construction is approximate only, its accuracy increasing as 00' 
becomes more nearly equal to zero. To obtain a second and very 
close approximation, repeat the construction by drawing a line 
through and the center of hi, thus finding a new point d, from which 
lay out a new point O as before. 
6 



To find the diameters for Case III proceed as follows, by Professor 
Moore {Amer. Mach., Feb. 26, 1903): First draw Fig. 6 in which R 
and r represent two of the mating radii. Draw the tangent ab and 
extend it by the distances ac, bd equal to the length of the arcs ae, bf. 

To do this use Rankine's approximate method (Machinery and 
Millwork) thus: Bisect the arc bf at g (because Rankine's method 
should not be used for arcs greater than 90 deg.). Draw the chord 



bg and extend it, making bh = 



From h as a center strike the arc 



gi giving bi = arc bg. Repeat bi giving bd = a,TC bf. Similarly, find 
c giving ac= arc ae when cd obviously equals one-half the length of 
the belt. 



82 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Lay off dj equal to 3.1416 to any scale and dk equal to unity to the 
same scale, dk making any angle with cd. Lay off d = 00' and draw 
jk and Im parallel to it giving dm, which is the radius of the middle 
step if the cones have an odd number of steps, or of a hypothetical 
middle step, which is not actually used in case the pulleys are to have 
an even number of steps. 

In Fig. 7 lay down this radius Rm, as shown (Figs. 6 and 7 are drawn 
to different scales), thus finding the point e. Lay down also the radii 
of the steps R, r given at the beginning. Through points aeh draw 
the arc of a circle, and proceed as in Fig. 3, Rn and rn being mating 
radii for speed en. As in Fig. 3 the line c/ gives the speed of the 
countershaft. 



That is, the capacity of Fig. 9 on the small step is 6| and on the 
large step, where the gain is most needed, 172 times that of Fig. 8. 

In the cases shown there' is a slight increase in the diameter of the 
large step but, without this increase, the gain would be nearly as 
large. So large an increase as the one shown is, of course, seldom 
needed. Many cone-pulley drives are, however, weak in capacity 
at the slow speeds and Mr. Norris's plan points out the remedy. 

The cone pulley shown in Fig. g gives a smaller total range of 
speeds than the one shown in Fig. 8, and, if the range of Fig. 8 is 
required, additional back gears are necessary. If double gears be 
used a five-step cone will give 15 speeds against 10 in Fig. 8, while a 
three-step cone will give 9. Fifteen are unnecessary and 9 are, in 




Fig. 6. — Finding the middle step for Case III. 



Fig. 7. — Finding the diameters for Case III. 



Figs. 6 and 7. — Graphical method of laying out cone pulleys. 



The arc ah gives the required compensation for the angle of the 
belt and provides that a belt length which is correct for one pair of 
steps will be correct for all others. The circle is not mathematically 
correct but is a remarkably close approximation. 

// the drive is too large to be laid down on the drawing board to a 
reasonable scale, the length of the belt may be obtained by calculating 
ah=On, Fig. 6 {00' and O'n of the right-angled triangle 00' n being 
known) and also calculating the quadrants pf and qe, leaving the 
arcs hp,aqXoht calculated or stepped off. These are such a small part 
of the whole that no material error will result from stepping them off 
on a small scale drawing. 

The High-power Cone Ptilley 

The power transmitted by cone pulleys may be greatly increased by 
increasing the diameter of the small step, but without increasing 
the over-all dimensions, as explained in a paper read before the 
Cincinnati Metal Trades Association in 1903 by H. M. Norris. 

Fig. 8 shows the standard and Fig. 9 the Norris design. The com- 
parative powers transmitted by the two constructions may be best 
shown by actual figures. Calling the highest belt speed in Fig. 8 — 
that obtained with the belt on the 4-in. step — 100, the slowest — that 
on the 12-in. step — will be 

iooXTiT = 333 
To maintain the same r.p.m., the highest belt speed in Fig. 9 must be 

100X^^^ = 2884- 



and the lowest will be 



5X^ = 255 + 



The smallest step of Fig. 8 is too small for a double belt, while the 
opposite is true for Fig. 9. To obtain the ratio of power capacities 
we must multiply the belt-speed ratio by a suitable ratio for this, 

say — ' and also by the ratio of the belt widths, -j' Doing this we 
7 25 

obtain: 

Power capacity Fig. 9 small step 288 10 4 

Power capacity Fig. 8 small step ~ 100 7 27 

Power capacity Fig. 9 large step 255 10 4 _ 

Power capacity Fig. 8 large step 33I 7 2J 



most cases, enough. It is this reduction in the number of steps that 
gives the increase in belt width. The additional back gear will, 
however, increase the over-all length of the headstock slightly if 
the entire gain is to be realized. 

The tendency of the belt to climb the side of the pulley against which 
it runs may be prevented by recessing the sides of the steps as shown 
in Fig. 10. The recess should be of ample depth to prevent the belt 
reaching its bottom. 



-i- 



ZVi Belt 

Fig. 8. — Conventional design 
of cone pulley. 





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T 




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Belt 





Fig. 9. — Norris design of 
cone pulley. 



Slide -Rule Solution 

The slide rule may be used for solving cone-pidley problems in cases 
which do not involve belt angles so large as to require compensation 
for belt length. The following explanation of this application of the 
instrument is by Robert A. Bruce (Amer. Mack., Aug. 18, 1904). 




Fig. 10. — Preventing the climbing tendency of belts. 



CONE PULLEYS AND BACK GEARS 



83 



The method is best explained by taking an actual case: Fastest 
speed 280; slowest speed 20; number of speeds 12. 

Lay off a straight line such that the length AB, Fig. 11, is equal 
to the distance between the points 20 and 280 on the 5-scale of the 
slide rule and divide it into eleven equal parts — i.e., one less than the 
total number of speeds. To do this draw BC of indefinite length 
and at any convenient angle. Space off eleven equal spaces of any 
convenient length. Join the last point D with A and, by a series 
of parallels through the remaining points, find the required divisions. 
The extreme and intermediate dividing marks will then form twelve 
graduations at equal intervals, and on applying the scale so that 20 
comes opposite the first and 280 opposite the last, as in Fig. 11, the 
numbers found opposite the remaining ten divisions will be the 
intermediate speeds required. The accuracy thus attained is suffi- 
ciently close for the purpose in view. A record of the speeds thus 
obtained may be made by writing opposite each graduation of the 
divided line the corresponding scale reading of the slide rule. 

The percentage rise or drop in changing from any speed to the one 
above or below may be at once obtained by inspection. For, apply- 



ing the scales so that ido of the slide-rule scale is opposite one grad- 
uation of the paper scale, it will be seen that the next graduation on 
the right falls opposite 127 on the scale, while the nearest graduation 
on the left lies against 78.5. The interpretation of these figures is 
that the percentage drop of speed is 100—78.5, or 2i| per cent., 
and the percentage rise of speed is 127 — 100, or 27 per cent. 

To obtain the ideal speeds so found the scheme to be adopted must 
be settled by the peculiar circumstances of the case, rather than by 
hard and fast rules. We wiU therefore assume two different cases: 

Let us first of all assume a single-speed countershaft and a cone 
with six steps and back gearing, the six quickest speeds being deliv- 
ered direct by coupling the cone to the spindle, and the slower speeds 
being secured by the use of back gearing. Let us also suppose that 
by the conditions of the problem the diameter of the largest cone step 
is fixed at 20 ins. The speed of the countershaft is equal (whether 
the number of cone steps is odd or even) to the geometric mean of 
the fastest and slowest driven cone speeds. If we therefore bisect that 
portion oi AB lying between 7 and B, that is, the lowest and highest 
direct-cone speeds, we obtain a line marked "Countershaft speed 




Fig. II- 





















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1st Gear ( B Scale") 


n 


i 
a 


Speed 




1 


Rat 


10 2nd Gear (B 


Scale) 


= 2.61 1 ■* 
= 6.82 


. 


1 





g 

Fig. 14. " 


U 




N 
ale) 


S .5 




S 


(Us 


e C Sc 


ail 
















m 








_ 


a 



Speeds 
(S Scale) 



B 






m 


A 










c 








H 






20" 


= 










is" 








16 




C 


U 










1 












B 


. A 











Fig. is. 



Figs, ii to 15. — Slide rule method of laying out cone pulleys and back gears. 



84 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



for six-step cone drive," which at once gives the scale reading 153 
for the countershaft speed. 

The ratio of hack gear is found by shifting the slide so that the 
point A comes opposite the point i on the scale, as in Fig. 12, when 
we shall have 4.25 as the scale reading opposite 7, this giving the 
ratio of the back gear, because i represents the lowest back-gear 
speed and 7 the lowest direct-cone speed which, obviously, is the 
back-gear ratio. 

The sizes of the cones now require fixing, and if lines C, D, E, mid- 
way between the main divisions i and 2, 2 and 3, 3 and 4, Fig. 12, 
are drawn with the slide in the position shown, we get scale readings 
of the ratios of the cones. 

This result follows from the fact that if the counter-cones and 
driven cones are similar, the ratio of the diameter of steps equidis- 
tant from the middle is the square root of the ratio of their respec- 
tive speeds. Thus in a three-speed cone where the diameter of the 
largest cone is twice the diameter of the smallest, the fastest speed 
is four times the slowest. 

Tu /- • t <.!, .• dia. cone 3 

Thus C gives 1.13 for the ratio -7-. 

dia. cone 4 

dia. cone 2 
JJ gives i.Ax tor the ratio -7: — 
° dia. 



E gives 1.82 for the ratio 



dia. 



cone 5 
cone 6 



dia. cone i 

Now cone 6 being fixed on as 20 ins. in diameter, cone i is 
found by direct proportion to be as nearly as possible 11 ins. 
In practice the diameters of the intermediate cones would usually 
be taken in arithmetical progression, the results so obtained being 
sufficiently near the values sought. But if closer results are required, 
remembering that the sum of the diameters of a pair of cone steps is 
constant and equal to 31 ins. (20 ins. + 11 ins.) if r is the ratio of any 
pair the diameter of one of them is 

r+T 
and that of the other 

31 
r+i 

If the cones are to have equal steps the lines C, Z), E, of Fig. 12, 
may be entirely dispensed with. The diameter of the largest cone 
being settled by practical conditions, that of the smallest can be 
found direct as illustrated in Fig. 13. Opposite 6 in the line AB 
place 20 (the diameter of the largest cone) of the C-scale of the slide 
rule, and opposite the first graduation ol AB find 10.97, or say 11 
ins. on the C-scale. 

It will be seen therefore that the above operations have involved: 
(a) the measurement of the distance between two points of the 
5-scale, (6) the transfer of this distance to the paper, (c) its division 
into eleven parts, and {d) the further bisection of one of these parts 
hyaline. The results obtained by direct reading are: The appro- 
priate speeds, the countershaft speed, the gearing ratio and the sizes 
of the cones, all of which are obtained without calculation. The 
method to be adopted for finding suitable gears to give the required 
ratio will be explained later. 

The second variation is to employ a four-speed cone delivering 
its motion either direct or through two changes of gearing. The 
first step would be exactly the same as before, the distance AB 
being the scale distance between fastest and slowest speeds on the 
5-scale, and the intermediate speeds being read direct as before. 
The further scheme of operations is shown in Fig. 14. 

The countershaft speed is obtained at the same time as the inter- 
mediate speeds by taking the scale reading of a line bisecting the 
fastest and slowest cone speed lines. 

The gear ratios are obtained on the 5-scale, from P to Q and from 
R to S, the unit of the slide-rule scale being placed respectively at 
P and R. The size of the smallest cone is given by the reading MN 
on the C scale, 20 {i.e., diameter of the largest cone step) being 



placed at TV and the diameter of smallest cone being given on the 
C scale at M as 14 ins. 

When finding the teeth in the wheels the clue is the easily remem- 
bered fact that in a pair of wheels in which the ratio of the faster 
divided by the slower is r the number of teeth in the pinion is 
sum of number of teeth 
^+^ 
Now, the sum of the number of teeth is always fixed when the centers 
of the wheels and the pitch have been determined. In compound 
gears the simplest case is back gearing where both pairs are equal. 
The total ratio of the gearing has been determined in the foregoing 
cases by a simple reading on the 5-scale, see Fig. 12, where the total 
ratio is 4.25. Using the C-scale, however, we should obtain the 
square root of this ratio and should thus have the ratio of each pair 
of wheels. Thus if i on the C-scale is put opposite A in Fig. 13 the 
line 7 would come opposite 2.06. And if the total number of teeth 
in each pair were 150, we should have as the number of teeth in 

. . 150 

pinion — -rj-- or 49 nearly, and of the wheel, 150—49 = 101. 

Returning to the case in Fig. 14, a convenient method of obtain- 
ing the changes of gear would be as in Fig. 15, where only three sizes 
of wheels are used, all of the same pitch but of decreasing breadths 
as we move from right to left. The fast or first gear is through the 
equal wheels AA and CB, the total ratio being 2.61. If the number 
of teeth as before is 150, then the number of teeth in C is obviously 
150 
— 7— or 43, and for B we have number of teeth= 150—43 = 107, 

whUe A has 75 teeth. The simplicity of the processes explained is 
obvious and the method can be modified by anyone understanding 
the principles of the logarithmic scale. 



Arithmetical Solution 

Arithmetical calculation may he used for solving cone pulley problems 
in cases which do not involve belt angles so large as to require com- 
pensation for belt length. The following systematic, procedure is 
by P. V. Vernon {Trans. Manchester Asso. of Engnrs., 1903). 

In the preceeding solutions the slowest speed was selected as the 
starting-point from which the others were obtained by working up- 
ward. Mr. Vernon inverts this process and begins with the fastest 
speed from which the others are obtained by working downward. 
Under the former method the ratio between the speeds is more 
than one; under the latter it is less than one, the two values being 
reciprocals. 

To calculate the ratio for the latter method, divide the slowest by 
the fastest r.p.m., and find the logarithm of the quotient. Divide 
this logarithm by the number of speeds less one and find the natural 
number corresponding to this logarithm, which number will be the 
required ratio. For the former method divide the fastest by the 
slowest r.p.m., and proceed as before. 

This calculation may he replaced by Table i of ideal speed ranges 
with sufficient accuracy for practical purposes, the greatest error 
introduced in the speed ratio being one-half of i per cent. 

To find the ratio and the speeds for the example shown in Fig. 16, 
in which it is required to find the correct proportions of gears and 
cone pulley for an ordinary back-geared headstock^ to produce 
twelve speeds varying from 280 down to 20 per min. The cone 
pulley is to have three steps, the largest 12 ins. diameter, and will, of 
course, be driven from a two-speed countershaft. This example is 
representative of a type of headstock largely used on medium-sized 
turret lathes. 

Referring to the table of ideal speed ranges it will be seen that the 
first number in each column is 1000, so that a corresponding range 
of speeds in the table would have twelve speeds varying from 1000 

1 The construction called back gear in the United States is in England called 
double gear. 



CONE PULLEYS AND BACK GEARS 



85 



1000X20 , , „ , , , 

down to a , or from 1000 down to 71.4. Refer to table 

200 

and look along horizontal line No. 12 for the figure nearest 71.4. 

This will be found to be 74.7 in the 21 per cent, column, which is 

probably near enough for the purpose, and fixes the common ratio 

required at .79, or 21 per cent, of drop from speed to speed. 

The percentage of drop from speed to speed will therefore be 21 
per cent, with 280 as a maximum. Plot out speeds as follows, 
either by calculation or slide rule: 280, 221, 174.7, 138, 109, 86.1, 
68, 53.7, 42.4, 33-5, 26.46, 20.9. 

To find these speeds by logarithms, note that the logarithm of the 
ratio is the common difference of the logarithms of the speeds, there- 
fore, find the logarithm of the highest speed and from it subtract 
the logarithm of the ratio, the result being the logarithm of the 



second speed. From this logarithm subtract again the logarithm 
of the ratio and the result will be the logarithm of the third speed, 
and so on. 

To find the correct proportions of gears and cone pulley to produce 
the above speeds: 

The set of speeds as obtained above may now be arranged as in 
Fig. 17, which represents in a simple way the general arrangement 
or typical form for obtaining geometrical ranges for all combina- 
tions of gears, cone pulleys, and countershaft changes. 

The arrangement consists of two main divisions of six speeds 
each, one division being entirely direct speed and the other entirely 
back geared, each division giving half the range and without any 
overlapping. 

Each main division consists of two sub-divisions of three speeds 



Table i. — Speeds in Geometrical Progression. 
Diminishing Speeds from the Highest as the Starting Point. 





10 


II 


12 


13 


14 


IS 


16 


17 


18 


19 


20 


21 


22 


23 


24 


25 


26 


27 


28 


29 


30 




per 


per 


per 


per 


per 


per 


per 


per 


per 


per 


per 


per 


per 


per 


per 


per 


per 


per 


per 


per 


per 




cent. 


cent. 


cent. 


cent. 


cent. 


cent. 


cent. 


cent. 


cent. 


cent. 


cent. 


cent. 


cent. 


cent. 


cent. 


cent. 


cent. 


cent. 


cent. 


cent. 


cent. 


I 


1000 


1000 


1000 


1000 


1000 


1000 


1000 


1000 


1000 


1000 


1000 


1000 


1000 


1000 


1000 


1000 


1000 


1000 


1000 


1000 


1000 


2 


900 


890 


880 


870 


860 


850 


840 


830 


820 


810 


800 


790 


780 


770 


760 


750 


740 


730 


720 


710 


700 


3 


810 


792 


774 


757 


740 


722 


706 


689 


672 


656 


640 


624 


608 


593 


578 


562 


548 


533 


S18 


504 


490 


4 


729 


705 


681 


658 


636 


614 


593 


572 


SSI 


531 


512 


493 


474 


456 


439 


421 


405 


389 


373 


358 


343 


5 


656 


627 


599 


573 


547 


522 


498 


474 


452 


430 


410 


3&9 


370 


3SI 


33Z 


316 


300 


284 


269 


254 


240 


6 


590 


558 


527 


498 


470 


444 


418 


394 


371 


349 


328 


308 


289 


271 


253 


237 


222 


207 


193 


180 


168 


7 


531 


497 


464 


433 


404 


377 


.351 


327 


304 


282 


262 


243 


225 


208 


193 


178 


163 


151 


139 


128 


118 


8 


478 


442 


408 


377 


348 


320 


29s 


271 


249 


229 


210 


192 


176 


160 


146 


133 


121 


no 


100 


91 


82.3 


9 


430 


393 


359 


328 


299 


272 


248 


225 


204 


185 


168 


152 


137 


123 


III 


100 


89.4 


81 


72.2 


64-5 


57.6 


10 


387 


350 


316 


284 


257 


231 


208 


187 


167 


150 


134 


120 


107 


95 


84-5 


75 


66.1 


59 


52.4 


45-8 


40.3 


II 


349 


312 


278 


247 


221 


197 


175 


155 


137 


121 


107 


94.6 


83.3 


73-2 


64. 2 


56.2 


48.9 


43 


37-4 


32. 5 


28.2 


12 


314 


277 


245 


215 


190 


167 


147 


128 


"3 


98.4 


86 


74-7 


65 


56.4 


48.8 


42. 2 


36.2 


i^-3 


26.9 


23 


19.8 


13 


282 


247 


215 


187 


163 


142 


123 


107 


92.4 


79-7 


68.7 


59-1 


50-7 


43-4 


37-1 


31.6 


26.8 


22.8 


19.4 


16.3 


13-8 


14 


254 


220 


189 


162 


140 


121 


103 


88. s 


75-7 


64.5 


55 


46.6 


39-5 


33-4 


28.2 


23-7 


19.8 


16.6 


13-9 


II. 6 


9-7 


IS 


229 


195 


167 


141 


121 


103 


87 


73-4 


62.1 


52.3 


44 


36.9 


30.8 


25-7 


21.4 


17.8 


14.6 


12.15 


10 


8.2 


6.8 


16 


206 


174 


147 


123 


104 


87.3 


73 


60.9 


50-9 


42:3 


35-2 


28.1 


24 


19.8 


16.3 


13-3 


10.8 


8.87 


7-23 


5.8 


4-7 


17 


i8s 


155 


129 


107 


89-3 


74-2 


61.3 


50.5 


41.7 


34-2 


28.16 


23 


18.7 


15-2 


12.4 


10 


8 


6.47 


5-21 


4.14 


3-32 


18 


167 


138 


"3 


93 


76.8 


63 


Si-S 


41-5 


34-2 


27.7 


22. s 


18.2 


14.6 


II. 7 


9-4 


7-5 


5-9 


4-7 


3-75 


2.94 


2.32 


19 


150 


122 


99-8 


80.1 


66 


53-6 


43-2 


34-8 


28 


22.4 


18 


14.3 


II. 4 


9 


7-14 


5.6 


4-4 


3-44 


2.7 


2.08 


1.63 


20 


135 


109 


87.8 


70.4 


S6.8 


45-5 


if'-i 


28.9 


23 


18. 1 


14.9 


II-3 


8.9 


6.9 


5-42 


4.2 


3-25 


2.51 


1.94 


1.48 


1. 14 


21 


121 


97 


77-3 


61.3 


48.7 


38.7 


30.5 


24 


18.9 


14.7 


ii-S 


8.9 


6.94 


5-4 


4.12 


3-2 


2.4 


1-53 


1-39 


1-05 




22 


109 


86.3 


68 


53-3 


41.9 


32.9 


25.6 


19.9 


iS-5 


II. 9 


9.2 


7-1 


5-4 


4-1 


3-13 


2.4 


1.78 


1-33 


I 






23 


98.4 


76.8 


59-8 


46.3 


36 


27.9 


21.5 


16. s 


12.7 


9.6 


7-37 


S.6 


4.22 


3-2 


2.38 


1.8 


1-31 


0.97 








24 


88.5 


68.3 


52.6 


40-3 


30.9 


23-7 


18 


13-7 


10.4 


7.8 


5-9 


4-4 


3-29 


2.46 


1. 81 


1-3 


0.97 










25 


79-7 


60.8 


46.3 


35 


26.6 


20. 1 


15-2 


II. 4 


8.S 


6.3 


4.72 


3-39 


2.56 


1.89 


1-37 


I 












26 


71.7 


54-1 


40.7 


30. s 


22.9 


17. 1 


12.7 


9-4 


7 


5-1 


3-78 


2.67 


2 


1-45 


1 .04 














27 


64.5 


48.1 


35-8 


26. 5 


19.6 


14. 5 


10.7 


7-7 


5-7 


4.14 


3.02 


2.1 


1.56 


I. II 
















28 


58.1 


42.8 


31-5 


23 


16.9 


12.4 


9 


6.5 


4-7 


i-ZS 


2.42 


1-57 


I. 22 


















29 


52.2 


38.1 


27.7 


20 


14-5 


10. S 


7-5 


5-4 


3-8 


2.71 


1-93 


1.32 


0.95 


















30 


47 


33-9 


24.4 


17-4 


12. s 


8.9 


^■i 


4-5 


3.16 


2. 2 


I -55 


0.82 




















31 


42.3 


30.2 


21.4 


iS-i 


10.7 


7-5 


5-3 


3-7 


2.6 


1.78 


1-25 






















32 


38.1 


26.8 


18.9 


13.2 


9.2 


6.4 


4-S 


3 


2. 12 


1.44 


0.99 






















33 


34-3 


23 -9 


16.6 


II. 4 


7-9 


S-4 


3,-7 


2-5 


1.74 


1. 17 
























34 


30.8 


21.3 


14.6 


9.9 


6.8 


4.6 


3-1 


2. 1 


1-43 


0.95 
























35 


27.7 


18.9 


12.8 


8.6 


5-86 


3-9 


2.6 


1 . 76 


1. 17 


























36 


25 


16.7 


II-3 


7-5 


5 


i-i 


2. 2 


1 .46 


0.96 


























37 


22.5 


14.8 


9.9 


6.5 


4-3 


2.8 


1.8 


I. 2 




























38 


20.25 


13.2 


8.7 


5-7 


3-7 


2.4 


i-S 


I 




























39 


18.2 


II. 7 


7.6 


4-9 


32 


2 


1-3 






























40 


16.4 


10.4 


6.7 


4-3 


2.75 


1-74 


1. 1 































86 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



each, one for the fast and one for the slow countershaft speed, the 
sub-divisions being in the same order iri each main division. Each 
sub-division consists of a group of three speeds, one for each step 
of the cone pulley, all in the same order and without overlapping. 
Suppose, In the example, that the countershaft and machine cones 
are identical; then the countershaft speeds will be 221 and 109 r.p.m. 



i 



i 



Fig. 16. — Lathe headstock with plain back gears, counter-shaft 
speeds 221 and 109 r.p.m., back gear ratio 4.1 1 to i. 



Direct Speed 
BMkGear ^"^^ 


Direct Speed 


Back Gear 


Couiil«r-8haft 
Fast or ^ow 


Fast 


Slow 


Fast 


Slow 


Btep of ( 
1 2 


3 


® 










1 


2 


3 


1 


2 


3 


1 


2 


3 


1 


2 


3 


.„)- > 




280 


221 


174.7 


138 


103 


80 


68 


53.7 


42.4 


33.5 


26.46 


20.9 













Fig. 17 












Direct Speed 


Direct Speed 


Back Gear 

4-11 to 1 


BacIcGeor 


CoUDler-abaft 


221 


109 


221 


109 




Step of Cone 
[7 2 3 










1 


2 


3 


1 


2 


3 


1 


2 


3 


1 2 


3 


Speeds -m^^ 


280 


221 


174.7 


138 


103 


86 


68 


53.7 


42.1 


33.5 28.5 


20.9 



Fig. 18 

Figs. 17 and 18. — Preliminary and final speed determinations for 
the case of Fig. 16. 

Note. — If the cone pulleys have an odd number of steps, the 
countershaft speed equals the speed of the driven cone with the 
belt on the middle step. If the number of steps is even, the counter- 
shaft speed 

= quickest speed of coneX 



4. 



slowest 



speed of cone 



{a) 



/quickest 

The largest diameter of the cone puUey is given as 12 in. The 
smallest diameter then is equal to 

largest diameter.^ countershaft 
of cone speed 



quickest speed of cone 

12X221 

= „o„ =9-47 ins. 



Q>) 



The middle step =half the sum of the other two steps = 10.73 '"s. 

The ratio of the back gears may be obtained by inspection of the 
range of speeds of Fig. 17, from which it will be seen that the ratio 
required is in the proportion of the highest direct speed to the 
highest back-gear speed, or as 280 : 68 = 4.11 to i. The gears should 
be proportioned to give this ratio to the nearest tooth. 

The required data are now complete, and the actual speeds may 
be laid out as in Fig. 18. 

If the gears which can be used will not give exactly 4. 11 to i, 
because of the necessity for using an integral number of teeth, the 
nearest approximation must be used. 



For double back gears, as shown in Fig. 19, the conditions chosen 
are that the cone pulley shall have three steps, the largest 18 ins. 
diameter, and to be driven Irom a two-speed countershaft. Eighteen 
speeds are required from 45 up to 300 r.p.m., the corresponding 
range in Table i of ideal speed ranges being 18 speeds, varying 
1000X4-5 



from 1000 down to- 



300 



= 15 



By looking along horizontal line No. 18, we find that 14.6 in the 
22 per cent, column is- the nearest figure to 15, and gives probably 
a near enough percentage for the purpose. If a greater degree of 
accuracy be required, the percentage of drop can be slightly changed 
to suit, but as the adjacent columns only vary from each other by 
a difference of i per cent., the table will be found to fulfill all 
practical requirements. 

The required speed range will then have a drop from speed to 
speed of 22 per cent., with 300 as a maximum. 

Plot out the speeds in a similar manner to the first example by 
calculation or slide rule, as shown in Fig. 20. 

It should be noted that Fig. 20 has exactly the same general form 
as in the first example, an extra division, however, being required 
for the extra gear ratio. The calculation is just as simple as in the 
first example, although rather longer, and it will be shown later that 
the method is equally applicable to the most complicated arrange- 
ments of gearing. 

The countershaft speeds will be 234 and in per min., as will be 
seen by inspection of Fig. 20, being equal to the second and fifth 
spindle speeds. 

The largest diameter of cone pulley is given as 18 ins. The small- 



est diameter will therefore be 



1 8X23 4 
300 



= 14.4, say 14 ins. The cone 



pulley will therefore have diameters of 14, 16 and 18 ins. 



r 














B 


=. 












= 












> 


1 


















1 




— 












1 








= 












1 


r-* 





























E 










^? 




1 




= 






A 




1 


1 




^ 


'— It. 


I 


1 




1 



Arrangement of speeds: 
1st group: 6 speeds, direct. 

A C 
2d group: 6 speeds, ^X"^ = 4-44 to i. 

E C 
3d. group: 6 speeds, ^X-^= 19-7 to I. 

Fig. 19. — Lathe headstock with double back gears. Countershaft 
speeds 234 and in r.p.m. 

The two gear ratios may be found by inspecting the table of speeds, 
from which it will be seen that the ratio required for the low gear 
is in the proportion of the first to the seventh speed. 
= 300:67.7=4.44 to I 
The ratio for the high gear is in the proportion of the first to the 
thirteenth speed. 

= 300:15.2 = 19.7 to I 
This gear ratio is exactly the square of the first gear ratio. The 
three divisions of speeds will thus have gears forming a geometrical 
progression with a common ratio equal to the first gear ratio thus: 
Single speed First gear Second gear 

I 1X4-44 1X4-44X4-44 

or I 4-44 19-7 

The above holds good for all arrangements planned by this method, 
no matter how many gear changes may be used. 



CONE PULLEYS AND BACK GEARS 



87 



The required data are now complete, and the speeds may be laid 
out as in Fig. 21. 

In the lowest line of Fig. 21 the percentage of drop from speed 
to speed is given, and it will be observed that this is very close to 
the 22 per cent, aimed at. If all the figures were worked out to 
sufficient places of decimals, exactly 22 per cent, would be obtained. 
It is not necessary, however, to overdo the calculations, or much 



The largest diameter of the cone pulley is given as 24 ins. 
smallest diameter then by formula (6) 

largest diameter of cone X fast countershaft speed 
quickest speed of cone 



The 



24X124 
150 



= 19. 





1st Group 


2iid Group 


3Td Group 


Direct bpecd or 
Low or EiBh'Gear 


Direct Speed 


Low Gear 


High Gear 


Counter- Shaft 
FaBt or Slow 


Fast 


Slow 


Fast 


Slow 


Fast 


Slow 


Btepof Cone 














1 


2 


3 


1 


2 3 


1 


2 


3 


1 


2 


3 


1 


2 


3 


1 


2 


3 


Speeds »»— 


300 


234 


182.6 


142.3 


111 86.5 


67.5 


62.6 


41.x 


32 


26 


19.6 


16.2 


11.8 


3.2 


7.2 


6.6 


4.4 



Fig. 20. 





1st Group 


2nd Group 


3rd Group 


Direct bpeed or 
low or Higb Gear 


Direct Speed 


Low Gear 

4,44 to 1 


High Gear 

19.7 to I 


Counter-Sbaft 
Speed i» - 


234. 


111. 


234. 


111. 


234. 


111. 


Step of Cone 
IT 2 3 














1 


2 


3 


1 


2 


3 


1 


2 


3 


1 


2 


3 


1 


2 


3 


1 


2 


3 


Speeds •^— >- 


300 


234 


182 


142.7 


111 


86.3 


67.7 


62.7 


40.9 


32.1 


26 


19.4 


16.2 


11.8 


9.5 


7.2 


6.6 


4.35 


^Brop -m-^ 




22 


22.2 


21.6 


22.2 


22.2 


21.6 


22.1 


22.3 


21.6 


22.1 


22.4 


21.6 


22.3 


22 


21.7 


22.2 


22.3 



The cone pulleys will therefore have steps 
19.840, 21.227, 22.614, and 24 ins. diameter, the 
diameters being in arithmetical progression with 
a common difference of 1.387 ins. 

If the drop in diameter between the steps of the 
cone is considerable, or if the drive is very short, 
it would be necessary to calculate the interme- 
diate speeds separately, as equal differences in 
diameter do not give equal percentages of speed 
change. The calculation is made as follows: 

Let X = diameter of required step of driven cone, 
y = diameter of required step of driving cone, 
a — largest diameter of cone pulley, 
h = smallest diameter of cone pulley, 
c = intermediate speed required, 
d = speed of driving cone. 

Then 

ad-\-bd 



c+d 



'y = a-\-b- 



FiG. 21. 
Figs. 20 and 21. — Preliminary and final speed determinations for the case of Fig. 19. 



time can be wasted without any corresponding gain. In many cases 
the gear ratios obtainable will introduce a slight error, which would 
more than extinguish the extra accuracy so obtained. In all prac- 
tical work approximations are permissible, providing that the errors 
are small and are known. The gear should be proportioned to give 
the above speeds as nearly as the pitches will allow. 

For more complex arrangements of back gears, as shown in Fig. 22, 
it is assumed that it is desired to find the correct proportions of gears 
and cone pulley for a headstock arranged to run direct or through 
any of four separate ratios of gearing, the cone pulley to have four 
steps, the largest diameter being 24 ins., and driven from a two- 
speed countershaft giving 40 speeds varying from i up to 150 per 
min. Fig. 22 shows the arrangement in diagrammatic form. The 
total ratio of speed range required being 150 to i, the corresponding 

■ = 6.66. 



To determine the gear ratios in the last example 
it will be seen by inspection of Fig. 23 that the 
ratios required for the four sets of gears are in 
the proportions of the first speed to the ninth, 
seventeenth, twenty-fifth and thirty-third respectively. These ratios 
are in geometrical progression with a common ratio of 2.783 to 1, and 
work out at 2.78, 7.74, 21.55, and 60. As the ratios of the various 
sets of gears are in geometrical progression, the first gear ratio only 
need actually be calculated, the second, third and fourth ratios being 
the square cube and the fourth power respectively of the first. 

Comparison of Figs. 23 and 24 shows that the desired range of 
speeds has been obtained within limits that are as accurate as the 
requirements. 



ISO 



range in the table will vary from 1000 down to 

By examining horizontal line 40 in the table we find that 6.7 in 
the 12 per cent, column is the nearest figure to 6.66, and is near 
enough for the purpose. 

The required speed range should then have a percentage of drop 
from speed to speed of 12 per cent., with 150 as a maximum. Plot 
out speeds, following the same method as in the previous examples, 
as shown in Fig. 23. 

The cone pulley in this example has four steps and the counter- 
shaft speeds must therefore be calculated, there being no middle 
step. 

The fast countershaft speed, by formula (a) 




= iSoX 



ISO 



= 123.6, 



say, 124 r.p.m. 

The slow countershaft speed may be found from the table of speeds 
given in Fig. 23, being equal to 

fast countershaft speed 



ratio of ist and sth speeds 
iSO ^ 
90 



A, B, C are equal gears. 

D F H K , . 

t;' ;^> -J' v are equal ratios = 2.78 to i. 

Arrangement of speeds: 
1st group: 8 speeds, direct. 

A D 
2d group: 8 speeds, r^X-p = 2.78 to i. 

A F D 
3d group: 8 speeds, pX 7;X •g = 7•74tOI• 
.4 ff i'" D 
4th group: 8 speeds, -gX -jX^X ^ = 21.55 to i. 

' , A R H F D ^ 
Sth group: 8 speeds, ^X-^X yX ^X -^=60 to i 



: 1244- 



^74-4 



Fig. 22. — Lathe headstock with multiple back gears, 
speeds 124 and 74.4 r.p.m. 



Countershaft 



88 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 





1st Group 


2nd Group 


3rd Group 


4th Group 


5th Group 


Direct Speed 

or im> *- 

lst,2nd,3rd or 4th Gear 


Direct Speed 


1st Gear 


2nd Gear 


3rd Gear 


4th Gear 


Counter-shaft 


Fast 


Slow 


Fast 


Slow 


Fast 


Slow 


Fast 


Slow 


Fast 


Slow 


Fast or Slow 


Stepof Cone 






















1 


2 


3 


4 


1 


2 


3 


4 


1 


2 


3 


4 


1 


2 


3 


4 


1 


2 


3 


4 


1 


2 


3 


4 


1 


2 


3 


4 


1 


2 


3 


4 


1 


2 


3 


4 


1 


2 


3 


4 




Speeds ^»— *- 


160 


182 


116 


I02.i 


90 


79.1 


09.6 


61.3 


63.9 


47.4 


41.7 


86.7 


82.3 


23.4 


ii 


ii 


I9.8i 


17 


16 


13.8 


11.6 


10.2 


8.98 


7.9 


0.90 


0.11 


6.38 


4.78 


4.16 


3.66 


3.22 


2.83 2.5 


8.19 


1.98 


1.69 


1.49 


1.3 


1.16 1.0l| 



Fig. 23. 





1st Group 


2nd Group 


3rd Group 


4th Group 


5th Group 


Direct Speed 

or ysm — >■ 
l3t,2nd,3rd or 4th Gear 


Direct Speed 


1st Gear 

2.78 to! 


2nd Gear 

7.74 to 1 


3rd Gear 

2155 to 1 


4th Gear 
eotoi 


Counter-shaft 
Speed x*«-^- 


124 


74.4 


124 


74.4 


124 


74.4 


124 


74.4 


124 


74.4 


Step of Cone 






















1 


2 


3 


4 


1 


2 


3 


4 


1 


2 


3 


4 


1 


2 


3 


4 


1 


2 


3 


4 


1 


2 


3 


4 


1 


2 


3 


4 


1 


2 


3 


4 


1 


2 


3 


4 


1 


2 


3 


i 




Speeds m» — >■ 


160 


132 


us 


102 


90 


79.2 


09.0 


61.2 


63.9 


47.4 


41.7 


30.7 


82.3 


28.4 


26 


22 


19.8^ 17 


16 


IS.li 


11.0 


10.2 


9 


7.9 


0.96 


0.12 


5.38 


4.73 


4.17 


3.07 


3.22 


2.88 


2.6 


2.2 


1.93 


1.7 


1.6 


1.8 


1.16 


i.oe 


% Drop;«i9 — s- 


22 


22 


22 


21.7 


22 


22 


22 


21.8 


22 


22 


22 


21.7 


22 


28 


22 


21.8 


-_ 


28 


22 


21.8 


28 


22 


22 


21.8 


22 


22 


22 


21.7 


22 


22 


28 


21.8 


22 


22 


22 


21.7 


22 


22 


22 



Fig. 24. 
Figs. 23 and 24. — Preliminary and final speed determinations for the case of Fig. 22. 



Table 2. — Speeds in Geometrical Progression 
Increasing Speeds from the Lowest as the Starting Point 





Ratio of 
progression 


1. 10 


I. II 


1. 12 


1. 13 


1. 14 


1. 15 


1. 16 


1. 17 


1. 18 


1 . 189 


1.20 


1 .21 


1.22 


1.23 


1.24 


1-25 


1.26 




I 


I 


I 


I 


I 


I 


I 


I 


I 


I 


I 


I 


I 


I 


I 


I 


I 


I 




2 


I .10 


1 .11 


1 . 12 


1. 13 


1. 14 


I. IS 


1. 16 


1. 17 


1. 18 


1. 19 


1 .20 


I. 21 


I .22 


1-23 


1.24 


I-2S 


1.26 




3 


I .21 


1.23 


1.25 


1.28 


1.30 


1.32 


1.34 


1.37 


1.39 


1. 41 


1.44 


1.46 


1.48 


1-51 


1.53 


1.56 


1.58 




4 


1.33 


1-37 


1 .40 


1.44 


1.48 


1.52 


1. 56 


1 .60 


1.64 


1.68 


1.72 


1.77 


I. 81 


1-86 


1.90 


1.95 


2 .00 




S 


I .46 


1-52 


I.S7 


1.63 


1 .69 


1.75 


1. 81 


1.87 


1.94 


2.00 


2.07 


2.14 


2.21 


2.28 


2.36 


2.44 


2.52 




6 


1. 61 


1.68 


1.76 


1.84 


1 .92 


2.01 


2.10 


2.19 


2.20 


2.38 


2.48 


2.59 


2.70 


2.81 


2.93 


3.0s 


3.17 




7 


1.77 


1.87 


X.97 


2.08 


2.19 


2.31 


2-44 


2.56 


2.70 


2.83 


2.98 


3.13 


3.29 


3.46 


3.63 


3.81 


4.00 




8 


1.95 


2.07 


2.21 


2.35 


2.50 


2.66 


2.83 


3-00 


3.18 


3.36 


3.58 


3.79 


4.02 


4.2s 


4. 51 


4.76 


S.04 




9 


2.14 


2.30 


2.47 


2.66 


2.85 


3.06 


3-28 


3-51 


3.75 


4.00 


4.30 


4.59 


4.90 


S.23 


5.59 


5-9S 


6.35 




10 


2.36 


2. 55 


2-77 


3.00 


3.2s 


3.51 


3-80 


4. II 


4.43 


4.76 


5. 16 


S.55 


5.98 


6.44 


6.93 


7-44 


8.00 




II 


2.59 


2 .84 


3-10 


3.39 


3-70 


4.04 


4.41 


4.81 


5. 23 


5.66 


6.19 


6.72 


7.29 


7.92 


8.59 


9-30 


10.08 




12 


2.8s 


3.15 


3-47 


3-83 


4.22 


4.65 


S-12 


5.62 


6.17 


6.73 


7.43 


8.13 


8.90 


9.74 


10.66 


11-63 


12 .70 




13 


3.14 


3-49 


3-89 


4-33 


4.81 


5.34 


5-94 


6.58 


7.28 


8.00 


8.91 


9.84 


10.86 


11.98 


13.22 


14 -54 


16-OI 




14 


3-45 


3-88 


4-36 


4-89 


S.49 


6. IS 


6-89 


7.70 


8.59 


9. SI 


10.70 


II .91 


13.25 


14.74 


16.39 


18.17 


20.17 




IS 


3 .80 


4-30 


4.88 


5.53 


6.26 


7.07 


7-99 


9.00 


10.13 


II .31 


12.84 


14.41 


16 . 17 


18.13 


20.32 


22.71 


25.42 




16 


4.18 


4.78 


5-47 


6-25 


7-13 


8.13 


9.27 


10.54 


11 .96 


13.45 


IS. 41 


17.43 


19.72 


22 .30 


25.20 


28.39 


32 .02 




17 


4 -39 


S-30 


6.12 


7.06 


8.13 


9.35 


10.75 


12.33 


14.11 


16.00 


18.49 


21 . 10 


24.06 


27.43 


31.25 


35. 49 


40.3s 




18 


S.OS 


S-88 


6.86 


7-98 


9.27 


10.75 


12.47 


14.42 


16.6s 


19 03 


22 .19 


2S.S3 


29.36 


33.73 


38.75 


44.37 


50.84 




19 


5.S6 


6-53 


7-68 


9.02 


10.57 


12 .36 


14.47 


16.87 


19.65 


22.63 


26.62 


30.89 


35.82 


41.49 


48.05 


55.46 


64.06 




20 


6. II 


7-25 


8.60 


10.19 


12.05 


14.22 


16.78 


19.74 


23.19 


26.91 


31.95 


37.37 


43-70 


51.04 


59.58 


69.33 


80.72 




21 


6,72 


8.05 


9-64 


II. SI 


13.73 


16.35 


19.46 


23.10 


27.36 


32.00 


38.34 


45.22 


53-31 


62.78 


73.89 


86.66 


101 .7 




22 


7.40 


8.93 


10.79 


13 .01 


15-65 


18.80 


22,58 


27 .02 


32.28 


38.05 


46.01 


54.72 


65-04 


77.22 


91 .62 


108.3 


128. I 




23 


8.14 


9-91 


12.09 


14.70 


17.8s 


21 .62 


26. 19 


31.62 


38.09 


45-25 


55.21 


66.21 


79-35 


94 98 


113. 6 


135.4 


i6i -4 




24 


8. 95 


II .00 


13-54 


16.61 


20.35 


24-86 


30.38 


36.99 


44.95 


53.82 


66.26 


80.12 


96.81 


116.8 


140.8 


169.2 


203-4 




25 


9.85 


12.21 


15.16 


18.77 


23-19 


28.59 


35.24 


43-28 


53.04 


64.00 


79.51 


96.95 


118. 1 


143.7 


174.6 


211 .5 


256.3 




26 


10.83 


13.56 


16.98 


21.21 


26.44 


32.88 


40.88 


50.64 


62.59 


76.11 


95.4 


117. 3 


144. 1 


176.7 


216.6 


264.4 


323 -0 




27 


II .91 


IS -OS 


19-02 


23.97 


30.14 


37.81 


47.42 


59.25 


73-86 


90.51 


114.4 


141. 9 


175.8 


217.4 


268.6 


330.6 


407-0 




28 


13 . 10 


16-70 


21 .30 


27 .08 


34 36 


43.48 


55.01 


69.32 


87.15 


107 .6 


137.3 


171. 7 


214.4 


267.4 


333.0 


413.2 


512.8 




29 


14.41 


18.54 


23-86 


30-60 


39.17 


50.01 


63.81 


81. II 


102 .8 


128.0 


164.8 


207.8 


261 .6 


328.9 


4130 


516.5 


646.1 




30 


15.86 


20.58 


26.72 


34-58 


44-66 


57. SI 


74-02 


94.89 


121. 3 


152 .0 


197.8 


251.4 


319.2 


404.5 


512. 1 


645.7 


814. 1 




31 


17.44 


22.84 


29.92 


39.07 


50.91 


66.13 


85.87 


III .0 


143. 1 


181 .0 


237.4 


304.2 


389.4 


497.6 


635.0 


807 .1 


I02S .0 




32 


19.19 


25-36 


33-52 


44-15 


58.04 


76.0s 


99-61 


129.9 


168.9 


215.3 


284.9 


368.1 


475.1 


612 .0 


787.4 


1008.0 


1292 .0 




33 


21 . 10 


28.14 


37.54 


49-89 


66.16 


87.46 


115-S 


151.9 


199.3 


256.0 


341.8 


445.4 


579.6 


752.8 


976.4 


1261 .0 






34 


23.21 


31.24 


42.04 


56.38 


75-43 


100.5 


134-0 


177.8 


235.2 


304.4 


410.2 


539-0 


707.2 


925.9 


1210.0 


1576.0 






3S 


25-54 


34-68 


47-09 


63.70 


85-99 


115 .6 


155-5 


208.0 


277.6 


362.0 


492.3 


652.2 


862.8 


1138.0 


1501 .0 








36 


28.09 


38.49 


S2-74 


71.99 


98.02 


133.0 


180.3 


243-4 


327. S 


430. S 


590.7 


789.1 


1052.0 


1400.0 










37 


30.90 


42.72 


59-06 


81.35 


III. 7 


IS2-9 


209.2 


284.8 


386. S 


512 .0 


708.9 


954-9 


1284.0 












38 


33-99 


47.42 


66.15 


91.92 


127.4 


175. 9 


242.7 


333-2 


4S6.I 


608.9 


850.7 


"55-0 


1566.0 












39 


37.39 


52.64 


74-09 


103.8 


145.2 


202.3 


281.5 


389.8 


538.2 


724.0 


1020.0 


1398 -0 














40 


41 13 


58.43 


82.98 


117. 3 


165. 5 


232 .6 


326. 5 


456.1 


635.1 


861. 1 


1225 .0 















CONE PULLEYS AND BACK GEARS 



89 



Table 2. — Speeds in Geometrical Progression— (Co«i«wMe(i) 
Increasing Speeds from the Lowest as the Starting Point 





Ratio of 

progression 


1.27 


1.28 


1.30 


1.32 


1-34 


1.36 


1.38 


1.414 


1.4s 


1.50 


1.55 


1 .60 




I 


I 


I 


I 


I 


1 


I 


I 


I 


1 


1 


1 


1 




2 


1.27 


1.28 


1.30 


1-32 


1-34 


1.36 


1.38 


1.41 


I -45 


1.50 


1-55 


1 .60 




3 


I. 61 


1.63 


1.69 


1.74 


1-79 


1.85 


1 .90 


2.00 


2.10 


2.25 


2.40 


2 .50 




4 


2 .04 


2.09 


2.19 


2 .29 


2.40 


2.51 


2.62 


2.83 


3.04 


3-37 


3-72 


4-09 




S 


2 .60 


2.68 


2.85 


3.03 


3.22 


3-42 


3.62 


4.00 


4.42 


5 -06 


S-77 


6.55 




6 


3.30 


3.43 


3.71 


4.00 


4-32 


4-65 


5.00 


S.66 


6.40 


7-59 


8.94 


10.48 




7 


4.19 


4-39 


4.82 


S.28 


5-79 


6-32 


6.90 


8.00 


9.29 


II-38 


13.86 


16.77 




8 


5.32 


S.62 


6.27 


6.98 


7.75 


8.60 


9.52 


11.31 


13.47 


17-08 


21.49 


26.84 




9 


6.76 


7.20 


8. IS 


9.21 


10.39 


11.70 


13.14 


16.00 


19.53 


25.62 


33-31 


42.95 




10 


8.59 


9.22 


10.60 


12.16 


13.93 


15.91 


18.14 


22.63 


28.33 


38.43 


51.63 


68.72 




II 


10.91 


11.80 


13.78 


16. OS 


18.66 


21.65 


25.03 


32.00 


41.08 


57.65 


80.03 


109.9 




12 


13-85 


15.10 


17.92 


21.19 


25.01 


29-44 


34-54 


45.25 


59.56 


86.48 


124.0 


175-9 




13 


17.60 


19.33 


23.29 


27.97 


33.52 


40.04 


47.67 


64.00 


86.37 


129.7 


192 .2 


281 .4 




14 


22.35 


24-75 


30.28 


36.92 


44-91 


54-46 


65.79 


90.51 


125.2 


194. S 


298.0 


450.3 


0) 

04 


15 


28.38 


31.68 


39.37 


48.73 


60.19 


74-06 


90.79 


128.0 


181. 5 


291.8 


461.9 


720.6 


"o 


16 


36.05 


40.5s 


SI. 18 


64-33 


80.65 


100.7 


125.2 


181.0 


263.3 


437-8 


715.9 


1152 .0 




. 17 


45.78 


SI. 91 


66.53 


84.92 


108.0 


136.9 


172.9 


256.0 


381.8 


656.7 


I I 09.0 


1844.0 


B 

2 


18 


58. 14 


66.44 


86.49 


112. 


144.8 


186.3 


238.6 


362.0 


553.6 


985.0 


1720.0 




19 


73.84 


85.0s 


112. 4 


147.9 


194-0 


253.3 


329-2 


512.0 


802.7 


1477-0 








20 


93.78 


108.8 


146. 1 


195-3 


260.0 


344-5 


454-3 


724.0 


1163.0 










21 


119. 1 


139.3 


190.0 


257.8 


348.4 


468.6 


627.0 


1024.0 


1687.0 










22 


151 .2 


178.3 


247.0 


340.3 


466.9 


637.3 


865-3 


1444.0 












23 


192 . 1 


228.3 


321. 1 


449-2 


625.6 


866.8 


1194.0 














24 


243-9 


292.2 


417.4 


592.9 


838.4 


1178.0 


1647.0 














25 


309.8 


374-0 


542.7 


782.7 


I123.0 


1603.0 
















26 


393-5 


478.7 


70s. 5 


1033-0 


1505.0 


















27 


499-7 


612.8 


917.2 


1363.0 




















28 


634-7 


784.4 


1192.0 






















29 


806.0 


1004.0 


1550. 






















30 


1023.0 


1285 .0 
























31 


1300,0 

























Increasing Speed Ratios 

In the above arithmetical solution the series begins with the high- 
est speed from which the others are obtained by working downwards. 
In the United States the opposite procedure is more common, the 
lowest speed being taken as the starting point and the others obtained 
by working upwards. Table 2 {Amer. Mack., March 2, 1916) is a 
companion to Table i, but arranged in accordance with American 
practice. 

The use of Table 2 is as follows : Find the desired ratio of the pro- 
gression at the top of one of the columns. The numbers in this 
column are multipliers which, multiplied by the lowest speed in 
r.p.m., give the other speeds of the series, the multiplier for the high- 
est speed being the over-all ratio of the set. More often the problem 
must be worked in the opposite direction, in which case divide the 
highest by the lowest desired speed, thus finding the over-all ratio 
of the set. In the left-hand column find the number of speeds 
desired and follow its line to the right until the nearest value to the 
over-all ratio is found. At the top of this column will be foutid the 
ratio of the progression, the numbers in the column forming, as be- 
fore, multipliers which, multiplied by the lowest speed, give the 
other speeds of the set as accurately as is possible without smaller 
ratio increments, those given being small enough for all practical 
requirements. 

A column based on Mr. Earth's ratio (\/2 = 1.189) is included. 
In this series each speed is exactly twice that of the fourth one above 
it. This relation is most convenient and the low value of the ratio 
of the progression permits close adjustment of the speed to the re- 
quirements. It, however, introduces more speeds than many de- 
signers will admit. In such cases the modified ratio (\/2 = r.4i) 
which is also included in the table may be used if it is not considered 
too large, which it usually should be. In this series each speed is 
exactly twice that of the second one above it. 



Prevailing values of the ratio between successive speeds formed the 
subject of an investigation by Prof. A. Lewis Jenkins {Amer. Mach., 
Apr. 13, 1916) who examined about 400 lathes and other tools. The 
average values found for lathes of American make are as follows: 

Average value 

Type of lathe of ratio 

Three-step cone, double back gear, 9 speeds 1.5 

Three-step cone, double back gear, 18 speeds 1.22 

Four-step cone, single back gear, 8 speeds 1.69 

Four-step cone, single back gear, 16 speeds 1.3 

Five-step cone, single back gear, 10 speeds i - 58 

Five-step cone, single back gear, 20 speeds 1.26 

All-geared head, 8 speeds 1.58 

All-geared head, 12 speeds 1.36 

All-geared head, 16 speeds 1.26 

All-geared head, 18 speeds i . 23 

Taking the average of these values for 8, 9 and 10 speeds gives 
1.58, and for 16, 18 and 20 speeds gives 1.25. 

For radial drilling machines having 20 spindle speeds the values 
vary from 1.27 to 1.35, the average being about 1.3. The value for 
vertical drilling machines having 8 spindle speeds varies with the size 
of the machine and is equal to r = .96755'^ where 5= size of machine 
which gives 1.49 for a 20-in. machine and 1.61 for a 36-in. machine. 
A constant ratio of 1.44 has been proposed for vertical drilling ma- 
chines having 10 spindle speeds. 

The high values accompany a low number of speeds and are the 
result of the attempt to cover a wide total range with an insufficient 
number of speeds . 

Planetary Back Gears 

The proportions of planetary back gears have been worked out by 
E. J. Lees {Amer. Mach., March i, 1906) as given in Table 3. Three 



90 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



idlers are used to give correct balance when locked up and driven 
direct. The general arrangement is shown in Fig. 25, which neces- 
sitates that the number of teeth in all gears shall be divisible by 
three. 

Table 3. — Proportions of Planetary Back Gears 



Size No. 



Diam. of pulley .... 

Face of pulley 

Width of belt 

Approx. h.p. at 300 

r.p.m. 

Shaft diam. D 

Pitch diam. pinion A 

No. teeth \n A 

Pitch diam. internal 

gear B. 

No. teeth in B 

Pitch diam. idler C. . 

No. teeth in C 

No. of idlers 

Diam. pitch of gears. 
Face of gears 



Ratio 



I 


2 


3 


4 


5 


6 


7 


8 


9 


10 


III 


8 


8 


10 


10 


12 


12 


IS 


15 


IS 


15 


18 


3 


3 


3i 


3i 


Ai 


4i 


Si 


Si 


6i 


61 


■ih 


2j 


2i 


3 


3 


A 


4 


5 


5 


6 


6 


7 


2h 


2i 


4i 


Ai 


7i 


7i 


II 


II 


13 


13 


18 


Ih 


li 


l^ 


Ij 


If 


If 


li 


li 


2 


2 


3 


2\ 


Ah 


2* 


5^ 


2 


5? 


2% 


6 


3 


9 


S 


18 


36 


18 


36 


18 


36 


18 


42 


18 


54 


IS 


9 


9 


107 


I0| 


10 


107 


I2f 


12I 


18 


18 


2S 


72 


72 


72 


72 


72 


72 


90 


90 


108 


108 


75 


3l 


2i 


3? 


2I 


3? 


2', 


5\ 


3? 


7i 


4i 


10 


27 


18 


27 


18 


27 


18 


36 


24 


4S 


27 


30 


3 


3 


3 


3 


3 


3 


3 


3 


3 


3 


3 


8 


8 


7 


7 


7 


7 


7 


7 


6 


6 


3 


l\ 


li 


li 


ih 


If 


If 


l| 


If 


2 


2 


3 


5 


3 


S 


3 


5 


3 


6 


3-142 


7 


3 


6 . 


to 


to 


to 


to 


to 


to 


to 


to 


to 


to 


to 


1 I 


I 


I 


I 


I 


I 


I 


I 


I 


I 


I 



12 



18 

75 

7 
18 

3 
13 
39 
25 

7S 
6 
18 
3 
3 
3 
2.923 
to 
I 




Fig. 25. — Planetary back gearing, 



Gear Ratios for Motor Drives 



Gear ratios for motor drives as pointed out by W. OwEN {Amer^ 
Mack., March 28, 1907) are frequently arranged to advance in multi- 
ples of the total speed ratio of the motor — a method which results in 
the highest speed with each gear in, duplicating the lowest with 




that gear out and in reducing the total range. These results are 
shown in Fig. 26, in which the motor ratio is three to one. Were 
the speeds made to advance as the gears are thrown out the result 

would be that shown in Fig. 27, which gives a total range of — 5-^ = 

ol 

1.51 times that of Fig. 26. 

The speeds of variable-speed motors are frequently arranged in 
arithmetical progression, as indicated in the illustrations which, while 
not as it should be, does not prevent the gear ratios being in geomet- 
rical progression thus giving most of the advantages of that system. 
Let 5 = highest motor speed, 
.J = lowest motor speed, 
« = number of speeds on motor, 
r = ratio of advance. 
Were the speeds of the motor arranged in geometrical progression 
the ratio of advance would be 

and, using this for the gear changes, the ratio of first gear change 



ratio of second gear change = 



- { (^ 



and so on for the other gear changes. 

Table 4 of back gear ratios for motor drives gives the correct gear 
ratios for most cases arising in practice. 

Table 4. — Back Gear Changes for Motor Drives 



^ .2 



V 






.Q 






l-s 


u 


a 



c 5 


c 


"i 

11 


^0 


.2 ^ 
■a "o 



3 ft 

2 " 



20.07 

20.7 

21.53 

22.53 

24.21 

26.84 
3145 
31.96 

34-43 
36.84 

38.9 
41.86 
45-24 
46.65 
SO. 52 

S6.I 

56.98 

63-8 

80.48 

80.91 

116. 7 
121 .9 
127-9 
127-9 
129. 1 

140.2 



40 


1.080 


2 


36 


1.090 


2 


32 


1 .104 


2 


28 


I. 122 


2 


24 


1.149 


2 


20 


I. 190 


2 


27 


I. 148 


3 


16 


1.260 


2 


30 


I. 130 


3 


24 


1. 169 


3 


21 


1.200 


3 


18 


1-247 


3 


12 


I.4IS 


2 


IS 


1.316 


3 


35 


I. 122 


2 


12 


1-442 


3 


30 


I -149 


2 


25 


I. 190 


2 


20 


1.260 


2 


9 


1-732 


3 


40 


I -130 


3 


36 


I. 148 


3 


IS 


I-415 


2 


36 


I -149 


2 


32 


I . 169 


3 


28 


1.200 


3 



10 
9 
8 
7 
6 

5 

9 

4 

10 



a rt rt 



Ratios of gear changes 



2.16 


4.667 


2.18 


4-753 


2.208 


4-873 


2.244 


5 035 


2.298 


5 -279 


2.380 


5-665 


3-444 


11.87 


2.S20 


6. 351 


3-39 


11-49 


3-507 


12.3 


3.6 


12.96 


3.741 


14.0 


2.830 


7-998 


3-948 


1559 


2.244 


5-035 


4.326 


18.72 


2.298 


S.279 


2.38 


5-665 


2.52 


6-351 


5-196 


27.07 


3-39 


11-49 


3-444 


H-87 


2.830 


7-998 


2 .298 


5-279 


3-507 


12.3 


3.6 


12. 96 



10.08 
10.36 
10.76 
11.30 

12,14 

13-48 

16.01 



22.63 
11.30 



12.14 

13-48 

16.01 



38.95 

40.86 
22.63 
12.14 
43-15 

46.65 



25-35 



27.87 
32.09 
40.33 



64. IS 
27.87 



64.04 



Fig. 26. Fig. 27. 

Figs. 26 and 27. — Back gear ratios for motor drives. 



Gear-box Construction 

The substitution of gear boxes for cone pulleys in feed gearing for 
machine tools has led to numerous constructions. The following 
analysis of some of the leading arrangements by A. M. SoSA (Amer. 
Mach., Feb. 15, 1906) will be of assistance to beginners in this field 
of work. 



CONE PULLEYS AND BACK GEARS 



91 



Fig. 33. 

Threads per Inch 

1 1>4 la 1?4 

2 2« 3 

3 5 

4 10 12 



3}i 
1 
11 



Fig. 28 shows the most usual way of applying the cone-gear 
mechanism, in which R indicates the driving and D the driven shaft. 
This arrangement is most economical for drives, where the power 
transmitted is constant. The largest gear gives the slowest speed 
and maximum torque, the smallest vice versa, and the pressure on 
all gear teeth, as well as lineal velocity, is constant. 

Fig. 29 is the inverse of Fig. 28. The cone is on the driving and 
the sliding pinion on the driven shaft, 
which may be connected to the feed 
screw, or it may be the feed screw 
itself. The sliding pinion is of the 
same diameter as the largest cone-gear, 
and gives the i to i rates, which is the 
fastest, and requires the maximum 
torque. 

This construction gives low speeds, 
as one revolution of the screw gives a 
feed equal to its pitch, and very coarse 
pitches are most in use. The power 
is a maximum for the coarsest feed 
and is transmitted through two large 
gears instead of two small pinions. 
The large gear on the screw reduces 
the pressure on the gear teeth, per- 
mits the use of small pitches and gives 
a compact arrangement. 

At the same time an increase of 
lineal velocity of the gear teeth is the 
result, but this is a rather desirable 
feature when low speeds are concerned. 
The ratio of speeds obtained by this 
method is as the ratio of the diameter 
of the largest and smallest cone-gears. 

For a ratio 4 to i, starting with a 
14-tooth pinion, the largest would be 
a 56-tooth gear. This difference in 
diameter of gears is cumbersome and 
marks the limit as to the cone ratio. 
Next to the feed-gear box or in some 
other part of the machine is usually 
found a second box containing four 
gears and a clutch, called the speed 
clutch box, which is nothing more than 
a back gear. 

It seems more economical, when 
possible, to place this back gear on the 
cone gear itself as shown in Fig. 30, in 
which the cone is in one piece and 
runs loose. The clutch slides, keyed 
on the shaft, and transmits the mo- 
tion to the cone directly or through 
the back gear. With gears 4 to i in 
diameter, as shown, a ratio of feeds of 
32 to I in round numbers is conven- 
iently obtained. 

Fig. 31 shows a combination for six 
feeds, speed ratio 12 to i, and gear 

ratio 3 to i. This is operated by one lever, the tumbler lever only. 
If the running speeds are low, it does not seem to be an objection 
to have all the gears running. This cone is in two parts; the three 
gears at the left are keyed to shaft R and the three at the right 
run loose, are in one piece on the same shaft and receive motion 
through the back gear as shown. Compounding two cones in this 
manner gives a very large ratio, with relatively small gears. 

Fig. 32 illustrates the use of four gears with a diameter ratio of 
2 to I only, giving a ratio for the cone of 8 to i. The arrangement 
is the same as in Fig. 31. 



In Fig. 33 the number of teeth are given, the smallest pinion 
having 14 and the largest gear 28 teeth. The first two gears at the 
left are keyed to shaft R, the other two running loose on a sleeve. 
The cone runs loose and is in one piece. Both clutches slide on keys 
on the cone shaft, and the other three gears run loose on the cone 
shaft. This arrangement seems very convenient for screw cutting. 
The table of threads per inch is shown in the figure. When the feeds 




Fig. 34 

20 tol 



















"■ 










r- 












































/ 




































Uj 




































/ 


































/ 




































/ 


































/ 
























?d 










/ 




































































/ 














































-_ 


-i 


31 


IW 


















/' 
























/' 












1 


































^ 






































i 


















_ 


— 


" 




' 








H 


L. 

















Fast 



4 6 8 10 12 14 
Speed Numbers 

Fig. 37. 
Speeds of Figs.36 and 39. 




Fig. 36. 




Fig. 39. 

Figs. 28 to 40. — Typical arrangements of geared feed boxes. 



per inch are arranged in successive groups, and each group is a multi- 
ple of the previous one, the gear ratios can be easily seen. The last 
group gives more directly the ratios of cone gears. The first number 
of first and second groups, gives the ratios for the clutch gears i to 
I — 2 to I. The first number of the third group (4), gives the back- 
gear ratio 4 to I. And the first number of the last group is the prod- 
uct of clutch gear (2) and back gear (4), and represents their 
combination. 

Fig. 34 represents a type of drive with cone and sliding gears. 
The compounding of cones suggested for feed gears would not seem 



92 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



advisable for the reasons previously explained, but compounding 
the tumbler gear would not change the conditions materially, and 
double the number of speeds may be obtained. What is generally 
the idler is, in this case, made of two gears in one piece, running loose 
on the stud. The ratio of these gears is equal to the percentage of 
increase of the two next speeds, in this case about 1.2 to i. The 
cone gears are spaced at a distance equal to the width of face plus 
clearance. The gears are drawn to scale, relative to each other, and 
represent a geometric progression of 16 speeds with a ratio of 20 to i. 
The total number of gears used is only nine. 

For heavy drives it is not possible to make combinations in the 
manner previously stated, and the back gears are more successfully 
grouped separately. 

Fig. 35 shows a combination of four gears. The ratios are as i, 
25, 6j and 15^, to i. Different considerations enter into this prob- 
lem, such as the distribution and the alignment of bearings, the 
elimination of sleeves and running fits under pressure or torsion 
and reducing the number of clutches, operating levers and 
interlocking devices. 

Other typical gear box arrangements are discussed by H. T. 
Millar {Amer. Mack., Dec. 14, 1905) as follows: 

It is possible to obtain twenty-one changes in geometrical pro- 
gression with twelve gears and four shafts. Fig. 36 is a develop- 
ment of the motion, the dimensions of which show the relative sizes. 
It is always better to lay the gears out in this manner first, then to 
figure the absolute sizes in consideration of the actual requirements 
of the case, which may limit the size of the largest or smallest of the 
gears. The speed ratio is 34 to i and the rises are shown in the 
chart, Fig. 37. Wheels J, / and K, Fig. 36, gear with corresponding 
wheels H, F and B. I, J and K are mounted on a splined shaft, 
which is the final shaft of the motion. The connection between 
pinion A and the train BC — H is made by the ordinary sliding wheel 
and tumble shaft, not shown; the third shaft of the motion M has 
seven speeds of revolution, corresponding to the gears mounted on 
it. For each of these the final shaft N has three, obtained by put- 
ting either I, J or K into gear. Obviously only one of these must 
be in mesh at a time, and it is an advantage to have only one handle 
to move them. If two handles are used they must be interlocked. 

Combinations of Gears 



F and /, having to keep a fixed center distance, and Fig. 39 shows 
the obvious remedy. In a similar manner four pairs of wheels 
could be used, giving four changes for every speed of the shaft 
above. The wheels on the driven shaft would need to be coupled 
together in pairs and moved in and out of mesh by a cam, or a pair 
of interlocking segments to be described later. 

When there are only two gears on the driven shaft, as on the 
Brown & Sharpe gear cutter, the segment A, Fig. 40, provides a neat 
method of moving the slow and fast gears. As is evident from the 



/debfcdcf bdadafaece\ 
bd ad af ae ce bf be cd ^" 



30 



20 




;o 



4^ 



G.5 to 1 



CF 



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 
Speed Numbers 

Fig. 41. — Speeds of Fig. 42. 

The details of a cam arrangement are shown in Fig. 38. The stir- 
nips slide loosely on the rod M, and are moved by the pins running 
in the cam, which turns half a revolution. 

Returning to Fig. 36, the dimensions of the gears BC — H rise in 
geometric ratio and need no comment. 

With some rises it is impossible to obtain a correct ratio between 





Fig. 42. — Typical arrangement of geared feed boxes. 

sketch it moves one gear out of mesh before putting the other in, 
and prevents the breakage which might occur through leaving both 
gears half in mesh. It also enables the gears to be put in while in 
motion if necessary. The rings R, kept from revolving by a rod 
passing through eyes at the back of the shaft, have semicircular 
pins set in them. Starting at the top sketch, the ring i?i is moved 
endwise until the pin 5 comes out of the recess in the segment. 
At that time the end of the segment comes in contact with the 
other pin T, moving it along; the pin S being now clear of the 
segment. 

The great advantage of all these arrangements is that no gear is 
in mesh except those actually doing the work and the number of 
these is kept at a minimum. 

If the speeds shown in Fig. 41 are near enough to correct pro- 
gression, the layout of Fig. 42 meets some cases. It is impossible 
to obtain absolutely correct rises, but those shown are near enough 
for the majority of purposes. As shown there are eighteen changes 
with a ratio of 34 to i. The three wheels keyed on the left-hand 
end of shaft G are stationary endwise. The sliding wheels CBA 
and DEF are moved on their shafts by cams laid out on both sides 
of plate P. Lever R fulfills a double purpose; it acts as an index for 
the nine different positions of the cam plate and it also stops the 
gears before any change takes place. When the plate is turned, it 
lifts lever R, which is connected by a shaft to clutch S. The clutch 
K puts in either the slow or fast gears. If the motion had to be 
placed in an inaccessible position a bevel or other gear could be 
mounted in place of the handle, and connections made by shafts 
to some convenient position. 

It is well to notice that the wheels in this motion should have 
varying widths. Each wheel has a different periphery speed and 
consequently a different load on the pitch line. 



I 



SPUR GEARS 



The movement away from the cpicycloidal and toward the in- 
volute system of gear-tooth profile has now reached the point where, 
for gears of small and moderate sizes, the involute system is prac- 
tically universal. The details of this system are, however, still a 
subject of controversy. For heavy mill gearing, whether cut or 
cast, the cpicycloidal system is still in large use. 

Fig. I illustrates the generation of an involute. Two rollers, the 
circumferences of which are the base circles, are connected by a 
tangent cord which carries a tracing point as shown. When the 
parts are moved as shown by the arrows, the cord being kept taut, 
the tracing point traces an involute on the card ahci attached to 
the lower roller. This involute is a correct tooth profile for the lower 
one of a pair of gears having pitch circles as shown. The profile 
for the upper gear is traced in the same way by attaching the card 
to the upper roller. 

In order to satisfy the geometrical conditions it is only necessary 
that the diameters of the rollers have the same ratio as the pitch 
circles. Since an indefinite num- 
ber of rollers having a given ratio 
are possible, it follows that an 
indefinite number of involutes 
and tooth profiles are possible 
for every pair of pitch circles. 

The line e} of the cord is the 
line of action and the angle egh 
between this line and the com- 
mon tangent to the pitch circles 
is the angle of obliquity or the 
pressure angle. This angle is 
obviously determined by the 
diameters of the base circles 
selected and the value of the 
angle is the leading subject of 
controversy. 




Fig. I. 



-Generation of an 
involute. 



Fig. 2. — Interference of involute 
gear teeth. 



nircnla r "Pitc h 



—J 




P = Circular Pitch 
p = Diametral Pitch 



A feature of the involute system is the interference between the pro- 
files when high- and low-numbered gears are in mesh. This inter- 
ference, for a pressure angle of 
14^ deg. and between a twelve- 
tooth pinion and a rack is illus- 
trated in Fig. 2, which shows how 
the outer end of the rack tooth 
cuts into the flank of the pinion 
tooth. The interference grows 
less as the number of teeth in 
the pinion is increased and in 
gears of the 145-deg. system 
having the usual addendum, it 
disappears with a pinion of 30 
teeth meshing with a rack. 

Fig. 2 shows the interference 
for true involute teeth of 145 deg. 
obliquity which, as a matter of 
fact, are not made. The discus- 



Table 



-Details of Involute Gear-tooth Systems 



Brown and Sharpe 



Sellers 



Hunt 
stub tooth 



Logue 
stub tooth 



Fellows 
stub tooth 



Adamson 
stub tooth 



Pressure angle. 
Addendum. . . . 



Working depth. 
Whole depth. . . 
Clearance 



145 deg. 

.21&3P OT 

. 6366P or 



. 6866P or 



.05P 



P 
2 

P 
2-1571 



20 deg. 
.3P 



14 i deg. 
25P or 



.7854 



20 deg. 



P 
1571 



■05P 



. 50P or 
• 5SP or 
. 05P or 



P 
•5708 



P 
1.7279 



P 
■ 1571 



. 2sP or 

■ 5°P or 

■ 55P or 

■ 05P or 



.7854 

P 
1-5708 

P 

1-7279 

P 

-1571 



20 deg. 

See below for 
values of 
addendum. 



15 deg. 

-25^^ 

-SoP 
■57^' 
.07P 



Values of Fellows Addendum 

Diametral pitch 4567 8 9 10 12 

Addendum, in ^ 4r ^ i x ff iS xV it 



Involute Tooth Systems 

The most common pressure angle is 14I deg., being that of 
the Brown and Sharpe system. "■ The pressure angle of the Sellers 
system — due to Wilfred Lewis — is 20 deg., which also is the 
angle of the Fellows^ and Logue stub-tooth systems. The Hunt 
stub-tooth system has an angle of 145 deg. 

' The Brown and Sharpe Mfg. Co. make cutters for other as well as for 
the standard angle. 

* The Fellows Gear Shaper Co. make cutters for a pressure angle of 14 J 
deg. also. 



sion of the angles has been obscured by its limitation to true 
involutes. As made by the Brown & Sharpe Mfg. Co. the 
tooth outlines, whatever the obliquity, are modified by rounding 
the points of the teeth in order to accommodate unavoidable im- 
perfections of workmanship and bring about more quiet action. 
This rounding also permits filling in the undercut of 14^ deg. low- 
numbered pinions, thus restoring most of the loss of strength due 
to the undercut of unmodified involute profiles. 

The advocates of unmodified involute profiles urge an increase 
of obliquity as a means of avoiding the undercut. As the angle 



93 



94 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



is increased the number of pinion teeth requiring modification for 
interference is reduced until, with an angle of 225 deg., the length 
of the tooth remaining unchanged, it disappears under the extreme 
condition of a 12-tooth pinion meshing with a rack. 

Most constructors have hesitated to adopt so large an angle and 
a compromise suggestion has been made to increase the angle to 20 
deg. and at the same time reduce the length of the teeth. The 
Fellows and Logue stub-tooth systems (the latter used by R. D. Nut- 
tall Co.) embody these features. 

The shorter tooth has also been strongly advocated for heavy 
mill gearing. It seems to have found greater adoption for this pur- 
pose in England than in the United States, the chief user in the latter 
country, so far as known to the author, being the C; W. Hunt Co. 
The object of this change in heavy gearing is to secure increased 
strength. Shortening the teeth without increasing the pressure angle 
will not avoid interference with pinions having as few as 12 teeth. 
The C. W. Hunt Co. prefer not to use pinions having less than 19 
teeth. 

Table i gives the principal details of the above-named systems 
and of the Adamson (British) system, the notation being given in 
the illustration above the table. ^ The Adamson {Jos. Adamson 6* 
Co.) system is based on the recommendations of Michael Long- 
ridge, the leading advocate of the stub tooth in Great Britain. 



Dimensions of Gear Teeth 

The dimensions of gears by diametral pitch and of the Brown and 
Sharpe standard may be determined from the well-known formulas 
of Table 2 by the Brown & Sharpe Mfg. Co. in which 



Table 2. — Formulas for Dimensions of Involute 
OF THE Brown and Sharpe Standard 
for a Single Wheel 



Formulas 



Examples 



P = 



N + 2 72-t-2 72-I-2 



D 



6.166 



TV 72 

^, DXN 6.166X72 , 
JJ =irf~; — = i = 6. 

iV-t-2 72-I-2 

p 12 
N = pD' = i2X6 = T2. 
N = pD — 2 = 12X6.166-2, or i2X6A-2 = 72. 

„ N+2 72-I-2 
D =-—^ = ^^—!- =6.166, or 61^^. 
p 12 ) 1- 

D = D'+~^ = 6+—, or 6-f-. 166 = 6.166. 
p 12 

p 12 '^^°' 

•' 10 10 ■°'^' 
Z)"4-y = .166+.013 = .179. 

7Z 3.I416 
P = ~i = - =.262. 



P = 



n ^ 3.1416 
P .262 



Gears 



(i) 
(2) 
(3) 

(4) 

(5) 
(6) 
(7) 

(8) 

(9) 
(10) 

(11) 
(12) 

(13) 
(14) 



p = diametral pitch or the number of teeth to i in. of diameter 

of pitch circle. 
P = circular pitch or the distance from the center of one tooth 

to the center of the next on the pitch circle, ins. 
T)' = diameter of pitch circle, ins. 
D = whole diameter, ins. I Larger 

iV = number of teeth [ wheel 

F = velocity J \ These wheels 

d' = diameterof pitch circle, ins. ] | run together 

<f = whole diameter, ins. ( Smaller 

n = number of teeth , wheel 

71 = velocity 

= distance between centers of the two wheels, ins. 

6 = number of teeth in both wheels. 

/ = thickness of tooth or cutter on pitch circle, ins. 
Z)" = working depth of tooth, ins. 

/ = amount added to depth of tooth for rounding the corners 
and for clearance, ins. 
Z)"H-/= whole depth of tooth, ins. 
^ =3.1416. 

The examples placed opposite the formulas are for a single wheel 
of 12 pitch 6.166 or 6x\ ins. diameter, etc., and in the case of the two 
wheels the larger has the same dimensions. The velocities are 
respectively i and 2. 

A list of tooth parts according to the Brown & Sharpe system will 
be found in Tables 6 and 8 and a list of useful multipliers in Table 5. 
See also Table 19. 

1 In one respect the notation of the illustration is not in universal use, 
some writers defining dedendum as the entire depth below the pitch line = 
dedendum 4- clearance as here defined. 



FOR A Pair of Wheels 



Formulas Examples 

fc = 2a^ = 2X4.5X12 = 108. 

bV 108X1 

n = — p- f> = = 36. 

v+V 3 

,- nv 36X2 

N = y=-^ = 72. 

NV 72X1 , 

n= = = 36. 

V 2 ^ 

bv 108X2 

iV = — r-f, = = 72. 

v+V 3 

pD'V 12X6X1 , 
n = ——= =36. 



«y 36X2 
^~N~ 72 "'■ 
_NV 72X1 
n 36 

pD'V 12X6X1 

v = = 2 = 2. 

n 36 

2a{n+2) 2X4-5X(72+ 2) 
Z) = 1 =• ^ =6.166. 



b = 



b 108 

2(l(«-(-2) 2X4-SX(3f + 2) 



b 



108 



= 3.166. 



_ b 108 

''-2i>-2Xl2-4-5- 

2av 2X4-5X2 

U =—7-7. = = 6. 

v+V 3 

J, 2aV 2X4-5X1 

'^%"+F=-l = 3- 

D'+d' 6+i 
a = — - — = —— = 4.5. 



(15) 
(16) 

(17) 
(18) 

(19) 
(20) 
(21) 
(22) 
(23) 
(24) 
(25) 
(26) 

(27) 
(28) 
(29) 



SPUR GEARS 



95 



Table 3 gives dimensions of gear teeth of the 145-deg. system 
and Table 4 gives similar dimensions of 20-deg. stub teeth both 
according to the practice of the Fellows Gear Shaper Co. 



pitch line to the base line. Draw straight radial flanks from the base 
line to the root line and round them into the clearance line. 



Table 3. — Dimensions of Gear Teeth of 14 J Degrees 

Pressure Angle with Standard Addendum — 

Fellows System 

Thickness of tooth =1.5708-7- diametral pitch. 

Addendum = i . 0000 -e- diametral pitch. 

Clearance, gear shaper gear = . 2 5 00 -j- diametral pitch. 

Clearance, milled gear = . 1571 -r- diametral pitch. 

Whole depth, gear shaper gear =2. 2500 -4- diametral pitch. 

Whole depth, milled gear = 2 . 157 1 H- diametral pitch. 



Table 4. — Dimensions of Gear Teeth of 20 Deg. Pressure 
Angle with Reduced Addendum — Fellows Stub-tooth 
System 

Thickness of tooth same as for i4j-deg. gear of same pitch as numer- 
ator of stub-tooth pitch fraction. 

Addendum, clearance, depth of space and whole depth of tooth 
same as for 14^-deg. gear shaper, gear of same pitch as denominator 
of stub-tooth pitch fraction. 





Thick- 
ness of 
tooth 


Adden- 
dum 


Clearance 


Whole depth 


Diametral 
pitch 


Gear 
shaper 
gear 


Milled 
gear 


Gear 

shaper 

gear 


Milled 
gear 


I 


1.5708 


I . 0000 


.2500 


.1571 


2.2500 


2.1571 


15 


1.0472 
.7854 
.6283 
•5236 


.6667 
.5000 
.4000 
•3333 




. IOA7 




1-4381 

1.0785 

.8628 


2 




.o78i; 




25 




.0628 




3 




.0524 




.7190 


4 


• 3927 


■ 2500 


.0625 


•0393 


•5625 


■5393 


S 


■ 3142 


. 2000 


.0500 


• 0314 


.4500 


• 4314 


6 


.2618 


.1667 


.0417 


.0262 


•3750 


■3595 


7 


■ 2244 


.1429 


•0357 


.0224 


.3214 


.3081 


8 


.1963 


.1250 


.0312 


.0196 


.2812 


.2696 


9 


• 1745 


.1111 


.0278 


•017s 


.2500 


•2397 


10 


■ IS7I 


. 1000 


.0250 


■0157 


• 2250 


■2157 


12 


.1309 


•0833 


.0208 


.0131 


•187s 


.1798 


14 


. 1122 


.0714 


.0179 


.0112 


.1607 


• 1541 


16 


.0982 


.0625 


.0156 


.0098 


.1406 


.1348 


18 


.0873 


•0555 


.0139 


.0087 


.1250 


.1198 


20 


.0785 


.0500 


.0125 


.0078 


.1125 


.1079 


22 


.0714 


•04S5 


.0114 


.0071 


.1023 


.0980 


24 


.0654 


.0417 


.0104 


.0065 


.0938 


.0899 


26 


.0604 


•0385 


.0096 


.0060 


.0865 


.0829 


28 


.0561 


•0357 


. 0089 


.0056 


.0804 


.0770 


30 


• 0524 


■0333 


.0083 


.0052 


.0750 


.0719 


32 


.0491 


.0312 


.0078 


.0049 


.0703 


.0674 



Diametral 
pitch 


Thickness 
of tooth 


Addendum 


Clearance 


Whole depth 
of tooth 


i 


• 3927 


. 2000 


.0500 


.4500 


\ 


.3142 


.1429 


• 0357 


• 3214 


f 


.2618 


.1250 


.0312 


.2812 


i 


.2244 


.iiii 


.0278 


• 2500 


A 


.1963 


. 1000 


.0250 


• 2250 


A 


• 1745 


.0909 


.0227 


•2045 


if 


• 1571 


•0833 


.0208 


.1875 


il 


.1309 


.0714 


.0179 


. 1607 



Approximate Gear-tooth Outlines 

Approximate gear-tooth outlines may be determined by the use of 
any of the various odontographs that have been proposed; of these 
probably the most accurate (in fact very accurate) and most generally 
available is that by Geo. B. Grant {Amer. Mack., May 22, July 3, 
1890, and A Treatise on Gear Wheels) which is repeated below in 
Tables 7 and 9. 

To draw the involute tooth first draw the pitch, addendum, root 
and clearance lines and space the pitch line for the teeth as in Fig. 3. 
Draw the basj line one-sixtieth of the pitch diameter inside the 
pitch line. 

Take the face radius from Table 7, multiply or divide it as called 
for by the table, take the resulting radius in the dividers and draw 
in the faces from the pitch line to the addendum line from centers 
on the base line. Take the tabular flank radius from the table, 
multiplying or dividing it as before, and draw in the flanks from the 




^^ 



Fig. 3. — Grant's odontograph for involute teeth. 



of 2 



Fig. 3 shows the resulting radii and centers for a pmion 
diametral pitch with 12 teeth and for a rack meshing with it. 

Special rule for the rack: Draw the sides of the rack tooth, Fig. 3, 
as straight lines inclined to the line of centers co at an angle of 15 
deg. Draw the outer half ah of the face one-quarter of the whole 
length of the tooth from a center on the pitch line and with 

radius = 



diametral pitch 
= .67X circular pitch 

If the gear is to have more than 30 teeth the rounding of the ends 
of the rack teeth is unnecessary. 

{Continued on page 92, first column) 



96 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 5. — Multipliers and their Logarithms for Finding Diameters oe Spur Gears from the Circular Pitch 

To find pitch diameter: Multiply the number of teeth by the multiplier for the pitch. 

To find outside diameter for standard (B. & S.) addendum: Add two to the number of teeth and proceed as before. 



P. 


Mult'r. 


Log. 


P. 


Mult'r. 


Log. 


P. 


Mult'r 


Log. 


P. 


Mult'r. 


Log. 


^" 


.019894 


s. 298722 


2// 

7 


.127324 


T. IO49IO 


lA" 


■377993 


1-577484 


2¥' 


.676408 


T. 830209 


tV 


.031831 


2.502850 


7 
16 


.139261 


T. 143829 


li" 


•397887 


T. 599760 


2V' 


.716197 


T. 855033 


v 


.035368 


"J. 548610 


¥' 


•159155 


T. 201820 


lA" 


.417782 


T. 620950 


2f" 


.755986 


1. 878514 


It/ 

8 


.039789 


^. 599763 


9 It 

T6 


• 179049 


T. 252972 


if" 


.437676 


T. 641153 


2¥' 


.795775 


X. 900789 


v 


•04S473 


^.657754 


1" 


■ 198944 


T. 298731 


lA" 


■457570 


T. 66045 7 


2f" 


■835563 


T. 921979 


¥' 


.053052 


2.724702 


2 It 
3 


.212207 


T. 236760 


^¥' 


.477465 


T. 678942 


2i" 


•875352 


T. 942183 


^" 


.059683 


2.775851 


w 


.218838 


T. 340132 


lA" 


•497359 


T. 696670 


2|" 


.915141 


T. 961488 


i" 


.063662 


^. 803880 


311 
i 


.238732 


T.3779II 


if" 


■517253 


T.713703 


3" 


.954930 


T.979971 


1" 


.070735 


^. 849634 


13 tt 
16 


.258627 


T. 412674 


1-16 


■537148 


T. 730094 


3¥' 


.994718 


T.997700 


It/ 


.079577 


2.900788 


t tt 
8 


.278521 


T. 444858 


li" 


■557042 


1.745888 


3i" 


1.034507 


•014733 


r 


.090945 


¥.958779 


15/' 
16 


. 29841S 


1.474821 


iH" 


■576936 


T.761128 


3f" 


1.074296 


.031124 


5 // 

16 


.099472 


¥.997701 


l" 


.318310 


T. 502850 


i|" 


■S96831 


1.775851 


32" 


I. 114085 


.046916 


¥' 


.106103 


T. 025728 


lA" 


.338204 


T. 529179 


iH" 


.616725 


T. 790092 








sit 
a 


.119366 


T. 076881 


i|" 


■358099 


1-554003 


2" 


.636619 


T. 803879 









Table 6. — Tooth Parts by Circular Pitch, Brown and Sharpe System. The 2D, qth and ioth Columns Relate 
also to Worms. From a Practical Treatise on Gearing by The Brown & Sharpe Mfg. Co. 

the size of any part of a circular pitch not given in the table, multiply the corresponding part of 1" pitch by the pitch 



To obtain 
required. 



'E. 

u 


u 

u 


S u 

u cd 



1 

s 
5 


c 



2 


B 


la 
I? 


en "^ 

ii 

a ^ 

D 


& 

-a 


s 


■d 
"0 

ft 


u 
u 

u 


u 
ft 

<u 

~^ Ui 
l-j aj 
1) 


-a 


"a, 

a 

5 



2 

.g 


s 

3 

Id 
a 
1) 

T3 
< 


"0 

■B 
& 

Id 

U) 

i: 
|2 


m 

"^ 

VM (J 
.-^ 

a ^ 

Q -^ 


"0 


1 

OJ 


■4~i 

1 

M 

-M 

"o 


p 


i" 
P 


p 


t 


s 


D" 


^+/ 


D"+/ 


PX 

■ 3095 


PX 

■3354 


p 


i" 


p 


i 


s 


D" 


^+/ 


D"+f 


PX 
-3095 


PX 

•3354 


2 


1 

z 


1.5708 


I . 0000 


.6366 


1.2732 


-7366 


1-3732 


.6190 


.6707 


i 


2 


6.2832 


.2500 


• 1592 


.3183 


.1842 


-3433 


• 1547 


.1677 


l| 


A 


1^6755 


-9375 


-5968 


I 


1937 


.6906 


1.2874 


-5803 


.6288 


4 
9 


2i 


7^0685 


.2222 


• I4I5 


.2830 


-1637 


-3052 


.1376 


.1490 


If 


f 


1^7952 


-8750 


-SS70 


I 


1141 


-644s 


I. 2016 


.5416 


-5869 


A 


2f 


7.1808 


.2187 


• 1393 


.2785 


. 1611 


.3003 


-1354 


• 1467 


if 


A 


1^9333 


-8125 


-S173 


I 


0345 


-5985 


1.1158 


-5029 


• 5450 


3 


2i 


7-3304 


-2143 


.1364 


.2728 


-1578 


.2942 


.1326 


•1437 


li 


2 
3 


2 . 0944 


-7500 


-4775 




9549 


-5525 


1.0299 


.4642 


■5030 


f 


2i 


7-8540 


.2000 


.1273 


.2546 


•1473 


.2746 


-1238 


• 1341 


lA 


M 


2^1855 


.7187 


-4576 




9151 


-5294 


.9870 


-4449 


.4821 


3 

8 


2f 


8-3776 


-1875 


.1194 


.2387 


.1381 


■2575 


. II6I 


-1258 


If 


A 


2 . 2848 


.6875 


-4377 




8754 


.5064 


.9441 


-4256 


.4611 


A 


2f 


8-6394 


.1818 


.1158 


.2316 


■ 1340 


.2498 


.1125 


. 1219 


I* 


3 

1 


2.3562 


.6666 


-4244 




8488 


.4910 


• 9154 


.4127 


• 4471 


1 
3 


3 


9.4248 


.1666 


. I06I 


.2122 


.1228 


.2289 


.1032 


.1118 


lA 


Vt 


2.3936 


.6562 


-4178 




8356 


-4834 


.9012 


.4062 


.4402 


A 


3i 


10.0531 


-1562 


• 0995 


.1989 


.1151 


. 2146 


.0967 


.1048 


li 


f 


2-5133 


• 6250 


■3979 




7958 


.4604 


■8583 


-3869 


•4192 


A 


3i 


10.4719 


-1500 


• 0955 


.1910 


• I 105 


.2060 


.0928 


. 1006 


lA 


1 6 


2.6456 


-5937 


-3780 




7560 


-4374 


-8156 


-3675 


■ 3982 


f 


3i 


10.9956 


.1429 


.0909 


.1819 


.1052 


.1962 


.0884 


.0958 


li 


f 


2.7925 


-5625 


-3581 




7162 


-4143 


.7724 


.3482 


■3773 


1 
4 


4 


12 . 5664 


-1250 


.0796 


•1591 


.0921 


.1716 


-0774 


.0838 


lA 


if 


2.9568 


-5312 


•3382 




6764 


-3913 


•7295 


-3288 


■3563 


1 


4i 


14.1372 


.IIII 


.0707 


• 1415 


.0818 


.1526 


.0688 


•0745 


I 


I 


3-1416 


.5000 


-3183 




6366 


.3683 


.6866 


-3095 


■3354 


i 


5 


15.7080 


. 1000 


.0637 


•1273 


• 0737 


-1373 


. 0619 


.0671 


IS 
16 


iiV 


3-3SIO 


-4687 


.2984 




5968 


-3453 


• 6437 


. 2902 


■3144 


A 


Si 


16-7552 


-0937 


• 0597 


.1194 


.0690 


.1287 


.0580 


.0629 


i 


if 


3-5904 


-4375 


-278s 




5570 


-3223 


.6007 


.2708 


•2934 


j\ 


Si 


17.2788 


.0909 


• 0579 


• 1158 


.0670 


.1249 


-0563 


.0610 


a 


I A 


3.8666 


.4062 


.2586 




5173 


-2993 


-5579 


-2515 


.2725 


4 


6 


18.8496 


.0833 


• 0531 


. 1061 


.0614 


-1 144 


.0516 


•0559 


f 


li 


3.9270 


.4000 


.2546 




5092 


• 2946 


-5492 


.2476 


■ 2683 


A 


6i 


20.4203 


.0769 


.0489 


.0978 


.0566 


-1055 


.0476 


.0516 


3 
4 


li 


4.1888 


-3750 


.2387 




4775 


.2762 


-5150 


.2321 


■2515 


f 


7 


21.9911 


•0714 


• 0455 


.0910 


.0526 


.0981 


.0442 


•0479 


■a 


itV 


4.5696 


-3437 


.2189 




4377 


-2532 


.4720 


.2128 


.2306 


A 


7i 


23-5619 


.0666 


.0425 


.0850 


.0492 


-0917 


-0413 


.0447 


1 


li 


4-7124 


■3333 


. 2122 




4244 


-2455 


-4577 


.2063 


.2236 


i 


8 


25-1327 


.0625 


-0398 


.0796 


.0460 


-0858 


-0387 


• 0419 


1 


If 


S-0265 


•3125 


.1989 




3979 


.2301 


.4291 


-1934 


. 2096 


i 


9 


28.2743 


■0555 


-0354 


.0707 


.0409 


.0763 


-0344 


• 0373 


f 


If 


5-2360 


.3000 


. 1910 




3820 


. 2210 


.4120 


-1857 


.2012 


A 


10 


31-4159 


-0500 


-0318 


.0637 


.0368 


.0687 


.0309 


• 0335 


f 


If 


5 -4978 


•2857 


.1819 




3638 


.2105 


-3923 


.1769 


. 1916 


A 


16 


50-2655 


.0312 


.0199 


• 0398 


.0230 


■-0429 


.0193 


.0210 


9 

T6 


II 


S-5851 


.2812 


• 1790 




3581 


. 2071 


.3862 


-1741 


.1886 


1 


20 


62.8318 


.0250 


-0159 


.0318 


.0184 


-0343 


-OIS5 


.0168 



SPUR GEARS 



97 



M 

H 

W 

H 

H 
O 



o 

<< 

M 
O 
O 
H 

?; 
o 

Q 

o 

< 



X 

-a 



CO 



lO 



=* a 



2 a 
a "^ 



U 



M 


















































c3 


















































3 
U 
t4 


■^ 


« 

3 


<N 


I^ 


tH 


Tl- OS 


CO t^ 


O -* t^ 


o 


CO 


SO 


Os w 


•* 


t~ O 0) 


>o 


00 


H 


CO so Os 


w 


CO 










rrt 


13 


Ol 


CN 


CO 


CO CO 


■5f Tj- 


lO lO lO 


so 


SO 


SO 


SO 1^ 


t^ 


t^ 00 00 


oo 


00 


OS 


Os Os Os 


O 


O 




























































li( 


n 


n 


o 


O 


O 


GOO 


O 


o 


o 





o 


O o 


o 


o 


O 


GOO 


M 


M 


Tl- 


00 H CO t^ 


sn 


w SO 00 












































OO 


Ti- so 00 O 


^ 


tH 01 C30 










































H 


W M H OJ 


Ol 


CO "* SO 


























































































43 a 


















































>. 


















































•^ 






CO 


o 


O 


CO t^ 


O CO 


SO Os CO 


SO 


OS 


M 


CO so 


00 


O CO to 


t^ 


Os 


M 


^ so 00 


Os 


M 










1 


'3 


t^ 


r^ 


00 


00 00 


Os Os 


Os Os o 


o 


o 


M 


M W 


H 


01 O) CSI 


Ol 


'"! 


cj 


CO CO CO 


cj 


^ 
























































^ 


1^ 


S 














































J^-fi 


-S 


en 


OS 


CO 


>o 


Os CSI 


>* so 


00 Os Os 


Os 


00 


SO 


vo -d- 


CO 


0) O OS 


t^ 


so 


lO 


CO H Os 


SO 


00 


o 


CO SO Tj- Ol 


Ol 


00 00 oq 


vn 


00 


OS 


o « 


to "* 


lO so t^ 


00 


OS 


o 


W CSI 


CO 


Tj- lO v> 


so 


r^ 


OO 


OS O O 


H 


Ol 


CSI 


so O t^ lO 


r^ 


t^ CO so 


^ ii 


cS 


T3 














































>, ^ 


P^ 


S 


o 


d 


o' 


H H 


H H 


H H H 


H 


" 


M 


01 OI 


cs 


01 CSI 0* 


01 


01 


01 


Ol CO CO 


C5 


ofj 


-* 


Th lo lo so 


t^ 


Os CO IH 
H 04 




































































































■> a 


















































<l> 


01 


no 


n 


H 


CSI N 


(N N 


« N W 


N 


„ 


Os 


t^ Tl- 


w 


00 lO Ol 


Os 


sn 


CO 


G t^ CO 


Os 


irt 










u 

ft. 


cs 




lO 


SO t^ 


00 Os 
M CSI 


*H Ol 
CO CO CO 


CO 
CO 


CO 


CO 


lo SO 
CO CO 


CO 


1:^ 00 OS 
CO CO CO 


Os 
CO 


o 
4 


4 


Ol Ol CO 

4 4 4 


CO 

4 


4 






















^ 






































O 

1 


u-> w O O 
«;i- lO so t^ 

1 1 1 1 

M so CSI H 




Os 

H 


GOO 

Ol 00 so 

H H CO 
1 1 1 

M M M 










































43 








































CO 


■;|- -Th to SO 


t^ 


0\ C^ 00 
















































H H 


S 
















































2 




o 


HI 


CN 


CO ^ 


lO SO 


<^ 00 Os 


o 


H 


M 


to "* 


«0 vO t>. 00 


Os 





M 


01 00 ^ 


lO 


so 










"o 




H 


IH 


tH 


M M 


M M 


M M (H 


p) 


<M 


a 


W N 


M 


M CSI N 


0) 


CO 


CO 


CO 00 00 


C3 


ofj 












d 

















































To obtain the 
pitch required. 



Table 8. — Tooth Parts by Diametral Pitch, Brown and Sharpe System. 
From a Practical Treatise on Gearing by The Brown & Sharpe Mfg. Co. 
size of any part of a diametral pitch not given in the table, divide the corresponding part of i diametral pitch by the 



i 
11 


o a 

•5- 


Thickness of 
tooth on 
pitch line 


— or the ad- 
P 
dendum 


Working 
depth of 
tooth 


Depth of 
space below 
pitch line 


4J 8 


"cS 
u 

8 
Q ^ 


u 

•73 ,a 

p (J 

a 

•a- 


Thickness of 
tooth on 
pitch line 


— or the ad- 
P 
dendum 


Working 
depth of 
tooth 


Depth of 
space below 
pitch line 




p 


p 


t 


5 


D" 


s+f 


D"+f 


P 


p 


t 


5 


D" 


s+f L 


"+/ 


i 


6.2832 


3.1416 


2 . 0000 


4 . OOGO 


2.3142 


4-3142 


15 


.2094 


.1047 


.0666 


■1333 


.0771 


1438 


3 
i 


4.1888 


2.0944 


1-3333 


2.6666 


1.5428 


2.8761 


16 


.1963 


.0982 


.0625 


• 1250 


.0723 


1348 


I 


3-1416 


1.5708 


I . 0000 


2 . 0000 


1-1571 


2.I57I 


17 


.1848 


.0924 


-0588 


.1176 


.0681 


1269 


li 


2.5133 


1.2566 


.8000 


I . 60GG 


-9257 


1-7257 


18 


-1745 


.0873 


■0555 


. nil 


.0643 


1198 


li 


2.0944 


1.0472 


.6666 


1-3333 


-7714 


I. 4381 


19 


-1653 


.0827 


■ 0526 


■1053 


.0609 


II3S 


l| 


I -7952 


.8976 


-5714 


I. 1429 


.6612 


1.2326 


20 


-I57I 


.0785 


.0500 


. 1000 


•0579 


1079 


2 


1.5708 


•7854 


.5000 


I . 0000 


-5785 


1.078s 


22 


.1428 


.0714 


■0455 


.0909 


,0526 


0980 


2i 


1-3963 


.6981 


-4444 


.8888 


■5143 


• 9587 


24 


.1309 


.0654 


.0417 


■0833 


.0482 


0898 


' 2i 


1.2566 


.6283 


.4000 


.80G0 


.4628 


.8628 


26 


.I2g8 


.0604 


•0385 


.0769 


-0445 


0829 


2i 


I. 1424 


■ 5712 


-3636 


■7273 


.4208 


-7844 


28 


,1122 


■ 0561 


-0357 


.0714 


.0413 


0770 


3 


1.0472 


-5236 


■3333 


.6666 


-3857 


.7190 


30 


.1047 


• 0524 


■0333 


.0666 


.0386 


0719 


3i 


.8976 


.4488 


-2857 


-5714 


.3306 


.6163 


32 


.0982 


.0491 


■ 0312 


.0625 


.0362 


0674 


4 


•7854 


■ 3927 


.2500 


.5000 


.2893 


■5393 


34 


.0924 


.0462 


.0294 


.0588 


.0340 


0634 


5 


.6283 


-3142 


. 2000 


.4000 


-2314 


-4314 


36 


.0873 


.0436 


.0278 


-OS55 


.0321 


0599 


6 


■5236 


.2618 


.1666 


-3333 


.1928 


-3595 


38 


.0827 


.0413 


.0263 


.0526 


.0304 


0568 


7 


.4488 


.2244 


.1429 


-2857 


-1653 


■ 3081 


40 


.0785 


-0393 


.0250 


.0500 


.0289 


0539 


8 


-3927 


.1963 


.1250 


• 2500 


.1446 


.2696 


42 


.0748 


•0374 


.0238 


.0476 


-0275 


0514 


9 


■ 3491 


■1745 


.iiii 


.2222 


.1286 


-2397 


44 


.0714 


-0357 


.0227 


•0455 


.0263 


0490 


lO 


.3142 


-1571 


.1000 


. 2000 


-1157 


-2157 


46 


.0683 


-0341 


.0217 


-0435 


.0252 


0469 


II 


-2856 


.1428 


.0909 


.1818 


-1052 


.1961 


48 


.0654 


• 0327 


.0208 


.0417 


.0241 


0449 


12 


.2618 


.1309 


-0833 


.1666 


.0964 


.1798 


SO 


.0628 


-0314 


.0200 


.0400 


.0231 


0431 


13 


.2417 


.1208 


.0769 


-1538 


.0890 


.1659 


56 


.0561 


.0280 


.0178 


-03S7 


.0207 


0385 


14 


.2244 


.1122 


.0714 


.1429 


.0826 


-1541 


60 


-0524 


.0262 


.0166 


-0333 


-0193 


0360 



98 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 9. — Grant's Odontograph for Epicycloidal Teeth 

Addendum = . 3 183 X circular pitch 

I 



Clearance = 



diametral pitch 
addendum 







For one 


For one inch 


Number of teeth 


diametral pitch 


circular pitch 


For any other pitch 


For any other pitch 


in tl 


le gear 


divide by that pitch 


multiply by that pitch 




Faces 


Flanks 


Faces 


Flanks 


Exact 


Intervals 


Rad. 


Dis. 


Rad. 


Dis. 


Rad. 


Dis. 


Rad. 


Dis. 


10 


10 


1.99 


0.02 


- 8.00 


4.00 


0.62 


O.OI 


-2-55 


1.27 


II 


II 


2.00 


0.04 


— 11.05 


6.50 


0.63 


O.OI 


-3-34 


2.07 


12 


12 


2.01 


0. 06 


CX! 


CO 


0.64 


0.02 


CO 


00 


13^ 


13-14 


2.04 


0.07 


15.10 


9-43 


0.65 


0.02 


4.80 


3.00 


isi 


15-16 


2. 10 


0.09 


7.86 


3-46 


0.67 


0.03 


2.50 


1. 10 


ni 


17-18 


2.14 


O.II 


6.13 


2.20 


0.68 


0.04 


1-95 


0. 70 


20 


19-21 


2. 20 


0.13 


5-12 


I-S7 


0.70 


0.04 


1.63 


0.50 


23 


22-24 


2. 26 


0-I5 


4.50 


I -13 


0.72 


0.05 


1-43 


0.36 


27 


25-29 


2-33 


0. 16 


4.10 


0.96 


0.74 


0.05 


1.30 


0. 29 


33 


30-3(> 


2.40 


0. 19 


3-8o 


0.72 


0.76 


0.06 


1.20 


0.23 


42 


37-48 


2.48 


0.22 


3-52 


0.63 


0.79 


0.07 


1 . 12 


0. 20 


58 


49-72 


2.60 


0. 25 


3-33 


0-S4 


0.83 


0.08 


1.06 


0.17 


97 


73-J:44 


2.83 


0.28 


3-14 


0.44 


0.90 


0.09 


1. 00 


0. 14 


290 


145-300 


2.92 


0.31 


3.00 


0.38 


0.93 


0. 10 


0-9S 


0. 12 


00 


Rack 


2.96 


0-34 


2.96 


0-34 


0.94 


0. II 


0.94 


0. II 



To draw the epicycloidal tooth, first draw the pitch, addendum, 
root and clearance lines and space the pitch line for the teeth as in 
Fig. 4. 

Draw the line of flank centers outside the pitch line at the tabular 
distance from it ("dis." in table) obtained from Table 9 and the line 
of face centers at the tabular distance ("dis.") inside of the pitch 
circle. Take the face radius ("rad.") in the dividers and draw the 
face curves from centers on the line of face centers. Take the flank 
radius ("rad.") and draw the flank curves from centers on the line 
of flank centers. 

Table 9 gives the distances and radii for 1 diametral and i in. 
circular pitch. For other pitches multiply or divide as directed in 
the table. 



Fig. 4 shows the resulting distances and radii for a pinion of 2 
diametral pitch with 20 teeth. 

Strength of Spur Gears by Calculation 

The working loads on spur gears are commonly determined from 
the formula proposed by Wilfred Lewis {Proc. Engrs. Club of Phila- 
delphia, 1892) as follows: 

W = SPfy 
in which I^ = pressure on teeth, lbs., 

5 = fiber stress, lbs. per sq. in., this stress being dependent 
on the speed in accordance with Tables 10 and 11 
below, 
P = circular pitch, ins., 
y = face, ins., 

y = a factor for difierent numbers and forms of teeth in 
accordance with Table 12. 

Tab^e 10. — ^Values of Factor S in the Lewis Formula for 
Strength of Gears 



Pitch-line, speed 
ft. per min. 


100 or 
less 


200 


300 


600 


900 


1200 


1800 


2400 


Cast-iron 

Steel 


8,000 
20,000 


6,000 
15,000 


4,800 
12,000 


4,000 
10,000 


3,000 
7,500 


2,400 
6,000 


2,000 
S.ooo 


1,700 
4.300 



The high-class, alloy-steel, heat-treated transmission gears of 
automobiles carry stresses materially in excess of those given in 
Table 10. F. M. Heldt, after analyzing the data of a large number 
of such gears, publishes the stresses of Table 11 as giving good results 
in intermittently meshed gears (The Horseless Age, Apr. 10, 1912). 
For constantly meshed gears the figures of the table should be reduced 
1 5 per cent. A piston speed of 1000 ft. per min. was assumed when 
calculating the pitch-line speed. 

Table ii. — Values of Factor 5 in the Lewis Formula Deduced 
FROM Automobile Practice 



Pitch-line speed, 
ft. per min. 


500 


600 


700 


800 


900 


1,000 


1. 100 


1,200 


Alloy steel case 


30,000 


27,000 


24,000 


21,000 


18,000 


15,000 






hardened. 


















Chrome nickel 


60,000 


53.000 


47,000 


42,000 


38,000 


34,000 


30.000 


27.000 


and chrome 


















vanadium steel 


















hardened all 


















through. 



















TAne of Flank Centers 




Fig. 4. — Grant's odontograph for epicycloidal teeth. 



The values of the factor y may also be obtained 
from Fig. 6 by Robert A. Bruce (Amer. Mack, 
Nov. 21, 1901) which gives these values for not only 
standard proportions of teeth but for stub teeth which 
are now coming into use. The stub teeth for which 
the chart is drawn are somewhat shorter than those 
of the Hunt and Logue systems, which see, but rea- 
sonable allowances may be made for the difference. 

The diagonal line shows a method of determining 
teeth of different systems but of equal strength. 
Thus, gears of 15 deg. obliquity, addendum .3183 
P, 16 teeth; 20 deg. obliquity, addendum .3183 P, 

P 
20 teeth; 14I deg. obliquity, addendum—, 29 teeth; 

P 
and 20 deg. obliquity, addendum-, 35 teeth have the 

same strength, the diameter, face and speed being the 



The Strength of Spur Gears by Graphics 

The working loads on spur gears may be determined 
graphically from Fig. 7 which has been constructed 



SPUR GEARS 



99 



Table 12. — Values of Factor y in the Lewis Formula for Strength of Gears 



Number 

of 

teeth 


Value of factor y 


Number 

of 

teeth 


Value of factor y 


Number 

of 

teeth 


Value of factor y 


Involute 
20" 


Involute 
15° 

cycloidal 


Radial 
flanks 


Involute 
20° 


Involute 

15° 
cycloidal 


Radial 
flanks 


Involute 
20° 


Involute 

15° 
cycloidal 


Radial 
flanks 


12 
13 
14 
15 
16 
17 
18 
19 


.078 
.083 
.088 
.092 
.094 
.096 
.098 
.100 


.067 
.070 
.072 
.075 
.077 
.080 
.083 
.087 


.052 
.053 

■ 054 
.055 
.056 

■ 057 
.058 
.059 


20 
21 
23 
25 
27 
30 
34 
38 


. 102 
. 104 
. 106 
.108 
. Ill 
.114 
.118 
.123 


.090 
.092 
.094 . 
.097 
.100 
. 102 
. 104 
.107 


.060 
.061 
.062 
.063 
.064 
.065 
.066 
.067 


43 

SO 

60 

73 

100 

150 

300 

rack 


.126 
.130 
.134 
.138 
.142 
. 146 
.150 
.154 


.110 
.112 
.114 
.116 
.118 
.120 

.122 
.134 


.068 
.069 
.070 

.071 
.072 
.073 
.074 
.075 



20 D. P. 

.1571 In. C. P. 



14 D. P. 
.2244 In. C. P 



jW7\ 

18 D. P. 
.1745 In. C. P. 



16 D. P. 
.1963 In. C. P. 




IVi D. P. 
1.2566 In. C. P. 




Wi D. P. 
2.0944 In. C, P. 



Fig. 5. — Gear teeth of full size, involute profile, 145 degrees pressure angle. 



to represent the Lewis formula by Robert A. Bruce {Amer. Mach., 
May SI, igco). 

The main chart, which applies to cast-iron and steel gears and to 
circular and diametral pitches, gives directly a preliminary false 



value for the load, which must be corrected by the use of the proper 
supplementary reduction scale below. 

The use of the chart is best shown by an example: Required the 
working load on a cast-iron spur gear of 30 teeth, 2-in. pitch, 5-in. 



100 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



face, at a pitch-line velocity of looo ft. per min., the teeth being of 
the iS-deg. involute form. Find 2-in. pitch on the left-hand vertical 
scale, trace to the right until the diagonal for 5-in. face is reached, 
then down for cast-iron or up for steel and read the preliminary false 
load of 10,000 lbs. for cast-iron or 25,000 lbs. for steel. Next apply 
the dividers to the reduction scale for is-deg. involute and cycloidal 



teeth and take up the distance between 30 teeth and 1000 ft. pitch- 
line velocity. Step this off to the left from the preliminary false load 
and read the answers, 3000 lbs. for cast-iron and 7500 lbs. for steel. 
Note that in the case of the reduction scale for 20-deg. involute 
teeth, the overlapping part indicates that with that tooth it is occa- 
sionally necessary to add to the preliminary false load. This is the 



^ 



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4 








t 








n L 









10 20 30 40 SO 60 70 80 

Number of Teeth 



90 100 110 120 



Fig. 6. — Values of y in the Lewis formula for the strength of gear teeth. 



Table 13.- — Maex and Gutter's Formulas and Gonstants for the Strength of Cast-iron Gears 



SYMBOLS, BOTH SYSTEMS 
pr = safe equivalent load at pitch line, lbs., 
5 = modulus of rupture = 36,000 lbs. per sq. in. for cast iron, 
i) = circular pitch, ins. = pitch arc, 
/= width of face of gear, ins., 
K = number of teeth in gear, 
^ = factor of safety, 
Suggested values: /fe = 4, for steady load, no reversal of stress 

k = 6, suddenly applied load, no reversal of stress 
/fe = 8, suddenly applied load, with reversal of stress 
V = velocity coefficient. See tables 
o = arc of action coefficient. See tables 



FORMULAS 
Brown & Sharpe i4M-deg. involute: 



stff 



1.26\ 



SpJ / X.2() \ 

W=-^{o.i5A —pa 

Fellows 20-deg. involute, stub tooth: 
spf/ 2^\ 

w=~Y\p.2^% — —pa 

Neither formula holds for values of « less than 12. 



Values of v 


Values of 


Pitch 

velocity, 

ft. per min. 


V 


Pitch 
velocity, 
ft. per min. 


I 




Teeth in 

engaging 

gears 


Corresponding a 


Brown & Sharpe 
i4M2-deg. 
involute 


Fellows 20-deg. 

involute 

stub tooth 


Brown & Sharpe 
i4H-deg. 
involute 


Fellows 20-deg. 

involute 

stub tooth 


Brown & Sharpe 
i4H-deg. 
involute 


Fellows 20-deg. 
involute 
stub tooth 


0000 


I .000 


1 .000 


1 100 


0.470 


0.540 


Single 
tooth engages 


1 .00 


1 .00 


100 


0.79S 


0.825 


1200 


0.4SS 


0.525 


12 


12 


1 .10 


1. 13 


200 


0.730 


0.7SS 


1300 


0.44s 


0.515 


20 


30 


1. 15 


1 .20 


300 


0.67s 


0.705 


1400 


0.43s 


0.505 


30 


30 


1.47 


1 .22 


400 


0.63s 


0.665 


1500 


0.430 


0.49s 


30 


40 


1 .60 


1.24 


500 


0.S9S 


0.635 


1600 


0.420 


0.48s 


30 


60 


1 .60 


I.2S 


600 


0.S6S 


0.61S 


1700 


0.41S 


0.47s 


30 


80 


1 .60 


1.26 


700 


O.S40 


0.595 


1800 


0.410 


0.470 


30 


100 


1 .60 


1,27 


800 


0.520 


0.580 


1900 


0.40s 


0.460 


30 


Rack 


1 .60 


1 .29 


900 


0.500 


0.565 


2000 


0.400 


0.4S0 


100 


100 


1 .60 


i.3t 


1000 


0.48s 


0.550 








100 


Rack 


1 .60 


1 .33 



SPUR GEARS 



101 



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102 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 




2.0 
LO 
1.8 
1.7 
1.6 
1.6 
1.4 
1.3 
L2 
1.1 
1.0 










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tact 




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y 























300 100 50 34 25 20 

Number ot Teetli 





2.0 
1.9 
1.6 
1.7 
1.6 

" 
1.3 
1.2 
1.1 
1.0 








- x-^- 1 


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o 


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Number of Teeth in Fiaion 

Fig. g. — Cut cast steel gears 



300 600 900 1200 IBOO 1800 2000 2400 2700 300 

Pitch Speed 



10 15 20 26 30 200 400 600 800 1000 1200 1400 1000 ISOO 20O0 10 15 20 25 30 

Number of Teeth in Pinion VKXch Speeds 

Fig. 8. — Machine molded cast steel gears. 
5a/e load at pilch line, lbs. = circular pitch, ins. X face, ins. X speed coefficient X shape coefficient X contact coefficient. 
If a fly wheel is close to the gear drive, figure with a peripheral speed from lo to 20 per cent, higher than the actual speed, depend- 
ing on closeness and mass of wheel. In figuring for strength use maximum, not average, load. For speeds exceeding 1000 ft. per min. 
for molded and 2000 for cut gears, take shape coefficient = .9 for all involutes and = 1.2 for cycloidal gears. The curves are based on 
a tensile strength of 65,000 lbs. per sq. in. Molded gears should be shrouded if possible. If set of shroud = I circular pitch. Figure 
the face as over the shroud. The usual maximum pitch is 7 in. 

Figs. 8 and 9. — The Mesta Machine Company's rules for the strength of steel rolling mill gears. 



case only when the selected point on the number-of-teeth scale is to 
the right of the selected point on the pitch-line-velocity scale. 

The working strength of shrouded gear teeth, according to Wilfred 
Lewis {Arrier. Mach., Jan. 30, 1902) is, for double shrouding to the 
full depth of the teeth, about 25 per cent, in excess of that of un- 
shrouded teeth. For very narrow faces an increased load up to 
about 50 per cent, may be used. On the other hand, for single 
shrouding, Mr. Lewis reduces the increase to 10 per cent. 

Many niackitie designers believe that gears are capable of carrying 
materially heavier loads than those determined by the Lewis formula — 
a belief that is supported by tests of gears to destruction by Profs. 
GuiDO H. Marx and Lawrence E. Cutter (Trans. A. S. M. E., 
Vols. 34 and 37). The gears tested. were of 10 diametral pitch, 
which is too small to justify positive deductions extending to heavy 
gears. These experiments are, however, the first in which gears 
have been tested to destruction and they supply the only definitely 
determined data on which to base a rational formula. 

The authors conclude that the Lewis formula underestimates the 
static strength of gears and overestimates the effect of an increase of 
speed. They also find it necessary to introduce a factor for the arc 
of action. The experiments included gears of the Bro wn and Sh arpe 
143^^ deg. standard form and the Fellows 20-deg. stub-tooth form — 
both with involute profiles and of cast iron. 

Table 13 gives the resulting formulas together with values of the 
velocity and arc of action coefficients. 

The Mesta Machine Company have developed methods of their 
own for determining the strength of heavy rolling-mill gears of steel. 



Table 14.- 



-Relation of Face and Pitch Line Speed in Mesta 
Heavy Steel Gears 



Machine molded gears 


Cut gears 


Face 


Pitch speed range, 
ft. per min. 


Face 


Pitch speed range, 


Circular pitch 


Circular pitch 


ft. per min. 


2>i 
3 

4 

4W 


From to 600 

400 to 900 

750 to 1300 

1 100 to 1500 

1400 to 1800 


3'/^ 

4 

4H 

S 

6 


From to 600 

450 to 1200 

1050 to 1900 

1700 to 2500 

2250 to limit 



These methods are given in Figs. 8 and 9 for machine molded and 
cut gears, respectively. This company makes the width of face to 
vary with the peripheral speed in accordance with Table 14. 

Table !■;. — Spur Gears that Failed in Service 





■a 









c 


c 
S 


"o 


in 

a 

u 

V 

& 



Pi 


g.s 

u 


en 

en 
U 
u 

a 






o 
<p 

"0 
■a 

ri 

u 
M 


-4-1 
'0 

u 

V a 
1- 


Velocity ratio, or ra 
of revs, of pin 
and wheel 


.^3 


.s « 


I.H.P. 


Ft. 


Lbs. 


Ins. 


Ins. 


No. 


Ratio 


Lbs. 


700 


2,280 


10,100 


4-5 


14 


42 


3.8 


160 


1,000 


2,356 


14,000 


5-4 


17 


41 


3-4 


153 


1,000 


2,334 


14,200 


5-0 


18 


43 


3-2 


158 


1,000 


2,261 


14,600 


5-5 


18 


41 


3-3 


148 


1,000 


2,241 


14,700 


5-6 


18 


46 


2.5 


146 


1,000 


2,208 


14,950 


5-0 


18 


47 


3-2 


166 


1,000 


2,200 


15,000 


5.62 


18 


46 


3-4 


149 


1,080 


2,318 


15,400 


5-25 


i6f 


43 


3-0 


176 


1,100 


2,401 


15,000 


5-0 


17 


47 


2.9 


177 


1,100 


2,406 


15,100 


5-5 


18 


50 


2.3 


153 


1,100 


2,410 


15,050 


5-0 


17 


47 


2.9 


177 


1,130 


2,242 


16,600 


5.8 


18 


43 


3-1 


160 


1,150 


2,320 


16,300 


5,75 


18 


46 


3-1 


158 


1,150 


2,320 


16,350 


5-75 


18 


47 


3-0 


158 


1,190 


2,323 


16,900 


5-7 


18 


47 


3-0 


163 


1,200 


2,209 


17,900 


5-0 


19 


52 


3-0 


189 


1,200 


2,418 


16,400 


4-75 


19 


52 


3-2 


182 


1,220 


2,209 


18, 20c 


5-0 


i8f 


49 


3-2 


19s 


1,360 


2,325 


19,300 


4-5 


18 


71 


2.8 


239 



SPUR GEARS 



103 



A list of spur gears that failed in service has been supplied by 
Michael Longridge {Proc. I. M. E., 1897). This Ust is repeated 
in Table 15 of which the last column has been added by the author. 

Strength of Bronze, Rawhide and Fabroil Gears 

The working loads of gears of hro7ize are less definitely known than 
those of iron or steel, but, for bronze of high quality, it is probably 
safe to impose loads one and one-half times those placed on cast-iron 
teeth of the same dimensions. 

The working capacity of rawhide gears, according to W. H. Diefen- 
DORF, Chief Engineer, New Process Rawhide Co. {Amsr. Mach., Apr. 
6, 1911) may be determined from the allowance of a pressure of isolbs. 
per in. of face for gears of i in. circular pitch. For other pitches 
the pressure allowance is to be taken in direct proportion, except that 
in no case should the pressure exceed 250 lbs. per in. of face. 

These figures are to be applied to gears made of the highest grade 
of rawhide only. For lower grades the unit pressure should be re- 
duced 15 per cent, or more. The figures are also intended for 
pinions having all rawhide working face. Pinions with bronze 
flanges having teeth cut through and forming part of the working 
face may be loaded with 10 to 25 per cent, greater pressures according, 
to the grade of the bronze and the thickness of the flanges. 

The unit pressure is not changed with the velocity or the number of 
teeth. 

The practice of the General Electric Company with rawhide gears, 
according to A. Schein {General Electric Review, Apr., 19 13), is to 
apply the Lewis formula using the stresses of Table i5 with the pro- 
viso that the dimensions must pass a further test because of the 
characteristics of these pinions due to heating. This test is embodied 
in the formula: 

WXV 



^•/••X 33,000 



c= 



FXN 



in which C = heating coefficient which must not exceed the values 
given in Table 1 7, 
TF = total load at pitch diameter, lbs., 
V = velocity at pitch diameter, ft. per min. , 
iV = number of teeth, 
F = width of face, ins. 

Table 16. — ^Values of Factor S in the Lewis Formula for 
Rawhide Pinions 



Speed at pitch dia. in ft. 


200 


400 


600 


800 


1000 


1200 


1400 


1600 


1800 


2000 


per min. 






















Stresses in lb. per sq. in. 


3600 


3300 


3100 


2800 


2600 


2400 


2200 


2000 1900 


1800 



Table 17. — Maximum 


Allowable 
Rawhide 


Values of 
Pinions 


C FOR Heating, 


Diametral pitch 

C 


I 


1400 


2 
1200 


2i 
1000 


3 
900 


3i 

800 


4 









Rawhide gears should have some degree of lubrication. The best 
lubricant is a mixture of graphite and lard oil or tallow — never 
mineral oil. 

The working capacity of fabroil gears, according to the General 
Electric Co., at whose works they originated, is, for most if not all 
services, equal to that of cast-iron gears of the same dimensions. 
The width of the cloth face should be equal to the face of the mating 
gear plus the aggregate end play of both shafts. The shrouds should 
never be permitted to run on the mating gear. 

The limiting peripheral speed of metallic spur gears is about 2000 ft. 
per min. Ordinary cut gears begin to be objectionably noisy at 
peripheral speeds of about 1200 ft. per min. 

Strength of Herringbone Gears 

The working capacity of herringbone gears, in accordance with the 
practice of the Falk Co., American makers of the Wuest herringbone 
gears, is given by the formulas: 



P = 



2.5 

in which h.p. = horse-power. 

P = tooth pressure, lbs. 

K= admissible stress, lbs. per sq. in. in accordance with 

Fig. 10. 
p = circular pitch, ins. For diametral pitch take nearest 

circular pitch. 
TF= total width of face, ins., including non-bearing 

width equal to i p. 
V = velocity of pitch circle, ft. per min. 












~1 






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600 


















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■ 




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[ — 


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^ 



200 400 600 800 1000 1200 UOO 1600 1800 2000 2200 2400 2600 2800 

K^ Velocity, Et.per Min, 

Fig. io. — Values of admissible stress in herringbone gears of various 

materials. 



Table i8 and Fig. lo have been prepared from this formula by 
J. E. HoLVECK {Mchy., June, 1913). The desirable speed limit, 
on account of noise, if the gears do not run in oil, is 1500 ft. per min. 
Much higher speeds may be used satisfactorily but with correspond- 
ingly increased wear and noise. The pinion is commonly of tougher 
. material than the gear, because of its greater wear, and the chart. 
Fig. 10, gives the admissible stress for various materials as developed 
by extended experience. The formulas and table are based on a 
tooth angle of 23 degrees. Note that, for the present purpose, the 
pitch is measured on the circumference — not on the normal. 



W18.000 

u 
o 

§15,000 

.a 

•-112,000 

I 8000 
m 

S 6000 

.a 



g 3000 



Fig. II. 





K 


s. 








s 


. 










'S 










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Ir 












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jn 










































1 































































































600 1200 1800 2400 3000 3600 

Pitch Line Velocity, Feet per Minute 



4200 



-Stress factor S in Bates's formula for strength of 
herringbone gears. 



The practice of the Fawcus Machine Company is thus given by 
W. C. Bates, Mech. Engr. of the company {Amer. Mach., Nov. 
18, 1915). 

The angle of the teeth with the axis is approximately 23 deg. ; the 
addendum = .2546 X circular pitch; the teeth have involute pro- 
files and the angle of obliquity is 20 deg. In order that the action 
of the teeth may overlap, the minimum over-all face width should 
be at least six times the circular pitch, this width including the central 
clearance space which, ordinarily, has a width equal to the circular 
pitch. 



104 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table i8. — Horse-power of Herringbone Gears 



Pitch 



Pitch dia. 
inches 



Face, 
inches 



Horse-power 



6 D.P. 
.5236" C.P. 

SD.P. 
.6283" C.P. 

4 D.P. 

.7854" C.P. 

3i D.P. 
.8976" C.P. 

3 D.P. 
1.0472" C.P. 

2i D.P. 
1.2566" C.P. 

2 D.P. 

1.5708" C.P. 

if D.P. 
1.7952" C.P. 

il D.P. 
2 . 0944" C.P. 



3t 



4-2 



Si 



8.4 



14 



4 
si 

5 

6i 
6i 



isi 



18 



16 



Velocity 
Cast iron 
Cast steel 
Cast iron 
Cast steel 

Cast iron 
Cast steel 
Cast iron 
Cast steel 

Cast iron 
Cast steel 
Cast iron 
Cast steel 

Cast iron 
Cast steel 
Cast iron 
Cast steel 

Cast iron 
Cast steel 
Cast iron 
Cast steel 

Cast iron 
Cast steel 
Cast iron 
Cast steel 

Cast iron 
Cast steel 
Cast iron 
Cast steel 

Cast iron 
Cast steel 
Cast iron 
Cast steel 

Cast iron 
Cast steel 
Cast iron 
Cast steel 



400 


600 


800 


1000 


1200 


1400 


1500 


1600 


1800 


7 


10 


12 


14 


15 


17 


18 


19 


20 


13 


18 


23 


27 


31 


35 


36 


37 


40 


9 


13 


16 


18 


20 


22 


24 


25 


26 


17 


24 


30 


35 


41 


46 


47 


49 


53 


10 


14 


17 


20 


23 


25 


26 


27 


29 


19 


27 


34 


40 


46 


51 


53 


55 


59 


13 


18 


21 


25 


29 


31 


32 


34 


36 


24 


34 


43 


so 


58 


64 


66 


69 


74 


16 


22 


27 


31 


36 


40 


41 


43 


46 


29 


41 


S3 


62 


71 


80 


83 


86 


93 


20 


27 


34 


39 


44 


SO 


SI 


53 


57 


36 


SI 


66 


77 


88 


99 


103 


107 


116 


21 


29 


36 


42 


47 


52 


55 


57 


61 


39 


S5 


70 


83 


95 


105 


no 


114 


123 


26 


36 


45 


52 


58 


65 


68 


71 


76 


48 


68 


87 


103 


118 


130 


136 


141 


154 


28 


39 


48 


55 


63. 


69 


71 


75 


80 


SI 


72 


92 


no 


126 


139 


146 


151 


I6S 


36 


SO 


61 


70 


80 


89 


90 


96 


102 


6S 


92 


117 


14c 


160 


177 


184 


192 


210 


41 


S7 


70 


80 


91 


lOI 


106 


109 


116 


74 


I OS 


134 


160 


185 


202 


211 


220 


240 


SI 


71 


87 


100 


114 


126 


132 


137 


145 


93 


130 


167 


200 


228 


253 


264 


274 


300 


64 


89 


109 


125 


143 


158 


166 


171 


182 


116 


164 


207 


250 


285 


316 


329 


342 


375 


79 


III 


136 


154 


178 


196 


206 


213 


226 


144 


204 


259 


309 


355 


393 


410 


425 


465 


86 


118 


146 


166 


190 


210 


219 


228 


241 


iSS 


218 


279 


331 


378 


420 


438 


455 


496 


106 


146 


180 


206 


234 


261 


272 


282 


300 


192 


271 


345 


413 


470 


521 


545 


562 


610 


no 


IS2 


187 


215 


244 


270 


282 


292 


311 


199 


280 


358 


426 


488 


540 


564 


585 


640 


^37 


190 


234 


269 


305 


338 


343 


366 


388 


249 


35° 


447 


534 


610 


67s 


707 


732 


800 



2000 

21 

42 
28 

55 

31 
63 
39 
79 

49 

99 

61 

122 

65 
131 

81 
162 



173 

107 
220 

122 
251 
152 
314 

19s 
392 
242 



258 
522 
320 
710 

333 
671 
406 
838 



The formula developed by Mr. Bates is a modification of the 
Lewis formula for spur gears as follows: 

in which 

1^ = working load on the teeth, lbs., 
5 = stress factor, taken from chart, Fig. 11, 

^ = circular pitch measured parallel to the edge of the face, ins., 
31 = the Lewis factor y for 15 deg. involute, standard addendum, 

spur gear teeth, for which see Table 12, 
/= total over-all face width, ins., 
X = a. factor dependent upon the variation of the maximum from 

the average torque for the gear cycle, 
Z = a factor for wear, depending upon the nearness with which 
lubrication conditions approach the ideal. 

The last two factors are important. The X factor must be ap- 



plied when the gears are used to drive reciprocating or other machin- 
ery in which the torque varies from maximum to minimum, one or 
more times during a single revolution. 

For instance, in a single cylinder pump or compressor, the torque 
varies from zero to maximum twice in one revolution of the gear. 
There will be almost no wear on the gear teeth at the dead center 
points, and at midposition of the plunger or piston the torque may 
be double the average calculated from the total horse-power require- 
ment of the machine. This must be taken into consideration in 
order that excessive wear may not occur on gear teeth at the position 
of maximum torque. For this reason the minimum value of X is 
I, and may be anything over this figure, depending upon the load 
cycle. 

The Z factor has also a minimum value of i, this condition being 
found when the gears run with a continuous supply of oil to the teeth. 
This supply may be obtained by allowing the lower portion of the 
gear to run partially submerged in an oil bath, and may be success- 



SPUR GEARS 



105 



fully used up to speeds of about 2000 or 2500 ft. per min. Above 
these speeds the centrifugal force throws most of the oil ofi, and the 
teeth should be sprayed with oil under a slight pressure, the stream 
being directed at the line of contact between gear and pinion on the 
entering side. For conditions of thorough grease lubrication, a 
factor of Z of 1. 10 to 1.20 is recommended; for rather scanty lubrica- 
tion and frequent inspection of the gears, 1.15 to 1.25 are good 
figures; for very indiSerent lubrication conditions, it should be in- 
creased to 1.20 or 1.30, to secure the best wearing conditions. 

A great many considerations enter into the problem of the selec- 
tion of the proper size of tooth to be used for given conditions of 
horse-power transmitted and pitch line velocity of the teeth. All 
other conditions being equal, the size of the teeth in gear and pinion 
determines the relative degree of quietness with which they will 
run at a given pitch-line velocity, the smaller the pitch the quieter 
the operation. 

Turbine reduction gears are distinguished by the exceptionally 
fine pitch teeth used for gear and pinion. The fine pitch is necessary 
that the gears may run quietly at high speeds, sometimes in excess 
of 6000 ft. per min. The face of the gears is made very wide propor- 
tionally to obtain the proper tooth strength and wearing qualities. 
At the other extreme are the conditions of rolling-mill service where 
low speeds are the rule, where absence of noise is not a factor and 
where sudden shocks call for gears with heavy teeth. 

As a general proposition, it is not considered good practice to use 
cast iron as a material for cut herringbone pinions. The additional 
cost of making the pinion of a steel forging is slight, and the benefits 
obtained by the use of the better wearing material will, in practically 
every case, justify the extra expense. It is also better practice to 
use a different grade of steel in the pinion than in the gear, when the 
latter is made of a steel casting. For general use, a .40 to .50 per cent, 
carbon steel forged pinion should mate with a gear from a .25 to .30 
per cent, carbon steel casting. This last must be thoroughly furnace 
annealed to diminish shrinkage strains and to obtain uniform hard- 
ness of steel throughout the casting. The use of materials of 
different textures prevents to a great extent the seizing or cutting 
of the two materials when under abnormally heavy loads. 

With further reference to design, the pinion should be made of a 
solid forging or casting in every case where the diameter does not in- 
volve excessive cost. The gears should be designed for rigidity both 
torsional and sidewise. Oval-arm gears should be avoided for these 
have little sidewise rigidity. If the diameter of the gear is relatively 
small, say under 30 or 40 in., and the face is proportionally narrow, 
say 3"^ to yi of the diameter, use cross-arm gears, but in every case 
where the width of face will permit, use doublearm gears, with a web 
section between the arms, forming an H-section. With the object 
of obtaining rigidity and absence of vibration in the drive, under no 
circumstances should a width of face be used for a cut herringbone 
gear, which is less than 3^10 the diameter of the gear. 

Pinions and gears may both be cut integral with their respective 
shafts. In fact, a pinion having (say) 16 teeth, 2 DP., with an 
outside diameter of 8.98 in., may be cut integral with a pinion shaft 
of the same diameter. The best practice is always to mount the pin- 
ion so that both the supporting bearings are close up to the faces of 
the pinion. The length of span between edges of bearings should 
not exceed four times the diameter of shaft supporting the pinion. 

The finished thickness of the rim under the teeth of a cut herring- 
bone gear is best designed as jr^ -\- 3^ in. for all gears. Where it is 

necessary to use a gear made in two pieces, and bolted together, the 
joint should be made through, not between, the arms, and should be 
entirely machined on the bearing surfaces, including a tongue and 
groove in the opposite halves extending from hub to rim. The rim 
should be split parallel to the tooth angle at a point midway between 
two teeth to prevent weakening them or interfering with their even- 
ness of wear. This construction of gear may be accurately re- 
assembled in its final position, with a minimum of care and expense. 



Regarding rigidity of mounting and accuracy of alignment of 
shafts, it may be said that indifferent methods will be surely attended 
by indifferent results. Cut herringbone gears generally run at much 
higher speeds than are considered practical for metal cut spur gears, 
and under these conditions require heavier and more rigid mountings, 
and more precise alignment in order that their full benefits may be 
realized. They are best mounted so that hubs on gears or pinions 
have only a running clearance between the end of bearings in which 
the shafts revolve. This eliminates end play of the shafts and pre- 
vents axial thrusts from an outside source being transmitted to the 
gear or pinion teeth. These axial thrusts, unless prevented, cause 
unnecessary wear on the teeth and are a frequent cause of noise. 

Dimensions of Spur Gear Parts 

The dimensions of the arms of sptir gears may be obtained from the 
following formulas and chart. The formulas are by Henry Hess 
{Amsr. Mach., Apr. 29, 1897) and represent the practice of the Niles 
Tool Works. They are based on an equal distribution of the pitch 
line load among the arms (which are assumed to act as cantilevers) 
and consider the sole load at the pitch line to be that given by the 
Lewis formula for the strength of gear teeth. For elliptical cross- 
sections: 






{N- 7)P^R 
20 A 

(jV^ tV^R 

20 A p^ 



for circular pitch 



for diametral pitch 



in which E = thickness of arm at hub, ins. 

2E= width of arm at hub, ins. 

N = number of teeth. 

P = circular pitch, ins. 

p = diametral pitch. 

R = ratio of face divided by circular pitch. 

A — number of arms. 
For other cross- sections: 

P32?(iV-7) 



Z = 



SO A 

n^RjN-y) 
Sop^A 



for circular pitch 



for diametral pitch 



in which Z = section modulus of arm at rim. 

The same results for elliptical cross-sections may be obtained from 
Fig. 12 by Prof. J. B. Peddle (Amer. Mach., Feb. 13, 1913). The 
use of the chart is explained below it. 



\^^d — >l 




Figs. 13 and 14. — Proportions of hollow arms for large gears. 



For large arms the designer will frequently prefer a cored section. 
A satisfactory one is that of Fig. 13, in which major and minor axes of 
both core and arm are relatively as 2 to i. By equating the moduli 
of resistance for solid and hollow elUptical sections of these propor- 



tions, it is found that E^ = 



D*-d* 
D ' 



in which E is the thickness of the 



solid arm as obtained by chart or formula; d and D are dimensions 
of the cored arm. See Figs. 13 and 14. 



106 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



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SPUR GEARS 



107 



In order to lessen the work of making the core box by substituting 
flat surfaces for curved ones, an approximation Hke Fig. 14 will add 
but slightly to the weight, as is shown by the ellipse dotted in for 
comparison. 

The outlines are formed of circular arcs struck from four centers, 
which approximate very closely to the true ellipse and look better. 
The construction of the core sides is readily apparent from the sketch. 

A suitable taper is i in 32 and 16, respectively, for the arm thick- 
ness and width; this gives a pleasing appearance for a moderately 
long arm, but it is not a hard-and-fast rule, as a greater or lesser taper 
may be employed to suit the designer's fancy without afiecting the 
strength of the arm, unless the taper is made so excessive as to bring 
the dimensions at the rim down to one-half of those at the base. 

As the tooth and arm are of the same material, the method is satis- 
factory for all cast gears, but this must not be interpreted to mean 
that this or any other formula will prevent shrinkage strain due to 
relatively large hubs or very heavy rims; where these occur, great 
care must be exercised in the foundry, and it will also not be amiss 
to add a generous amount of metal to the arms. 




^ = diametral pitch. 
P = circular pitch, ins. 
t = thickness of tooth at pitch line, ins. 
iV^ = number of teeth. 

Fig. 15. — Notation of C. H. Logue's formulas for the dimensions 
of spur gears. 

Other dimensions of spur gears may be obtained from the following 
formulas by C. H. Logue {Amer. Mach., Sept. 30, 1909) the notation 
being given in connection with Fig. 15. 

M =^4^, or 1.25P 



M' = 54^.ori.6oP 



R' =—^, or .913-P 

W' = ^^, or .6866P 
P 

R=l:f , or. s6sP 



Mean cross-section of arm = i.3X/XP 
A 



(the same multiplier being used when finding both A and E) 

D =d-\-^\/NF if reinforcements are used opposite key ways. 
Otherwise 

D =2d 

The number of arms may be as follows: 

Diameter No. of arms 

Up to 60 ins 6 

60 to 80 ins 8 

Over 80 ins 10 

For convenience of chucking without distortion the most desirable 
number of arms is either 6 or 9. 

The width of face of spur gears according to traditional rules, 
should be from two and one-half to three times the circular pitch. 
The increased pressures put upon steel gears would, however, seem 
to call for greater widths since the increased strength of steel as 
compared with cast-iron is not accompanied by correspondingly 
increased wearing properties. Accordingly we find the gears of street 
railway cars with faces of five times the pitch. 

Chordal Pitch 

Chordal pitch is the pitch measured on the chord as spaced by the 
dividers instead of on the arc of the pitch circle. It is used only when 
laying out chain sprockets and large gears and segments that cannot 
be cut on a gear cutter. When laying out such large gears the 
chordal pitch must be used, as the chordal pitches of two mating 

Table 19 — Pitch Diameters for i In. Chordal Pitch 
For other pitches multiply by the pitch 



-4 



mean section Xi-sy 



2, 24 or 3 
E =(2, 2i 01 s) A 



No. 


Pitch 


No. 


Pitch 


No. 


Pitch 


No. 


Pitch 


teeth 


diameter 


teeth 


diameter 


teeth 


diameter 


teeth 


diameter 


4 


1-414 


39 


12.427 


74 


23.562 


109 


34-701 


S 


1 .701 


40 


12.746 


75 


23.880 


no 


35. 019 


6 


2 .000 


41 


13.064 


76 


24. 198 


III 


35.337 


7 


2.30s 


42 


13.382 


77 


24-517 


112 


35.655 


8 


2.613 


43 


13-699 


78 


24-835 


113 


35.974 


9 


2.924 


44 


14.018 


79 


25-153 


114 


36.292 


10 


3.236 


45 


14-335 


80 


25-471 


115 


36 ,610 


II 


3-549 


46 


14-653 


81 


25-790 


116 


36.929 


12 


3.864 


47 


14.972 


82 


26.108 


117 


37.247 


13 


4-179 


48 


15-291 


83 


26.426 


118 


37.565 


14 


4-494 


49 


15-608 


84 


26.744 


119 


37.883 


IS 


4.810 


50 


15.927 


85 


27.063 


120 


38,202 


16 


5-126 


51 


16.244 


86 


27-381 


121 


38.520 


17 


5-442 


52 


16.562 


87 


27.699 


122 


38.838 


18 


5-759 


53 


16.880 


88 


28 ,017 


123 


39 156 


19 


6.076 


54 


17 .200 


89 


28,335 


124 


39.475 


20 


6-392 


55 


17.516 


90 


28.654 


125 


39-793 


21 


6.710 


56 


17-835 


91 


28.972 


126 


40 . 1 1 1 


22 


7.027 


57 


18.152 


92 


29.290 


127 


40.429 


23 


7.344 


58 


18.471 


93 


29.608 


128 


40 . 748 


24 


7.661 


59 


18.789 


94 


29.927 


129 


41 .066 


25 


7.979 


60 


19.107 


95 


30.24s 


130 


41.384 


26 


8.297 


61 


19.42s 


96 


30,563 


131 


41.703 


27 


8.614 


62 


19.744 


97 


30,881 


132 


42 .021 


28 


8.931 


63 


20,062 


98 


31 .200 


133 


42.339 


29 


9.249 


64 


20.380 


99 


3I-S18 


134 


42.657 


30 


9.567 


65 


20.698 


100 


31.836 


135 


42.976 


31 


9.884 


66 


21 ,016 


lOI 


32.154 


136 


43-294 


32 


10.202 


67 


21.335 


102 


32.473 


137 


43-612 


33 


10.520 


68 


21.653 


103 


32.791 


138 


43-931 


34 


10.838 


69 


21.971 


104 


33 109 


139 


■ 44 - 249 


3S 


11.156 


70 


22,289 


105 


33.428 


140 


44-567 


36 


11.474 


71 


22 .607 


106 


33.740 


141 


44.890 


37 


II. 791 


72 


22 ,926 


107 


34.058 


142 


4S . 204 


38 


12. IIO 


73 


23.244 


108 


34376 


143 


45. 522 



108 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



gears of different sizes differ, although the circular pitches are equal. 
The chordal pitch is given by the equation: 

P' = Z>' sin 

n 

in whichP'= chordal pitch, 
Z)' = pitch diameter, 
w = number of teeth. 
Table ig is calculated from this equation. 

Gear Cutter Sets 

The production of strictly correct gear tooth curves involves the use 
of a special cutter for each pitch and for each number of teeth. 
In practice, cutters for each pitch are made in numbered sets, each 
cutter being used through a considerable range as follows: 



No. of 
cutter 


To cut gears 
having teeth from 


No. of 
cutter 


To cut gears 
having teeth from 


I 
2 
3 

4 


135 to a rack 
55 to 134 
35 to 54 
26 to 34 


5 
6 

7 
8 


21 to 25 
17 to 20 
14 to 16 
12 to 13 



The tendency at present is to reduce these ranges by introducing 
intermediate cutters in order to obtain greater accuracy as follows : 



No. of 


To cut gears 


No. of 


To cut gears 


cutter 


having teeth from 


cutter 


having teeth from 


I 


135 to a rack 


S 


21 to 22 


m 


80 to 134 


sH 


19 to 20 


2 


55 to 79 


6 


17 to 18 


2H 


42 to 54 


6H 


15 to 16 


3 


35 to 41 


7 


14 


3^i 


30 to 34 


7H 


13 


4 


26 to 29 


8 


12 


4H 


23 to 25 







Gaging Gear Teeth 

The most common method of gaging gear teeth is by the use of the 
Brown and Sharpe gear tooth vernier caliper. This instrument 
measures the chordal thickness which, for large pitches and small 
diameters, differs sensibly from the arc or circular thickness. When 
using the caliper it is necessary to know the perpendicular distance 
from the outer circumference to the chord measured. Accuracy of 
blank diameter is assumed and necessary. Table 20 gives the 
necessary constants for one diametral pitch and for i in. circular 
pitch. 



Table 


20. — Chordal Thickness of Involute Gear Teeth 


One diametral pitch. 
For other pitches divide by the pitch 


i-In. circular pitch. 

For other pitches multiply by the 

pitch 






Chordal 


Perp. 






Chordal 


Perp. 


No. of 


No. of 


thickness 


dis. 


No. of 


No. of 


thickness 


dis. 


teeth 


cutter 


at pitch 


circum. 


teeth 


cutter 


at pitch 


circum. 






line 


to chord 






line 


to chord 


12 


8 


1.5663 


1.0514 


12 


8 


.4986 


• 3347 


13 


7W 


1.5670 


1.0474 


13 


7K2 


.4988 


.3334 


14 


7 


1.567s 


I .0440 


14 


7 


.4990 


.3323 


15 


6\i 


1.5679 


I .0411 


15 


6 1/2 


.4991 


.3314 


17 


6 


1.5686 


I .0362 


17 


6 


.4993 


.3298 


19 


SW 


I .5690 


1.0324 


19 


5^4 


4995 


.3286 


21 


5 


1.5694 


I .0294 


21 


5 


• 4996 


.3277 


23 


4H 


I .5696 


1.0268 


23 


4W 


.4997 


.3268 


26 


4 


1.5698 


l.0?37 


26 


4 


.4997 


■ 3258 


30 


3\i 


I. 5701 


I .0208 


30 


3H 


.4998 


■ 3249 


35 


3 


1.5702 


I .0176 


35 


3 


.4998 


■ 3239 


42 


2K2 


1.5704 


I. 0147 


42 


2M 


.4999 


.3230 


55 


2 


1.5706 


1.0112 


55 


2 


.5000 


• 3219 


80 


iK 


1.5707 


1.0077 


80 


iH 


.5000 


.3208 


135 


I 


1.5708 


I .0046 


135 


I 


.5000 


.3198 



The following method of gaging involute spur gear teeth makes use 
of the ordinary vernier caliper, gages for uniformity of spacing as 
well as for thickness of teeth and is independent of the accuracy 
with which the blanks are turned. It is due to H. E. Taylor 
{Amer. Mach., June 26, 1913). The dimension used is measured 
on a tangent to the base circle and is the maximum dimension 
over two or more teeth. 

In Fig. 16 the pitch and base circles are shown, the latter -being 
determined by the angle of obliquity, a. The manner in which the 
involute profile is generated involves the equality of any arc of the 
base circle and the corresponding tangent. This implies that the 
base circle arc spanning one tooth and one-half a space shall equal 
the corresponding tangent and that two such arcs spanning two 
teeth and one space shall be equal to the combined tangents t, 
Fig. 16. It is on this fact that the system is based. 




Fig. 16. — Gaging gear teeth with the vernier caliper. 

If the base circle is larger than the root circle the tangent will be 
the maximum dimension over two teeth. With large obliquities, 
as in some systems, the base circle falls within the root circle and in 
such cases it may be necessary to gage over more than two teeth. 
The caliper may be applied at various angles as in the illustration, 
the readings being identical if the curves are true involutes. In 
this way the truth of the curves may be tested, and, by applying the 
instrument over different pairs of teeth, the uniformity of the spacing 
may be tested. 

As the result of a mathematical analysis which it is not necessary 
to repeat here, Mr. Taylor finds that: 

K 



lier reading = 3^ 7 — 5 — ttt 

° dimetral pitch 



in which K = a, constant to be taken from Table 21. 

The use of the method and table is best shown by an example: 
Assume a gear of four diametral pitch, twenty teeth and 14H deg. 
obliquity. In the table, opposite twenty teeth and in the 14M deg. 
column, we find 2^ = 4.66654 and 



vernier reading = 



4.66654 



This reading makes no allowance for backlash. If .001 in. back- 
lash is to be provided for, it should be subtracted from the reading, 
giving 1. 1656 in. 

The same table may be used for gaging metric gears, in which the 



SPUR GEARS 



109 



module is analogous to (not identical with) the diametral pitch in the 
English system. 

pitch diameter in mm. 



Table 21. — Constants for Gaging Involute Gear Teeth 



and 



module = - , r . , 1 

number of teeth 

venier reading = module X iiT 



Assume a gear of four module, twenty-eight teeth and 14H deg. 
obliquity. Opposite twenty-eight teeth and in the 14M deg. column 
we find: 

X = 4. 70822 

and vernier reading = 4. 70822X4 

= 18.833 niDi- 

As before, this reading makes no allowance for backlash. 

The factoring of numbers is frequently required in the calculation 
of trains of gearing. Table 22 by John Parker {Amer. Mach., Dec. 
5,1907) gives the smallest prime factors of all numbers between i and 
9999. The top horizontal line gives the thousands and hundreds, 
the left vertical column the tens and units, and the body of the table 
the smallest prime factors. To find the other factors divide by the 
factor found and consult the table again. If no factor appears 
opposite a given number, the number is prime. Numbers divisible 
by 2 and 5 are omitted. Such numbers should be divided by 2 or 5 
before consulting the table. 

Example. — Required the smallest prime factor of 979. In the 
first table find 9 in the top horizontal line and 79 in the left vertical 
column. At the intersection is 11 — the smallest prime factor. 
Similarly, consulting the table again, we find no entry for 971, show- 
ing that number to be prime. 



Angle of 
obliquity 


14M2 


18 


20 


22 


No. of teeth 
gaged over 


2 


2 


2 


3 


2 


3 




Values of K 



13 

14 

15 
16 

17 
18 
19 



23 
24 

25 

26 
27 
28 
29 

30 
31 
32 

33 

34 

35 
36 
37 
38 
39 

40 
41 
42 
43 

44 

45 
46 
47 
48 
49 



4.61444 
6196s 
62486 
63007 
63528 

64049 
64570 
65091 
65612 
66133 

66654 
67175 
67696 
68217 
68738 

69259 
69780 
70301 
70822 
71343 

71864 
72385 
72906 
73427 
73948 

74469 
74990 
75511 
76032 
76553 

77074 
77595 
78116 
78637 
79158 

79679 
80200 
80721 
81242 
81763 

.82284 



4-57958 
58936 
59914 
60892 
61870 

62848 
63826 
64804 
65782 
66760 

67738 
68716 
69694 
70672 
71650 

72628 
73606 
74583 
75562 
76540 

77518 
78496 
79474 
80452 
81430 

82408 
83386 
84364 
85342 
86320 

87298 
88276 
89254 
90232 
91210 

92188 
93166 
94144 
95122 
96100 

■97078 



4 56070 
•57395 
.58720 

4.60045 
.61370 

.62695 
.64020 
•65345 
.66670 
•67995 

•69320 
4^70645 
.71970 
•7329s 
.74620 

•7S94S 
.77270 
.78595 
.79920 
4^81245 

.82570 
•8389s 
•85220 
•86545 
.87870 

.89195 
4^90520 
•91845 
•93170 
•94495 

.95820 
.97145 
■98470 
.99795 
S.0II20 

.02445 
.03770 
•05095 
.06420 
•07745 

.09070 



7.51282 
.52607 
•S3932 
•55257 
.56582 

•57907 
•59232 
7^60557 
.61882 
.63207 

•64532 
•65857 
.67182 
.68507 
.69832 

7.711S7 
.72482 
■73807 
■75132 
■76457 

■77782 
•79107 
7^80432 
•81757 
.83082 

.84407 
.85932 
•87057 
.88382 
.89707 

7.91032 
•92357 
•93682 
.95007 
•96332 

•97657 

.98982 

8.00307 

. 1 63 2 
.02957 

.04282 



4.54284 
.56020 
•57756 

•59492 
4.61228 

.62964 
.64700 
■66436 
■68172 
■69908 

4.71644 
.73380 
■75116 
.76852 
.78588 

4.80324 
.82060 
■83796 
■85532 
.87268 

.89004 
4.90740 
.92476 
.94212 
■95948 

■97684 
.99420 
S.01156 
.02892 
.04628 

.06364 
.08100 
.09836 
5-11572 
.13308 

■ 15044 
■16780 
.18516 
S .20252 
.21988 

.23724 



7-45568 
•47304 
•49040 

7^50776 
•52512 

.54248 
•55984 
-57720 
•594S6 
7 .61192 

.62928 
.64664 
.66400 
.68136 
.69872 

7.71608 
•73344 
•75089 
.76816 
•78552 

7.80288 
.82024 
.83760 
.85496 
.87232 



7.90704 
•92440 
•94176 
•95912 

•97648 
.99384 
8.01120 
.02856 
.04592 

.06328 
.08060 
.09800 
8.I1536 
.13272 

.15008 



110 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 





























Table 


25 


•- 


-Prime 


Factors 


OF 


Numbers 


FROM I 


TO 


3999 
































Nos. I to 2499 


Nos. 2501 to 4999 


|0|l|2|3l4l5|6|7|8|9 il0lll|l2|l3il4lis|lbll7|l8|l9|20|2l|22|23|24l 


|2S|26|27|28|29l30|3l|32|33l34l3Sl36|37l38|39l40|4ll42|43l44l45l46|47l48|49 


01 






3 


7 




3 






3 


17 


7 


3 






3 


19 




3 






3 


II 


31 


3 


7 


01 


41 


3 


37 




3 




7 


3 




19 


3 


13 




3 


47 




3 




II 


3 


7 


43 


3 




13 


03 






7 


3 


13 




3 


19 


II 


3 


17 




3 




23 


3 


7 


13 


3 


II 




3 




7 


3 


03 




19 


3 






3 


29 




3 


41 


31 


3 


7 




3 




11 


3 


13 


7 


3 






3 




07 






3 




II 


3 




7 


3 




19 


3 


17 




3 


11 




3 


13 




3 


7 




3 


29 


07 


23 


3 




"7 


3 


31 


13 


3 






3 




II 


■3 






3 


7 


59 


3 




17 


3 


II 


7 


09 


3 


•• 


II 


3 






3 






3 






3 


'7 




3 






3 


23 


7 


3 


47 




3 


09 


13 




3 


53 




3 






3 


7 


II 


3 




13 


3 


19 


7 


3 


31 




3 


II 


17 


3 




II 




3 






3 


7 


13 


3 






3 


11 


7 


3 


17 




3 


29 




3 






3 






11 


3 


7 




3 


41 




3 


13 


7 


3 




23 


3 


37 




3 






3 


II 


13 


3 


7 


17 


3 


13 






'3 




7 


3 




23 


3 


II 




3 




13 


3 


17 




3 


7 




3 






3 


19 


13 


7 


3 




29 


3 


23 


II 


3 






'3 




47 


3 


7 




3 


II 


19 


3 




7 


3 




17 


17 




'3 


7 




3 


II 




3 


19 


7 


3 






3 


13 


37 


3 


17 


23 


3 




29 


3 


7 




17 


3 




11 


3 




7 


3 




31 


3 






3 


II 




3 


23 




3 


7 




3 


53 




3 


19 


•• 


7 


3 


II 




3 






3 






3 


23 




3 


7 




3 


17 


19 


3 


13 


7 


3 


41 


19 


11 


3 






3 






3 




13 


3 


7 




3 






3 


•• 


7 


3 




31 


3 


61 




21 


3 


11 


13 


3 






3 


7 




3 




19 


3 




7 


3 






3 


17 


43 


3 




II 


3 


21 






3 


7 


23 


3 






3 


11 


7 


3 


61 


3 


3 




13 


3 


29 




3 






3 


7 


23 




3 




17 


3 




7 


3 




13 


3 






3 






3 






3 


7 


11 


3 


23 




23 


3 


43 


7 


3 


37 




3 


II 




3 


13 




3 






3 


7 


41 


3 






3 




7 


3 


27 


"3 






3 


7 


17 


3 






3 


13 


'7 


3 






3 




II 


3 


41 




3 


17 


13 


3 


27 


7 


37 


3 


11 




3 


S3 


7 


3 


23 




3 




43 


3 






3 




19 


'3 


7 


29 


3 


13 


29 . 


•• 


3 




7 


3 


23 


17 


3 






3 






3 




II 


3 


7 


31 


3 






3 


17 


7 


29 


3 


11 




3 


29 


13 


3 






3 




19 


3 


7 




3 






3 


43 


7 


3 




II 


3 


31 




. . 


3 




, , 


3 


, , 


17 


3 


7 




3 


, , 


II 


3 




7 


3 






3 




23 


3 


II 


31 




3 




19 


3 


7 


31 


3 




47 


3 




7 


3 




29 


3 




61 


3 


23 


II 


3 






33 


3 


7 




3 




13 


3 




7 


3 




II 


3 


31 




3 


23 




3 




19 


3 


7 




3 


i3 


17 




3 




7 


3 


13 


Si 


3 






3 






3 


37 




3 


7 


11 


3 


41 




'3 




37 






3 




19 


3 


7 


II 


3 




17 


3 




7 


3 


29 




3 


II 


13 


3 






3 




37 


43 


3 


7 




3 






3 


47 


'7 


3 




37 


3 


31 


11 


3 


19 




3 


13 




'3 


7 




39 


3 






3 




7 


3 






3 




17 


3 


13 




3 


II 


37 


3 


7 




3 






3 


39 




7 


3 


17 




3 


43 


41 


3 


19 




3 




II 


3 


7 




3 




23 


3 




7 


3 


II 


41 




3 




II 


3 




. , 


3 


29 




3 


7 


17 


3 


II 


23 


3 




7 


3 


13 




3 




, . 


41 


3 


19 




3 


17 




3 


7 


13 


3 




II 


3 


23 


7 


3 


41 




3 




19 


3 


II 


47 


3 


43 




II 


3 


7 




3 






3 


23 


7 


3 


II 


17 


3 




31 


3 


19 


29 


3 






3 


7 


43 




3 


13 




3 


17 


7 


3 




11 


3 




19 


3 




13 


3 




43 


3 


7 




3 


29 




47 




3 


13 




3 






3 


7 




3 


31 


29 


3 




7 


3 






3 


23 


19 


3 






47 


3 




41 


3 


7 


11 


3 


17 




3 




7 


3 






3 


II 


31 


3 






3 


47 


37 


3 


49 


7 




3 






3 


II 


7 


3 


13 




3 




19 


3 




17 


3 


43 




3 


7 


13 


3 


31 


49 




3 




7 


3 




47 


3 


17 




3 


41 


23 


3 


II 




3 


7 


•• 


3 


•• 




3 


13 


7 


SI 


3 


, . 




3 


11 


19 


3 


, , 


23 


3 






3 


7 




3 


13 


17 


3 




7 


3 






3 


51 




II 


3 




13 


3 


23 




3 


7 


53 


3 


11 




3 


.. 


7 


3 


19 




3 






3 




53 




3 


II 




3 


7 




3 






3 




7 


3 






3 




17 


3 






3 


13 


II 


S3 


3 


7 




3 




43 


3 




7 


3 


II 


13 


3 




59 


3 






3 


6i 


29 


'3 


'7 


23 


3 


57 


3 






3 






3 






3 


7 


13 


3 


23 


31 


3 




7 


3 


19 


II 


3 


37 




3 


57 






3 






3 


7 




3 






3 


13 


7 


3 






3 






3 




67 


3 




59 




3 


"1 




3 


13 




3 




7 


3 


19 




3 






3 




II 


3 


29 


17 


3 


7 




59 


3 




31 


3 


11 


7 


3 






3 






3 


17 


37 


3 






3 


'7 


47 


3 




43 


3 


61 




7 


3 


19 




3 


. . 


, . 


3 


31 




3 


13 




3 


7 


II 


3 




37 


3 




7 


3 


23 


61 


13 


3 


11 




3 


. , 


29 


3 






3 


7 




3 


17 


31 


3 


.. 


7 


3 




59 


3 




II 


63 


3 






3 






3 


7 




3 






3 


29 


7 


3 




41 


3 


13 




3 


31 


17 


3 


63 


II 




3 


7 




3 




13 


3 




7 


3 


53 




3 


17 


23 


3 






■3 




II 


3 


7 


67 






3 






3 


23 


13 


3 




II 


3 


7 




3 






3 




7 


3 


II 




3 




67 


17 


3 




47 


3 






3 


7 




3 


19 




3 




7 


3 


17 


II 


3 




13 


3 


31 




69 


3 


13 




3 


i 




3 




II 


3 




7 


3 


37 


13 


3 




29 


3 


II 




3 


•• 


23 


3 


69 


7 


17 


3 


19 




3 




7 


3 




43 


3 




S3 


3 


13 


II 


3 


17 


41 


3 


7 


19 


3 




71 




3 




7 


3 




II 


3 


13 




3 




31 


3 




, , 


3 


7 




3 


19 


13 


3 




7 


71 


3 




17 


3 




37 


3 






3 






3 


7 


II 


3 


43 




3 


17 


7 


3 


13 




3 


73 






3 




II 


3 






3 


7 


29 


3 


19 




3 


II 


7 


3 






3 


41 




3 




73 


31 


3 


47 


13 


3 


7 


19 


3 




23 


3 




7 


3 


29 




3 






3 


17 




3 


II 




77 


7 


3 




13 


3 






3 






3 


11 




3 


7 


19 


3 






3 


31 


7 


'3 






77 


3 






3 


13 


17 


3 


29 


II 


3 


7 




3 




41 


3 




7 


3 


11 


23 


3 


17 




II 


79 


•• 


•• 


3 






3 


'7 


19 


3 


II 


13 


3 




7 


3 




23 


3 


•• 


■• 


3 




43 


3 


37 


79 




3 


'7 




3 




II 


3 


31 


7 


3 


13 




3 


23 




3 


II 


29 


3 


19 




3 


7 


3 


. 81 


3 






3 


13 


7 


3 


II 


, , 


3 


23 




3 






3 


41 


13 


3 


7 




3 






3 


81 


29 


7 


3 


43 


11 


3 




17 


3 


59 




3 


19 




3 


7 


37 


3 


13 




3 


31 


7 


3 


73 


83 


.. 


3 






3 


II 




3 






3 


7 




3 






3 




7 


3 




37 


3 




13 


83 


3 




II 


3 


19 




3 


7 


17 


3 




29 


3 


II 


7 


3 


47 




3 






3 




19 




87 


3 


II 


7 


3 






3 






3 






3 


19 




3 


7 




3 






3 




'7 


3 


87 


13 




3 




29 


3 




19 


3 


11 


17 


3 


7 


13 


3 


61 


53 


3 


41 


7 


3 


43 




3 


17 


89 


•• 


3 


17 




3 


19 


13 


'3 


'7 


23 


3 


29 




3 




7 


3 






3 




11 


3 




19 


89 


3 






3 


7 




3 


II 


•• 


3 


37 


7 


3 






3 


59 




3 


67 


13 


3 






3 


91 


7 




3 


17 




3 




7 


3 






3 




13 


3 


37 


19 


3 


31 


11 


3 


7 


29 


3 


47 


91 




3 




7 


3 


II 




3 






3 




17 


3 


13 




3 


7 




3 


.. 




3 


67 


3 


93 


3 






3 


17 




3 


13 


19 


3 






3 


7 




3 




II 


3 




7 


3 






3 


93 






3 


II 


41 


3 


31 


37 


3 


7 




3 




17 


3 




7 


3 


23 




3 


13 




3 




97 






3 




7 


3 


17 




3 






3 




11 


3 






3 


7 




3 


13 




3 


II 


97 


7 


3 






3 


19 


23 


3 


43 


13 


3 






3 


7 


17 


3 






3 




7 


3 


59 




99 


_3 




13 


_3 






3 


17 


29 


3 


Jl 


11 


3 






_3 




7 


3 






3 


II 




^ 


99 


11. 




^ 


13 




3 


7 




3 


. . 59 


^ 


29 


7 


3 




13 


_3 


53 


11 


_A 


37 




3 


99 



Nos. 5001 to 7499 


Nos. 7S0I to 9999 


l5olsi|52|53lS4lSSl56|S7l58|59|6o|6i|62|63|64|65 166167 168 I69I 70 J71I72I73I 74 


7Si76|77l78l79l&o|8l|82|83|84|85|86t87|88|89l90|9l|92|93l94l9Sl96|97l98|99 


01 


3 




7 


3 


II 




3 






3 


17 




3 




37 


3 


7 




3 


67 




3 


19 


7 


3 


01 


13 


II 


3 


29 




3 




59 


3 


31 




3 


7 


13 


3 




19 


3 


71 


7 


3 




89 


3 




03 




3 


II 




3 




13 


3 


'7 




3 


17 




3 


19 


7 


3 






3 


47 




3 


67 


11 


03 


3 






3 


7 


53 


3 


13 


19 


3 


11 


7 


3 




29 


3 






3 




13 


3 


31 




3 


07 


3 




41 


3 






3 


13 




3 




31 


3 


7 


43 


3 




19 


3 




7 


3 






3 


07 






3 


37 




3 


II 


29 


3 


7 


47 


3 






3 




7 


3 


41 


23 


3 


13 


17 


3 




09 


•• 


3 






3 


7 


71 


3 


37 


19 


3 


41 


7 


3 


13 


23 


3 




II 


3 


43 




3 




31 


09 


3 


7 


13 


3 


11 




3 


•• 


7 


3 


67 




3 


23 


59 


3 






3 


97 


37 


3 


7 


17 


3 


11 




19 


3 


47 


7 


3 


31 




3 


23 




3 






3 


17 


11 


3 


7 




3 


13 




3 




II 


7 


3 


II 


73 


3 






3 




13 


3 


79 


31 


3 


7 




3 


61 




3 




7 


3 




II 


13 


3 




13 


3 




37 


3 


29 




3 


'7 




3 


59 


11 


3 


17 


7 


3 


31 




3 




71 


3 


13 


II 


23 


3 


13 


41 


3 


7 


43 


3 


47 




3 




7 


3 




13 


3 


67 




3 




II 


3 


23 


17 


29 


7 


3 


13 




3 


41 




3 


61 


II 


3 






3 


7 


13 


3 


17 




3 


11 


7 


3 




17 




3 






3 






3 




19 


3 


7 


23 


3 


37 


71 


3 


13 


7 


3 


31 


59 


3 




47 


19 


3 




17 


3 


•• 


•• 


3 


7 


II 


3 


13 


29 


3 


71 


7 


3 






3 


II 




3 




13 


3 


19 


73 


19 


3 


7 




3 


23 




3 




7 


3 






3 


29 


11 


3 






3 




•• 


3 


7 


21 




3 


23 


17 


3 




7 


3 




31 


3 






3 






3 


II 


19 


3 


7 




3 


, , 


41 


21 


3 




7 


3 


89 


13 


3 




S3 


3 




37 


3 




II 


3 


7 




3 






3 




7 


3 


23 




47 


3 




II 


"3 




59 


3 




19 


3 


7 




3 


II 


37 


3 




7 


3 


17 


31 


3 


13 


23 




'3 






3 


71 




3 


7 




3 




11 


3 




7 


3 


23 




3 


89 




3 


II 




27 


II 


3 




'7 


3 




17 


3 






3 


11 


13 


3 




61 


3 


7 




3 






3 


17 


7 


27 


3 


29 




3 




23 


3 


19 


II 


3 






3 


7 


79 


3 






3 


11 


7 


3 


71 


31 


3 


29 


47 


23 


3 


73 


61 


3 


13 


17 


3 


7 




3 






3 




7 


3 




13 


3 






3 


17 


29 




3 


59 




3 


7 


II 


3 






3 


•• 


7 


3 






3 


11 


19 


3 


13 




3 






31 


3 


7 




3 






3 


II 


7 


3 


37 




3 


13 


59 


3 


19 


53 


3 


29 


79 


3 


7 




3 


31 


17 


13 


3 


41 


7 


3 


47 




3 




19 


3 






3 


II 


23 


3 


7 




3 




37 


3 




33 


7 


3 






3 


II 


43 


3 


19 


17 


3 




23 


3 


7 


47 


3 






3 


13 


7 


3 






33 


3 


17 


II 


3 




29 


3 




13 


3 


7 


89 


3 


II 




3 




7 


3 






3 






3 


37 


3 


11 




3 




7 


3 




13 


3 




17 


3 




41 


3 






3 


7 


31 


3 




11 


3 


37 




7 


3 


17 




3 


79 




3 


II 




3 






3 


7 




3 






3 


23 


7 


3 


19 


39 




3 


13 


19 


3 


29 




3 






3 


7 


17 


3 


47 


13 


3 


23 


7 


3 




II 


'3 


41 


43 


39 


3 




71 


3 


17 




3 


7 


3 


3 




S3 


3 




7 


3 


13 




"^ 






3 






3 


41 


71 


53 


3 


7 




3 






3 


13 


7 


3 


79 


17 


3 


31 


29 


3 




II 


3 


37 


13 


3 


7 


41 




3 






3 


II 


7 


3 


19 


23 


3 






3 






3 




. . 


3 


7 


31 


3 


13 




43 


3 


37 


7 


3 




23 


3 






3 






3 




17 


3 


7 


II 


3 


53 




3 




7 


3 


43 


19 




3 


11 


13 


3 


17 




3 






3 


7 


37 


3 




41 


3 




7 


3 






3 


61 


47 


7 




3 




13 


3 




7 


3 


19 




3 




II 


3 




17 


3 


41 




3 


7 




3 


II 


47 




3 


61 


7 


3 


13 




3 


17 




3 






3 


23 


83 


3 


7 


13 


3 




11 


3 


43 


7 


49 


3 


19 


29 


3 




31 


3 






3 


23 


11 


3 


7 




3 


61 


17 


3 




7 


3 


II 




3 


49 






3 


47 




3 


29 


73 


3 


7 


83 


3 


13 




3 




7 


3 




11 


3 






3 




51 




3 


59 




3 


7 




3 




11 


3 




7 


3 






3 


43 


13 


3 


11 




3 






SI 


3 


7 


23 


3 




83 


3 


37 


7 


3 


17 


41 


3 


53 




3 




11 


3 


13 




3 


7 




3 


53 


31 




3 


53 


7 


3 




II 


3 






3 


13 




3 






3 


7 


17 


3 


23 




3 


29 


53 


7 


3 






'3 




31 


3 




79 


3 


17 




3 


7 


11 


3 


19 


47 


3 


41 


7 


3 


59 


37 


57 


13 


3 


7 


11 


3 






3 




7 


3 


47 




3 


II 


79 


3 


29 




3 




17 


3 


7 




57 


3 


13 




3 


73 


7 


3 


23 


61 


3 


43 


11 


3 


17 


13 


3 






3 


7 


19 


3 


II 




3 


59 




7 


3 


23 


S3 


3 




13 


3 


59 


73 


3 


11 




3 


7 




3 


19 




3 




7 


3 




59 




3 




29 


3 




41 


3 


13 


11 


3 


7 


19 


3 


17 




3 


47 


7 


3 


II 


13 


3 




23 


61 


3 


13 




3 


43 


67 


3 


7 




3 


11 


61 


3 




7 


3 






3 




23 


3 


S3 


17 


3 


61 




47 


3 


7 


19 


3 




II 


3 




7 


3 


.. 




3 


13 




3 


II 




3 




43 


3 


7 


63 


6i 


3 


19 


31 


3 




7 


3 


11 


67 


3 






3 


23 




3 






3 


7 


13 


3 


37 


17 


63 


3 


79 


7 


3 




11 


3 






3 






3 






3 


'7 


59 


3 




73 


3 


13 


7 


3 


67 


3 




23 


3 


7 


19 


3 


73 




3 




7 


3 




29 


3 


59 


67 


3 




37 


3 


13 


53 


3 


67 


7 


II 


3 




31 


3 




7 


3 




13 


3 


II 




3 




89 


3 


17 




3 


7 




3 


■ • 


69 


37 


3 


11 


7 


3 






3 




47 


3 


31 




3 






3 


7 




3 




67 


3 




7 


69 


3 




17 


3 


13 




3 






3 


11 




3 


7 




3 


S3 


13 


3 


17 


7 


3 




71 


3 


'71 


11 




3 


41 




3 


S3 


29 


3 


7 


13 


3 




23 


3 




7 


3 






3 


71 


II 


3 


31 


71 


67 


3 


19 


17 


3 


7 




3 


II 


43 


3 


13 


7 


3 




47 


3 


73 




3 


17 


19 


3 




13 


73 


3 


7 




3 


13 




3 


23 


7 


3 






3 






3 




13 


3 


19 


II 


3 


7 


73 


3 


73 






3 




7 


3 


II 




3 


37 




3 


31 
67 


19 


3 


43 




3 


7 




3 


17 


29 


3 




77 




31 


3 


19 




3 


7 


53 


3 


43 


S9 


3 




7 


3 




11 


3 


13 




3 




19 


3 




77 




3 


7 




3 


41 


13 


3 




7 


3 




3 


47 


29 


3 






3 


61 




3 


7 


II 


79 


3 






3 




7 


3 






3 




37 


3 




II 


3 






3 


7 




3 


29 


47 


3 


79 


II 


7 


3 




79 


3 




17 


3 


61 


23 


3 




13 


3 


7 


67 


3 


83 




3 




7 


3 


17 


81 




3 






3 




13 


3 






3 


7 


II 


3 






3 




7 


3 


73 


43 


3 


II 




81 


3 




31 


3 


23 




3 


7 


17 


3 






3 


83 


7 


3 






3 


19 


II 


3 




41 


3 


83 


13 


71 


3 


'7 




3 






3 


31 


7 


3 


61 


13 


3 


29 


41 


3 






3 


II 




3 


7 


83 




3 


43 




3 


59 


7 


3 


83 


17 


3 


19 




3 


13 


31 


3 




11 


3 


7 


23 


3 




67 


87 




3 


17 




3 


37 


II 


3 


7 




3 


23 




3 


13 


7 


3 


II 


71 


3 


19 




3 


83 




87 


3 




13 


3 


7 




3 






3 


31 


7 


3 




II 


3 




37 


3 


53 




3 






3 


89 


7 




3 


17 


II 


3 




7 


3 


S3 




3 


19 




3 


II 




3 


83 


29 


3 


7 


37 


3 




89 




3 




7 


3 




19 


3 




13 


3 




II 


3 


89 


61 


3 


7 


41 


3 


43 




3 


11 


7 


91 


3 


29 


II 


3 


17 




3 




43 


3 




41 


3 


7 




3 






3 




7 


3 


23 


19 


3 


91 






3 


13 


61 


3 






3 


7 


II 


3 


59 


17 


3 




7 


3 






3 


II 




3 


97 


93 


II 


3 


67 




3 


7 




3 


71 


13 


3 


II 


7 


3 


43 


19 


3 




61 


3 


41 




■3 




59 


93 


3 


7 




3 






3 




7 


3 


13 




3 




17 


3 


29 




3 


II 


53 


3 


7 


13 


3 


97 


3 






3 


23 


29 


3 


II 




3 


7 




3 




73 


3 


37 


7 


3 




47 


3 




13 


3 


97 


71 


43 


3 


53 


II 


3 


7 




3 


29 




3 


19 


7 


3 


II 


17 


3 






3 




97 


3 


13 


99 




_3_ 


_7 




3 


11 


41 


3 


12 


7 


3 






3 


67 




3 


13 




_3 


31 


23 


_3 


_7 




99 


3 




II 


3 


12. 


_7 


3 


43 


37 


3 






3 


II 




3 




17 


^ 


_7 


29 


_3 


41 


19 


3 



SPUR GEARS 



111 



O 



Bi 


H 


< 


O 


1-1 


M 


& 




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to 


w 





rt 


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w 


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rt 


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r* 


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ro 


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fN 




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112 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



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" M 



BEVEL GEARS 



The dimensions and angles of bevel gears may be calculated from the 
formulas of Tables i and 2 which have been arranged by John 
Edgar {Amer. Mach., Apr. 13, 1905). The notation of the formulas 
is given in Figs, i, 2 and 3. 

Table i. — Formulas for Dimensions and Angles of Bevel 
Gears with Shafts at Right Angles 



Table 2. — Formulas for Dimensions and Angles of Bevel 
Gears with Shafts Not at Right Angles 



Diametral pitch . . . . 
Number of teeth . . . 

Pitch diameter 

Outside diameter . . . 

Diameter increase . 



Center angle .... 

Angle increment. 
Face angle 



Cutting angle 

Number of teeth for which to 
select cutter. 

Backing increment'. . . '. 



Pinion 



P 

Ni 
Di 
ODi 

di 




Gear 



N" 



p 




Ni 


DiXP 


Di 


N2^P 


OD2 


D2 + di 


di 


2 sin ij> 
P 


e 


go-<t> 


3 






e + d 




e-3 



1 Note that backing increment is the same as one-half the diameter increase 
of the mating gear. 

For Notation see Fig. i. 

The dimensions and angles of bevel gears having shafts at right 
angles may, in most cases, be obtained from Table 3 by Chas. Watts 
{The Engr., Aug. 13, 1909). 

A difference of practice prevails regarding the cut angle. By some 
the angle decrement is made larger than the angle increment while, by 
others, the angles are made equal — clearance being given by cutting 
the bottoms of the spaces parallel with the tops of the teeth of the mat- 
ing gear. 

When using revolving cutters the latter method is preferable 
while generated and planed teeth are cut by the former. The formu- 
las of Tables i and 2 are based on the latter practice and Tables 
3 and 4 on the former. To use Tables 3 and 4 with the latter method, 
it is only necessary to use the angle increment for the angle decrement, 
this angle subtracted from the center angle giving the cut angle. 

The proportion of the gears, that is, the ratio of the number of 
teeth in the large gear divided by the number of teeth in the pinion, 
being known, the center angles are read directly from columns B. 



\ Pinion 


Gear 




P 




P 




Number of teeth. . 


Ni 
Di 
ODi 


DiXP 

Ni^P 
Di+di 


N2 
D2 
OD2 


D2XP 


Pitch diameter ... ... 


N2^P 


Outside diameter 


D2 + d2 


Diameter increase 


di 


2 cos (j) 


d2 


2 COS B 


Center angle — shafts at less 
than 90 deg. 


•t- 


p 

, sin 

^+cos 


e 


p 

0-<i> 


Center angle — shafts at great- 


4> 


cos V 

tan ?> = Ti 


e 


0-4- 


er than 90 deg. 























V 


0-90° 
. 2 sin (b 
tan 3 r^ — - 






Angle increment 


3 






Face angle * 




<j,-3 

Ni 




e + s 








e-3 


Number of teeth for which to 


N' 


N" 


N2 


select cutter. 


cos <l> 


COS S 


Backing increment 


Bi 


sin (fi 
P 


Bi 


sm 8 




P 



For Notation see Figs. 2 and 3. 

To find the outside diameters, add the diameter increment to the 
pitch diameters. The diameter increment is found by dividing the 
quantity in column F, for the large or small gear respectively, by 
the diametral pitch. 

To find the face angles add the angle increment to the center 
angles. The angle increment is found by dividing the quantity in 
column E by the number of teeth in the large gear. 

To find the cutting angles subtract the angle decrement from the 
center angles. The angle decrement is found by dividing the quan- 

{Continued on page 116, first column) 





Fig. 2 Shafts at less than 90 Degrees 



Pinion 



Fig. I -Shafts at Right Angles Backim 




Fig. 3 Shafts at niore than 90 Degrees 



Figs, i to 3. — Notation of formulas for dimensions and angles of bevel gears. 

113 



114 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 3. — Dimensions and Angles of Bevel Gears 

For notation see page 107, 



With Shafts at Right Angles 



Proportion of 


F 


77 




n 




B 


wheels 


Large wheel 


Small wheel 


Hi 


u 


Large wheel 


Small wheel 


I to I 


1.414H 


-/ 


1-414- 


-/ 


80° -36'- 


- K 


93°-2o'- 


- K 


45° 


45° 


I to 1.020 


1 . 400 -^ 


-/ 


1.428H 


-/ 


8i°-i8'- 


- K 


94° -14'- 


- K 


45°-34' 


44° -26' 


I to 1 . 040 


1.386- 


-/ 


1.442- 


-/ 


82°- 0'- 


- K 


9S°- 9'- 


- K 


46°- 7' 


43°-53' 


I to 1.050 


I-379-: 


-/ 


1.448-^ 


- / 


83°-i6'- 


- K 


95°-35'- 


- K 


46°- 24' 


43°-36' 


I to I . ICO 


1-345^ 


-/ 


1-479- 


-/ 


85°- 4'- 


- K 


97°-39'- 


- K 


47°-43' 


42°-i7' 


I to I . I II 


I.338H 


-/ 


1.486- 


- / 


85°-27'- 


- K 


98°- 5'- 


- A' 


48° -00' 


42°— 00' 


I to 1. 125 


1.328H 


- / 


1.494- 


-/ 


85°-57'- 


- K 


98°-39'- 


- K 


48°-22' 


4i°-38' 


I to 1. 143 


1-317- 


-J 


1-505- 


-/ 


86°-3i'- 


- K 


99°-i9'- 


- K 


48° -48' 


4I°-I2' 


I to 1. 150 


1.312- 


-J 


1.509- 


-/ 


86°-4S'- 


- K 


99°-3S'- 


- K 


48°-59' 


41°- I' 


I to I . 166 


1.301- 


-J 


1-518- 


-/ 


87°-i8'- 


- K 


ioo°— 13'- 


' K 


49°- 24' 


40°-36' 


I to 1. 180 


1.293- 


-J 


1.526- 


-J 


87° -43'- 


- K 


ioo° — 41'- 


- K 


49° -43' 


40° -1 7' 


I to 1 . 200 


1.280- 


-J 


1-535- 


-J 


88°-2i'- 


- K 


ioi° — 24'- 


- K 


50° — 12' 


39°-48' 


I to 1 . 240 


1-255- 


-J 


I-S57- 


-J 


89°-32'- 


- K 


I02°— 46'- 


- K 


51°- 7' 


38°-53' 


I to 1.250 


1.249- 


-J 


1-561- 


- J 


89°-47'- 


' K 


103°- 4'- 


- K 


5I°-20' 


38° -40' 


I to 1 . 280 


1.231- 


-J 


1.576- 


-J 


9o°-37'- 


- K 


104°— l'- 


- K 


52°-oo' 


38°-oo' 


I to 1 . 285 


1.227- 


-J 


I-S79- 


-J 


90°-47'- 


- K 


104°— 12'- 


- K 


52°- 8' 


37°-52' 


I to 1 . 300 


1 . 219- 


- J 


1-585- 


-J 


91°- 9'- 


- K 


i04°-38'- 


- K 


52°-26' 


3 7° -34' 


I to I .320 


1.208- 


-/ 


1-594- 


-J 


9i°-39'- 


- K 


105° — 12'- 


- K 


52°-5i' 


37°- 9' 


I to 1.333 


1 . 200 - 


-/ 


I .600- 


-J 


9i°-59'- 


' K 


io5°-35'- 


- K 


53°- 7' 


36°-S3' 


I to 1.350 


1.190- 


- / 


1.607- 


-J 


92°- 24'- 


- K 


106° — 4'- 


- K 


53°-28' 


36°-32' 


I to 1 . 400 


1.162- 


-/ 


1.627- 


-J 


93°-4i'- 


- K 


io7°-25'- 


- K 


54° -28' 


3S°-32' 


I to 1.420 


1-151- 


- J 


1-635- 


-J 


94°- 2'- 


- K 


io7°-56'- 


- K 


54°-5i' 


35°- 9' 


I to 1.428 


1.147- 


-J 


1-638- 


-J 


94°-i2'- 


- K 


io8°— 7'- 


- K 


S5°-oo' 


35° -00' 


I to 1 . 440 


1.141- 


-J 


1 . 642 - 


-J 


94°-27'- 


- K 


io8°-25'- 


- K 


55°-i3' 


34° -47' 


I to I . 450 


1-135- 


-J 


1.646- 


-/ 


94°-39'- 


- K 


io8°-39'- 


-K 


55°-24' 


34°-36' 


I to I . 480 


I. 120- 


-J 


1-657- 


-/ 


95°-i6'- 


- K 


109° — 22'- 


- K 


■55°-57' 


34°- 3' 


I to I . 500 


I. 109- 


-/ 


1.664- 


-/ 


9S°-4i'- 


- K 


109° -50' - 


- K 


56°-i9' 


33°-4i' 


I to 1.520 


1.099- 


- / 


■ 1.670- 


-/ 


96°- 5'- 


■r- K 


iio° — 16'- 


- K 


56°-4o' 


33°— 20' 


I to 1.550 


1 . 084 - 


-/ 


1.680- 


-/ 


96°-37'- 


- K 


iio°-54'- 


- K 


57°-io' 


32°-So' 


I to I . 560 


1.079- 


-/ 


1.684- 


-/ 


96° — 48'- 


=- K 


iii°- 7'- 


- A' 


57°-2o' 


3 2° -40' 


I to 1 . 600 


1.060- 


-/ 


1.696- 


-/ 


97°-3i'- 


^ K 


iii°-56'- 


- K 


58° -co' 


32°-oo' 


I to 1 . 640 


1.041- 


-/ 


1.706- 


-J 


98° — 11'- 


- K 


II2° — 42'- 


- K 


S8°-38' 


3I°-22' 


I to 1.650 


1.036- 


-/ 


1.710- 


-J 


98°-2i'- 


- K 


ii2°-53'- 


- K 


58°-47' 


3i°-i3' 


I to 1 . 666 


1.029- 


-/ 


1-715- 


- J 


98°-36'- 


- K 


113°— 11'- 


- A 


S9°- 2' 


30° -58' 


I to I . 680 


1-023- 


- / 


1.718- 


-J 


98°-49'- 


- K 


ii3°-25'- 


- A 


59° -14' 


30° -46' 


I to 1 . 700 


1.014- 


-/ 


1.724- 


- J 


99°- 7'- 


- K 


ii3°-46'- 


- A 


59°-32' 


30°- 28' 


I to I. 720 


1 . 005 - 


-/ 


1-.730- 


-J 


99°-2S'- 


r-K 


114°- 7'- 


- A 


59°-5o' 


30° -10' 


I to I . 750 


.992- 


-/ 


1.736- 


-J 


99° -50'- 


^ K 


ii4°-36'- 


-A 


60°- 15' 


29°-4S' 


I to 1.760 


.988- 


-/ 


1-739- 


-J 


ioo°— 0'- 


- K 


ii4°-46'- 


- A 


60°— 24' 


29°— 36' 


I to i.8co 


.971- 


-/ 


1.748- 


-J 


ioo°— 31'- 


- K 


ii5°-23'- 


- A 


6o°-S7' 


29°- 3' 


I to I . 840 


• 955- 


- / 


1-757- 


-J 


ioi°— 3'- 


- K 


ii6°— 0'- 


- A 


6i° — 29' 


28°-3i' 


I to 1.850 


-951- 


-/ 


1-759- 


-J 


ioi° — 10'- 


- K 


ii6°- 8'- 


- A 


6i°-3/ 


28° -23' 


I to I . 880 


.940- 


-/ 


1-765- 


-/ 


ioi° — 30'- 


- K 


ii6°— 32'- 


- A 


6i°-59' 


28°- i' 


I to I . 900 


-932- 


-/ 


1.770- 


-/ 


ioi°-45'- 


- K 


ii6°-47'- 


- A 


62° -14' 


27° -46' 


I to 1.920 


■ 924- 


-/ 


1-774- 


-/ 


102°— 0'- 


- K 


117°- 3'- 


- A 


62° — 29' 


27°-3i' 


I to 1.950 


.912- 


-/ 


1.779- 


- J 


102° — 19'- 


- K 


ii7°-27'- 


- A 


62°-5i' 


27°- 9' 


I to 1 .960 


.908- 


-/ 


1.781- 


-J 


I02°-26'- 


- K 


ii7°-34'- 


- A 


62°-58' 


27°- 2' 


I to 2 . 000 


.894- 


-/ 


1.789- 


-J 


102° — 51'- 


- K 


ii8°- 3'- 


-A 


63°-26' 


26° -34' 


I to 2.040 


.880- 


-/ 


1.796- 


^ J 


io3°-i5'- 


- K 


ii8°-3i'- 


- A 


63°-S3' 


26°- 7' 


I to 2.080 


.866- 


-/ 


1.802- 


- J 


io3°-39'- 


- K 


ii8°-s8'- 


-A 


64° — 20' 


2 5° -40' 


I to 2. 100 


■859- 


-/ 


1.805- 


-J 


103°— 49'- 


- K 


119° — 10'- 


- A 


64° -3 2' 


25°-28' 


I to 2. 120 


-853- 


-/ 


1 . 809 - 


'J 


104°— 0'- 


- K 


119° — 23'- 


- A 


64° -45' 


25°-iS' 


I to 2 . 1 60 


.840- 


-J 


I-81S- 


-J 


104° — 21'- 


- K 


119°— 46'- 


-A 


65°- 9' 


24°-Si' 


I to 2 . 200 


-830- 


-J 


1.821- 


-J 


104°— 40'- 


- K 


120°— q'- 


- A 


65°-33' 


24°-27' 


I to 2 . 24c 


-815- 


-J 


1.826- 


-J 


105°— 0'- 


- K 


I20°-32'- 


-A 


6s°-57' 


24°- 3' 


I to 2.250 


.812- 


-J 


1.827- 


-J 


105°- 5'- 


- K 


i2o°-37'- 


- A 


66°— 2' 


23°-58' 


I to 2 . 280 


■ 803- 


^J 


1.831- 


-J 


io5°-i9'- 


- K 


i20°-53'- 


- A 


66° — 19' 


23°-4i' 


I to 2 . 300 


■797- 


-J 


1.834- 


-J 


io5°-27'- 


- K 


I21°- 2'- 


-A 


66° -30' 


23° -30' 


I to 2.333 


.788- 


-J 


1.838- 


-J 


io5°-42'- 


- K 


I2I° — 19'- 


- A 


66° -48' 


23° -1 2' 


I to 2.360 


-780- 


■^ J 


1.841- 


■r-J 


io5°-52'- 


- K 


I2I°-3l'- 


- A 


67°- 2' 


22°-58' 



BEVEL GEARS 



115 



Table 3- 


—Dimensions and Angles of 


Bevel Gears With Shafts at Right Angles — {Continued) 


Proportion of 


F 


E 


D 


B 


wheels Large wheel 


Small wheel | 


Large wheel 


Small wheel 


I to 2.400 


769- 


-/ 


1 . 846 -i- 


-/ 


io6°— g' -¥ K 


I2I°-50'4 


- K 


67°-23' 


22°-37' 


I to 2 . 440 


758-^ 


-/ 


1.850-r 


-/ 


io6° — 24'-;- K 


I22°- 8'-f 


- K 


67°-43' 


22°-I7' 


I to 2.480 


747-^ 


- / 


1.855- 


-/ 


io6°-39'-- K 


I22°— 26'-f 


- K 


68°- 3' 


2i°-S7' 


I to 2 . 500 


743-^ 


-/ 


1.857- 


-/ 


io6°-46'-- K 


I22°-34'-^ 


- K 


68°-i2' 


2i°-48' 


1 to 2.520 


738- 


-/ 


1.859- 


-/ 


ic6°- S3'- K 


I22° — 4o'-f 


- K 


68°-2i' 


2i°-39' 


I to 2 . 560 


727-7 


- / 


1.863- 


-/ 


107°- 7'^ K 


I22°-57'4 


- K 


68° -40' 


2I°-2l' 


I to 2.600 


718- 


-/ 


1.866-7 


-/ 


io7°-i&'^ K 


I23°-Il'- 


- K 


68°-57' 


2I°- 3' 


I to 2.640 


7084 


- / 


1.870-7 


-/ 


107°— 32'-^ K 


I23°-25'- 


- K 


. 69°-i5' 


20° -45' 


I to 2 .666 


702-7 


-J 


1.872-T 


- / 


ic7°-39'^K 


i23°-34'- 


-K 


69° — 26' 


20° -34' 


I to 2 . 700 


694-7 


-J 


1.875- 


-/ 


ic7°-5o'-f- K 


i23°-47'-i 


-K 


69°-4i' 


20° — 19' 


I to 2.720 


690 -: 


- J 


1.877- 


-/ 


■ io7°-S7'- K 


i23°-53'- 


- K 


69°-49' 


20° — 11' 


I to 2 . 760 


681^ 


-J 


1.880-; 


- / 


io8°- 7'-- K 


124°— 6'- 


- K 


70°- 5' 


I9°-SS' 


I to 2 . 800 


672-: 


- J 


1.883- 


-/ 


io8°-i7'--i<: 


i24°-i8'- 


- K 


70° — 21' 


i9°-39' 


I to 2 . 840 . 


664^ 


-J 


1.886- 


- / 


ic8°-28'-- K 


1 24° -30' -7 


- K 


7o°-36' 


19° -24' 


I to 2 . 880 


656-7 


-J 


1.889-^ 


-/ 


. io8°-37'-^ K 


i24°-4i'- 


- K 


7o°-5i' 


19°- 9' 


I to 2.900 


651- 


-J 


1.891-= 


- / 


io8°-43'-- K 


i24°-53'- 


- K 


70° -59' 


19°- I' 


I to 3 . 000 


632^ 


-J 


1.897- 


-/ 


109°- 6'-^ K 


i25°-i4'- 


- K 


7i°-34' 


i8°-26' 


I to 3 . 100 


614- 


-J 


1.903- 


- / 


iog°-26'^K 


i2S°-37'- 


- K 


72°- 7' 


i7°-53' 


I to 3 . 200 


596- 


'J 


1.909- 


-/ 


■ i09°-45'- K 


126°- 0'- 


- K 


72°-39' 


I7°-2l' 


I to 3.333 


575- 


-J 


1-915- 


- / 


iio°- 8'^ K 


I26°-25'- 


- K 


73°- 18' 


i6°-42' 


I to 3 . 400 


564- 


-J 


1.919- 


-/ 


II0°-20'-7- K 


i26°-38'- 


- K 


73°-37' 


i6°-23' 


I to 3.450 


557- 


-J 


1.921- 


- / 


110°-2(>'-i- K 


i26°-46'- 


- K 


73°-5c' 


1 6° -10' 


I to 3 . SCO 


549- 


-J 


1.923- 


-/ 


iio° — 34'-H K 


i26°-S5'- 


- K 


74°- 3' 


iS°-57' 


I to 3.550 


542- 


-J 


1.925- 


-/ 


iio°— 41'-^ K 


127°- 3'- 


- K 


74° -16' 


1 5° -44' 


I to 3 . 600 


535- 


-J 


1.927- 


-/ 


iio°-48'-h K 


I27°-Il'- 


- K 


74°- 29' 


i5°-3i' 


I to 3.631 


531- 


-J 


1.928- 


- / 


iio°-S2'-¥ K 


i27°-i5'- 


- K 


74° -36' 


i5°-24' • 


I to 3 . 684 


524- 


-J 


1.930- 


-/ 


IIo°-S9'-^ K 


127° — 23'- 


- K 


74° -49' 


i5°-ii' 


I to 3.736 


S17- 


-J 


1.932- 


-/ 


iii°- s'^K 


i27°-3o'- 


-K 


7S°- I' 


14° -59' 


■ I to 3.777 


512- 


-J 


i^934- 


-/ 


iii°-io'-- K 


127°— 36'- 


- K 


75°- 10' 


14° -50' 


I to 3 . 789 


Sio- 


- J 


1-934- 


- / 


iii°-ii'-¥ K 


i27°-37'- 


- K 


75°-i3' 


i4°-47' 


1 103.833 


505- 


-J 


I-93S- 


-J 


iii°-i6'-- K 


i27°-43'- 


- K 


7S°-23' 


i4°-37' 


I to 3.888 


496- 


-J 


1-937- 


- J 


III°-22'-- K 


i27°-48'- 


- K 


75°-35' 


i4°-25' 


I to 3.944 


492- 


-J 


1.938- 


-J 


iii° — 27'-^- K 


i27°-53'- 


- K 


75°-46' 


14°- 14' 


I to 4 . 000 


485- 


-J 


1.940- 


-/ 


iii°-33'^K 


128°- 3'- 


- K 


75° -S8' 


14°- 2' 


I to 4. Ill 


472- 


-J 


. 1-943- 


-/ 


iii° — 4S'-^ K 


i28°-i6'- 


- K 


76°-2i' 


i3°-39' 


I to 4.176 


466- 


-J 


I-94S- 


-/ 


iii°-5o'-- K 


I28°-22'- 


. K 


76°-32' 


i3°-28' 


I to 4.235 


459- 


-J 


1.946- 


-/ 


iii° — 54'-- K 


128°— 28'- 


h K 


76°-43' 


i3°-i7' 


I to 4312 


452- 


^ J 


1.948- 


7-/ 


II2°- I'^K 


i28°-35'- 


h K 


76°-S7' 


13°- 3' 


I to 4.375 


446- 


■rJ 


1.949- 


H/ 


II2°- 6'H-X 


i28°-4o'- 


¥ K 


77°- 7' 


i2°-53' 


I to 4.428 


.440- 


i-J 


I-9SI- 


7-/ 


II2°-I0'-- K 


i28°-45'- 


hK 


77°-i7' 


i2°-43' 


I to 4. 500 


• 434- 


h/ 


1.952- 


¥ J 


ii2°-is'-Hii: 


i28°-5i'- 


r- K 


77°-28' 


I2°-32' 


I to 4.571 


.427- 


hJ 


1-954- 


¥ J 


ii2° — 2o'-^ K 


i28°-57'- 


r- K 


77°-4o' 


I 2° -20' 


I to 4.666 


.419- 


h J 


1-955- 


r- J 


ii2°-26'-f: K 


129°- 3'- 


- K 


77° -54' 


I2°- 6' 


I to 4 . 800 


.408- 


f-/ 


1.958- 


¥ J 


ii2°-35'-^K 


i29°-i3'- 


r- K 


78° -14' 


ii°-46' 


I to 5.000 


■392- 


f- J 


1 .961 - 


- / 


ii2°-4s'-Z 


129° — 26'- 


¥K 


78°-4i' 


II° — 19' 


I to 5.142 


.382- 


^J 


1.963- 


¥ J 


ii2°-53'--ii: 


i29°-34'- 


¥■ K 


79°- 0' 


II°— 0' 


I to 5-230 


•375- 


^ J 


1.964- 


¥ J 


ii2°-57'H-ii: 


i29°-39'- 


¥ K 


79°-ii' 


10° -49' 


I to 5. 385 


.365- 


^ J 


I .966- 


- / 


113°- 3'^K 


129° — 46'- 


T- K 


79°- 29' 


io°-3i' 


I to 5.461 


.36c- 


¥J 


1.967- 


-/ 


113°- 6'^K 


i29°-49'- 


¥ K 


79°-37' 


IO°-23' 


I to 5.538 


•355- 


¥ J 


1.968- 


- / 


ii3°-io' H-i? 


i29°-53'- 


7- K 


7 9° -46' 


io° — 14' 


I to 5.666 


• 348- 


i-J 


1.969- 


7-/ 


ii3°-i6'-7- K 


i29°-59'- 


¥ K 


79°-59' 


io°- l' 


I to 5.750 


■342- 


¥ J 


1.970- 


7- / 


ii3°-i8'--i!C 


129°— 2'- 


7- K 


80°- 8' 


9°-S2' 


I to 5 . 833 


.338- 


¥ J 


1.971 


7-/ 


ii3°-2o'-;- K 


13°°- 5'- 


¥ K 


80°- 16' 


9° -44' 


I to 5.916 


•333 


7- / 


1.972 


T- / 


ii3°-23'^K 


130°- 9'- 


¥ K 


80° -2 5' 


9°-35' 


I to 6 . 000 


.328 


¥ J 


1.972 


-/ 


ii3°-26'-f- K 


I30°-I2' 


¥ K 


8o°-33' 


9°-27' 



I 



116 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



tity in column D by the number of teeth in the large gear. The 
cutting angle thus obtained gives the standard Brown and Sharpe 
depth for involute teeth. 

The use of the table is best shown by an example: 
Gears of 60 and 30 teeth; 8 diametral pitch 
Proportion 60 to 30 = 2 to i 

. 60 .894 60 .894 

Outside dia. of large gear =pitch dia.+F=-^-l — l~~~o' — o~ 

= 7i+-ii2 = 7.6i2 ins. 
= 63° 26' 

102° 51' 



Center angle of large gear 
Face angle of large gear 



= 63° 26' + £=63° 26'+- 



K 



= 63° 26'+ 



102 51 



60 



Cutting angle of large gear 



63° 26' + !° 43' 
118° 3' 



= 65° 9' 
= 63°26'-D=63°26'- 



= 63°26'-i^|^=63°26'-i°59' 



K 



= 61° 27' 

The dimensions and angles of miter gears, of diametral pitch may, 
in most cases, be taken from Table 4 by Wm. G. Thumm {A mer. Mach., 
June 13, 1907). 

The profiles of the teeth of bevel gears are laid out on the developed 
backing cones as indicated in Fig. 5. The number of teeth contained 
in the circumference of the developed cone is to be calculated by 
dividing the actual number of teeth in the gear by the cosine of a, 
or by multiplying twice 40 by the diametral pitch. The number 
of teeth thus found is to be used when consulting Grant's odonto- 
graph, which see, for the various radii, the profile being drawn as 
for this number of teeth and as for a spur gear of pitch radius OA . 

Strength of Bevel Gears by Calculation 

The working loads on bevel gears may be determined from the 
formula proposed by Wilfred Lewis {Proc. Eng. Club of Philadel- 
phia, 1892) as follows: 

d 
W= SPfy- 

in which TF = pressure on teeth, lbs., 

5 = fiber stress, lbs. per sq. in., this stress being dependent 
on the speed in accordance with Table 10 given in con- 
nection with the Lewis formula for spur gears, which 
see, 

P = circular pitch, ins., 

/=face, ins., 

y = a factor for diSerent numbers and forms of teeth in 
accordance with Table 12 given in connection with 
the Lewis formula for spur gears, which see. In se- 
lecting this factor for bevel gears the actual number 
of teeth is to be multiplied by the secant of one-half 
the pitch cone apex angle, the result being the equiva- 
lent number of teeth for which y is to be selected, 

(i = inside pitch diameter, ins.. 

Z) = outside pitch diameter, ins.. 
The formula presupposes that d is not less than f £), which it should 
never be. 

Strength of Bevel Gears by Graphics 

The working loads on bevel gears may be determined by the follow- 
ing method, by Robert A. Bruce {Amer. Mack., May 31, 1900) 
which is based on and gives the same results as the Lewis formula: 

First, find the face of a spur gear equivalent to the actual face on 
the bevel gear by the use of Fig. 6. Instructions for use are given 
below the chart. 

Second, find the number of teeth of a spur gear equivalent to the 
actual number of the bevel gear by the use of Fig. 7, for which 
instructions for use wiU be found below it. 




r^_ 



Fig. 4. — Needed shop dimensions of bevel gears. 




Fig. 5. — Profiles of bevel gear teeth. 

Third, using the equivalent face width and number of teeth, 
follow the instructions for Mr. Bruce's chart for the strength of spur- 
gear teeth. Fig. 7 of the section on Spur Gears. 

If desired, the Lewis factor y may be found by tracing downward 
to the second curve of Fig. 7 of this section and thence horizontally 
to the value of the factor at the right, as shown. 

Selecting Bevel Gears from Stock Lists 

Commercial or listed bevel gears for shafts at right angles may 
frequently be used for shafts at other angles, especially if the re- 



BEVEL GEARS 



117 




G (Pitch Dla.) 
H (OatsideDla.1 



A = Cutting angle =B—D 

B = Center angle 

C = Face angle =B+E 

D = Angle decrement 

E = Angle increment 

F = Diameter increment 

G = Pitch diameter 

H = Outside diameter =G-\-F 

J = Diametral pitch 

K = Number of teeth in large wheel 

L =From pitch line to outside angle =§ diameter increment of 

mating wheel 
G+F=H 

Angle 5+angle E = angle C 
Angle 5— angle Z) = angle A 
The values of F, E, D, B are shown in table 
Notation of Table 3. 













Table 4 


. — Dimensions and Angles 


OF MiTEE Gears 












No. of 


Pitch diameters | 


Outside diameters | 


Face 
angle 


Cut 


teeth 


2 P. 


3 P. 


4 p. 


SP- 


6 p. 


7 P. 


8 P. 


10 P.! 


2 P. 


3 P. 


4 P. 


5 P. 


6 P. 


7 P. 


8 P. 


10 P.| 


angle 


12 


6 


4 


3 


2i 


2 


IT 


1} 


I A 


6.71 


4.48 


3.35 


2.68 


2.24 


1.92 


1.68 


1-34 


51° 43' 


37° 14' 


13 


6i 


A\ 


3l 


2! 


2i 


I? 


If 


lA 


7.21 


4.80 


3.60 


2.88 


2.40 


2.06 


1.81 


1-44 


51° 12' 


37° 49' 


14 


7 


Al 


3i 


25 


2I 


2 


If 


lA 


7.71 


S.14 


3.8S 


3.08 


2.57 


2.20 


1.93 


1.54 


so" 47' 


38° 20' 


IS 


7i 


S 


3i 


3 


28 


2j 


i5 


I A 


8.21 


S.46 


4.10 


3.28 


2.73 


2.35 


2.06 


1.64 


50° 23' 


38° 46' 


16 


8 


S\ 


4 


3i 


2S 


2j 


2 


1 10 


8.71 


5. 80 


4-35 


3.48 


2.90 


2.49 


2.18 


1.74 


50° 03' 


39° 09' 


17 


8i 


5! 


4i 


3I 


2S 


2t 


2j 


ii^i, 


9.21 


6.14 


4.60 


3.68 


3-07 


2.63 


2.31 


1 .84 


49° 45' 


39° 30' 


18 


9 


6 


4l 


3l 


3 


2^ 


2i 


11% 


9.71 


6.48 


4-8S 


3.88 


3.24 


2.77 


2.43 


1-94 


49° 29' 


39° 48' 


19 


9i 


6J 


4l 


35 


3J 


2? 


2i 


1."^ 


10.21 


6.80 


5. 10 


4.08 


3.40 


2.92 


2.56 


2.04 


49° is' 


40° 04' 


20 


10 


6! 


S 


4 


3t 


2? 


2i 


2 


10.71 


7.14 


S-35 


4.28 


3.57 


3.06 


2.68 


2.14 


49° 03' 


40° 19' 


21 


lOi 


7 


si 


45 


38 


3 


2f 


21^ 


II. 21 


7.46 


5-60 


4.48 


3-73 


3.20 


2.81 


2.24 


48° 51' 


40° 32' 


22 


II 


7i 


Si 


4i 


3^ 


3\ 


2f 


2-,% 


II. 71 


7.80 


S-8S 


4.68 


3.90 


3.3s 


2.93 


2.34 


48° 41' 


40° 45' 


23 


III 


7l 


5l 


4§ 


3I 


3? 


2i 


2X% 


12.21 


8.14 


6.10 


4.88 


4.07 


3-49 


3 -06 


2.44 


48° 31' 


40° S6' 


24 


12 


8 


6 


41 


4 


3r 


3 


21*0 


12.71 


8.48 


6.3s 


5. 08 


4.24 


3.63 


3-18 


2.54 


48° 22' 


41° 06' 


25 


I2J 


8J 


6} 


s 


4i 


3^ 


3J 


2,"n 


13-21 


8.80 


6.60 


5. 28 


4.40 


3-77 


3.31 


2.64 


48° 14' 


41° IS' 


26 


13 


8! 


6J 


s5 


4§ 


37 


3i 


2 A 


13.71 


9.14 


6.8s 


5. 48 


4.57 


3-92 


3-43 


2.74 


48° 07' 


41° 24' 


27 


I3J 


9 


6| 


si 


4I 


3? 


3l 


2^\ 


14.21 


9.46 


7.10 


5-68 


4.73 


4.06 


3-56 


2.84 


48° 00' 


41° 32' 


28 


14 


9l 


7 


si 


4^ 


4 


3i 


2A 


14.71 


9.80 


7.3s 


5-88 


4.90 


4.20 


3-68 


2.94 


47° 54' 


41° 39' 


29 


I4i 


9i 


7i 


si 


4i 


4? 


3f 


2l% 


IS.2I 


10.14 


7.60 


6.08 


5 -07 


4.35 


3-81 


3-04 


47° 47' 


41° 46' 


30 


IS 


10 


7l 


6 


s 


4? 


3! 


3 


15-71 


10.48 


7.8s 


6.28 


S-24 


4-49 


3-93 


3-14 


47° 42' 


41° 53' 


31 


15J 


lOj 


7i 


6J 


si 


4? 


3l 


3 15 


16.21 


10.80 


8.10 


6.48 


S-40 


4.63 


4.06 


3-24 


47° 37' 


41° 59' 


32 


16 


io§ 


8 


61 


si 


4t 


4 


3A 


16.71 


11.14 


8.35 


6.68 


S.S7 


4-77 


4.18 


3-34 


47° 32' 


42° 04' 


33 


ibl 


II 


8i 


6| 


si 


4? 


4i 


315 


17.21 


11. 46 


8.60 


6.88 


S.73 


4-92 


4.31 


3-44 


47° 27' 


42° 10' 


34 


17 


Ilj 


8§ 


61 


St 


4l 


4i 


3A 


17.71 


11.80 


8.8s 


7.08 


S.90 


5.06 


4-43 


3-54 


47° 23' 


42° is' 


35 


I7i 


III 


81 


7 


s§ 


S 


4l 


3A 


18.21 


12.14 


9. 10 


• 7.28 


6.07 


5-20 


456 


3-64 


47° 19' 


42° 19' 


36 


18 


12 


9 


7b 


6 


S^ 


4§ 


3 A 


18.71 


12.48 


9.35 


7.48 


6.24 


5.3s 


4.68 


3.74 


47° 15' 


42° 24' 


37 


18J 


I2i 


9i 


7i 


6i 


s? 


4f 


315 


19.21 


12.80 


9.60 


7.68 


6.40 


5. 49 


4.81 


3-84 


47° 11' 


42° 28' 


38 


19 


12| 


9J 


7i 


6§ 


s? 


4i 


3l"5 


19-71 


13-14 


9.8s 


7.88 


6.57 


5-63 


4-93 


3-94 


47° 08' 


42° 32' 


40 


20 


13! 


10 


8 


6S 


s^ 


S 


4 


20.71 


13-80 


10.35 


8.28 


6.90 


5 -92 


5-18 


4.14 


47° 01' 


42° ■Sg' 


42 


21 


14 


lOj 


81 


7 


6 


Si 


4to 


21.71 


14-48 


10.85 


8.68 


7.24 


6. 20 


5-43 


4-34 


46° 56' 


42° 46' 


44 


22 


14! 


II 


81 


7§ 


6? 


SJ 


4i*o 


22.71 


15-14 


11.35 


9.08 


7.57 


6.49 


5-68 


4-S4 


46° so' 


42° 52' 


46 


23 


isl 


Hi 


9J 


7S 


6^ 


si 


4l5 


23.71 


15.80 


11.85 


9.48 


7.90 


6.77 


5-93 


4-74 


46° 46' 


42° 58' 


48 


24 


16 


12 


9i 


8 


6? 


6 


415 


24.71 


16.48 


12.35 


9.88 


8.24 


7.06 


6.18 


4-94 


46° 42' 


43° 06' 


SO 


25 


i63 


I2i 


10 


8.3 


7^ 


6J 


s 


25-71 


17-14 


12.85 


10.28 


8.57 


7.35 


6.43 


S-14 


46° 37' 


43° 12' 


S4 


27 


18 


I3J 


loj 


9 


7? 


61 


SA 


27.71 


18.48 


13.85 


11.08 


9-24 


7-92 


6.93 


S-54 


46° 31' 


43° 18' 


58 


29 


195 


I4J 


II5 


9% 


8? 


1\ 


sA 


29.71 


19.80 


14-85 


11.88 


9.90 


8.49 


7-43 


5-94 


46° 24' 


43° 24' 


60 


30 


20 


IS 


12 


10 


8^ 


1\ 


6 


30.71 


20.48 


IS.3S 


12.28 


10.24 


8.77 


7-68 


6. 14 


46° 21' 


43° 30' 



quired gears have a ratio of unity, and the bevel-gear ratio may 
frequently be made unity when the speed ratio is not by adding a 
pair of spurs to the train. The following explanation of this fact 
and of the method of selecting the gears from gear-maker's lists is 
due to W. C. Conant {Amer. Mach., June 13, 1901). 

In Fig. 8, AB and CD are right-angle pairs having the same cone 
apex, pitch and tooth system. Inspection wiU show that B and C 
will mesh together properly. Given the shaft angle NOM and the 
ratio of B and C, the problem is to find the right-angle pairs AB and 
CD from which to select B and C. Going further, it is equally evi- 



dent from Fig. 9 that gear A may piesh with an indefinite number of 

gears BCDE, provided that the pitch cones intersect at the common 

point 0, and the gears BCDE may all be members of right-angle 

pairs, each combination AB, AC, AD, AE giving different angles of 

A A A A 
shafts and different speed ratios -s> -7^' -7;' ■?;• 

BCDE 

The conditions given in practice are, two shafts intersecting at 
any angle to run at any speed ratio, to find from standard lists bevel 
gears that will meet the requirements. 

Referring to Fig. 8, let OM and ON be the center lines of two 



118 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



shafts and let C and B be the pitch hnes of any two bevel gears 
that win give the required speed ratio. Draw OP perpendicular 
to OM and dotted line from R perpendicular to OP; this latter line 
to represent the pitch line of a gear mating with B to form a right- 
angle pair. By a similar construction draw D to form a right-angle 
pair with C. It is evident from the figure that any diameter gear 

B having a ratio with its mate ^ of -j and any diameter gear C 

C 

having a ratio with its mate Z? of -^ will run correctly together pro- 

vided ratio -p; is a constant, the most favorable case being that when 
p::;=i, gears B and C being identical. To solve the problem, it is 



and 



therefore 



Let 



when 



0R = 



sin ROP 
B 



cos MOR 
A 



sin MOR cos MOR 
A _ cos MOR 
B~sm MOR 
A 



cos MOR 



sin MOR 

In the same manner it can be shown that 
D_ ,_ cos NOR 
C~^ "sin NOR 



(c) 



id) 



Example. — ^Required a pair of gears to connect two shafts at an 



Breadth of Face of Equivalent Spiir Wheel, Ins. 
876 54321 


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^ ^- ^ ^ V ^-v ^v^^^\S^5^^5\v^»-^-C; -t- I 


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.10 



.50 



.15 .20 .25 .30 .35 .40 .45 

Values of r= Breadthof Face of Bevel Wheel Divided by Length- of Slant Side of Pitch Cone 
Divide the width of face by the length of slant height of pitch cone, the result being r; find the value of r on the base line, trace vertically 
to the curve, horizontally to the diagonal for the actual face width and vertically to the top where read the equivalent face width. The ex- 
ample is for ?•= . 25 and actual face width =5 ins., the equivalent face width being 3.85 ins. 

Fig. 6. — The face of bevel gears reduced to the equivalent face of spur gears. 



7? /"i 

only necessary to find the ratios -j and y; and select the proper 

member from each pair. We have, 

Angle JkrOiV = angle 71/02? -I- angle .ZVO-R (o) 

B , „„ C 



Since 

therefore 

or, 

Let 

when 



0R = 



sin MOR 

C 

sin MOR~ sin NOR 



and OR-- 



sin NOR 



B 



sin NOR = ^sin MOR 



B 



=R 



sin N OR = R sin MOR 



(b) 



From a table of natural sines select such values for the angles 
MOR and NOR that their sum is MON and that sin NOR is R times 
sin MOR, thus satisfying equations (a) and (b). If the gears are 
to be equal, R becomes i and the angles MOR and NOR are equal. 



We have next to find the values of -5 and of 



D. 
C 



Since 



0R = 



B 



sin MOR 



angle of 60 deg., the speed ratio ^ =-R = i, giving at once JlfOi? = 

iVOi? = 30 deg. Substituting the sine and cosine of 30 deg. in (c) 
or (<f) we have: 



.866 
".500 



= 1-73 



and we have only to find a pair of right-angle gears of the required 
strength having this ratio; for example, gears having 24 and 42 teeth 
give a ratio of 1.75, which is sufficiently accurate for the class of work 
for which cast gears are used. Of these we may obviously use either 
the pinion or the gear. 

C 2 

If the required gears are to have a different ratio, or say ■^=R = -' 

we find by inspection of a table of natural sines angles to satisfy 
equations (a) and (6). Thus we find that 

Sin 4i°( = .6s6) = 2 sin I9° = (2X.326) nearly, and that 41° -|- 19° 
= 60°, which is to say that angle N0R = 4i° and angle MOR = ig°. 

Substituting the values of the sines and cosines in (c) and {d) we get 

•9455 



and 



•3255 
•7547 
.6560 



= 2.9 



= 1-15 



BEVEL GEARS 



119 



From a catalogue list we now select the pinion of a pair of right-angle 
gears having a ratio of 2.9 and we will suppose that a pair found 
having 1 7 and 50 teeth answer the requirements of strength. 

Since the speed ratio of the shafts is to be 2, the gear to run with 
the 1 7-tooth pinion must have 17X2=34 teeth and it must come from 



10 



Eauivalent Number of Spur Teeth 
20 30 40 50 60 70 



i l | l I I I I I I I , 
■Nimber^f^th 



90 100 



Lci' 



n-Bevel-Wfaeel 



J -dULLi T]I71.LL 




g: 



10° 15 20 2-5" 30 35° 40" 
Half Angle of Fitch Cone in Degrees 



45° 



50° 



Find the half angle of the pitch cone on the base line ; trace 
vertically to the curve of secants, horizontally to the diagonal 
for the actual number of teeth, and vertically to the equivalent 
number of spur teeth at the top. The example is for half cone 
angle = 3i| deg. and actual number of teeth = 34, the equivalent 
number of teeth being 40. 

Fig. 7. — The number of teeth of bevel gears reduced to the equiva- 
lent number of teeth of spur gears. 

a pair in which it mates with a gear having 34X1.15=39 teeth. 
Therefore the 1 7-tooth pinion selected from the first pair and the 
34-tooth pinion from the second are the gears required. 

In order to determine whether the pinion or gear of a given pair 
is to be used, observe the following rule: 

When the angle MOR is less than 45 deg., use the pinion of the pair 
B-A, and conversely when the angle MOR is greater than 45 deg., 
use the gear of the pair B-A. As a corollary it follows that the 
same rule applies to the pair D-C, using the angle NOR as the 
critical angle. 

As Fig. 8 is drawn with angle MON less than 90 deg.. Fig. 10 is 
drawn to show angle MOiV greater than a right angle. It will be ob- 
served that the obtuse angle uses larger gears than the acute angle, 
of which advantage may be taken in cases where strength is needed 
and consequently large gears required. 

While it will generally be possible to find a pair containing one of 
the required gears, it will usually be found more difficult to find a 
pair from which the mating gear can be selected. That is to say, 
having found the 1 7-tooth pinion, it will be difficult to find a 34- 
tooth pinion belonging to a pair having a ratio of r.15 with the same 
pitch and face. There is also the danger that the teeth of the two 
pairs from which the mating gears are selected may have been de- 



signed for difierent systems and that therefore the mating gears will 
not run well together. For these reasons it is better to use two gears 
of the same size and make the required speed change at some other 
point. At the worst, if only a single stock gear is used, that much 
pattern work will be saved. 




Fig. 10. 
Figs. 8 to 10. — Stock bevel gears for shafts at any angle. 

Cutting Bevel Gears with Rotary Cutters 

High class bevel gears cannot he made with rotary cutters. The diame- 
ter of a gear and the pitch of its teeth diminish as the cone apex is 
approached and the tooth profiles should change to correspond but 
this condition cannot be met with revolving cutters. Nevertheless, 
for many purposes, gears made with such cutters are entirely satis- 
factory and they will, no doubt, continue in use for the indefinite 
future. 

The Offset Method 

When the usual, or oSset, method is employed, a cutter having 
the correct profile for the outer ends of the teeth is selected. The 
thickness of this cutter must, however, be such as to pass through 
the spaces at the inner ends, and since cutters for spur teeth would 
cut spaces due to the pitch at the outer ends, such cutters cannot be 
used for bevel teeth. Since the sides of the spaces between the teeth 
radiate from the cone apex, they must be cut separately, the blank 
and cutter being adjusted for each cut. 

The general nature of these adjustments may best be studied by 
considering a case having exaggerated proportions and, for this 
purpose, cutters with straight sides may be substituted for those 
having involute profiles. Taking first a cutter with parallel sides, 
Fig. II, it is clear that by offsetting the cutter to the left one-half its 
thickness, the right side of the cut will be radial with the pitch cone 
apex of the blank. If the cutter be then adjusted to the right by 
its thickness. Fig. 12, the blank being turned to the right by the width 
of the space, the left side of the cut will be radial. 

With a tapering cutter. Figs. 13, 14, 15 and 16, the action is quite 
different. As the cut proceeds, its depth decreases. The pitch 
line ah, Fig. 13, must lead to the cone apex o. At the beginning of 
the cut, its point a is cut by the corresponding point a' of the cutter, 
Fig. 14, this being found by measuring up from the end of the cutter 
the distance of a, Fig. 13, from the bottom of the cut. The cutter 
being offset by one-half its thickness at this point a', the point a, 



120 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Fig. 14, will be on the center line of the blank. At the end of the 
cut, however, the inside pitch point b, Figs. 13 and 14, is cut by the 
point b' of the cutter, this point being found in the same manner 
as a'. The pitch line ab, Fig. 14, therefore does not lead toward 
the cone apex, but to the left of it. In order that it may so lead, 
as it must, the cutter has to be adjusted to the right, as in Fig. 15, 
until the points ab are radial with the cone apex. 

This adjustment reduces the offset of Fig. 14. The actual adjust- 
ment is usually determined by trial and, for those having experience 
in the work, that is a satisfactory way. Those without experience 
would, however, prefer a more definite method, and the amount of 
the adjustment is easily determined by calculation. 

This adjustment must be such as to shift the pitch line, ab, Fig. 14, 
to the right by the distance cd such that the pitch line extended passes 



The original offset of Fig. 14 being -, the final offset of Fig. 15, 



which equals cd becomes 

Final offset = 



e-7) 



apex distance 
face 



The values of / and /' are to be determined by measuring the thick- 
ness of the cutter at the outer and inner pitch points with a micro- 
meter or gear-tooth caliper. These values determined and the offset 
calculated, the cutter is adjusted to the offset and the gear is cut 
once around. The cutter is then adjusted to the right of the cutter 
line to the same offset, Fig. 16, the blank is adjusted to correspond 
and the cut is made a second time around. 

The movement of the blank from the position of Fig. 15 to that of 




Offset 


..< 




m. 


1 
1 





Fig. II. 
Figs, ii and 



Fig. 12. 



12. — Action of a cutter with parallel sides. 




"Apex Distance 

Fig. 13. 



Fig. 14. Fig. 15. 

Figs. 13 to 16. — Diagrams of the offset method. 



Fig. 16. 



through the cone apex, as in Fig. 15. Calling the thickness of the 
cutter at the outer pitch point t and at the inner pitch point t', it is 
clear from Fig. 14 that 



cd = beX- 



Inspecting Fig. 14 we see that 



ie = — 



and for — we may place the equal ratio, 



apex distance 



face 



Fig. 3, giving 



-(^-9 



apex distance 
face 



' The slight error in this formula is so small as to be negligible. 



Fig. 16 is also calculable, but the calculation involves finding the 
lineal value of the movement at the pitch circle and then the trans- 
lation of this value into angular measure. The determination of the 
adjustment by trial is therefore preferable. With the cutter prop- 
erly offset, the second cut at the pitch line will always be radial, the 
only thing remaining being to so turn the blank as to insure the cor- 
rect thickness of the teeth. This thickness for the outer end is to 
be taken from a table of tooth parts for spur gears, the blank being 
adjusted by trial until this thickness is obtained. 

The result is a set of teeth which are correct throughout their length 
at the pitch line, but not elsewhere. The correct radius of curvature 
at the outer is greater than at the inner end of the teeth, whereas the 
cutter being selected for the outer end, continues the larger curvature 



BEVEL GEARS 



121 



throughout the length, the result being a surplus of metal outside 
the pitch line which increases as the inner end is approached. This 
surplus is removed by filing, as indicated by the dotted lines of Fig. 
17, which is taken from one of the Brown and Sharpe publications. 
Care must be used that this filing does not reduce the teeth at the 
pitch line. 

The number of the cutter is usually selected as for a spur gear of 
radius af, Fig. 13, the number of teeth in this spur being equal to 
2 XafX diametral pitch. If, however, the cut angle is less than about 
30 deg., the rotation of the blank from the position of Fig. 15 to that 
of Fig. 16 leads to an undue narrowness of face at the outer diameter 
and a cutter one or two numbers lower than the one given by the 
rule may be selected. The use of the lower numbered cutter neces- 
sitates more filing of the teeth and, if the operator is not experienced 
in filing, the rule may be followed. 



stub teeth, but this will lead to criticism by the unthinking only. 
Those without experience will obtain better results from this than 
from the offset method. 

The method takes advantage of the fact that there is no geometrical 
necessity for the face and working depth cones having a common apex 
with the pitch cone, this common apex being nothing more than an 
outgrowth of the custom of maintaining a uniform ratio of depth to 
pitch throughout the length of the teeth. This ratio was adopted 
in the case of spur gears as a necessary feature of interchangeable 
sets running from a twelve toothed pinion to a rack. This considera- 
tion does not enter the case of bevel gears, each pair of which is a 
thing by itself, and there is hence no necessity for adhering to the 
customary ratio or to common cone apices, whicli do no more than 
provide a fixed ratio throughout the length of the teeth. 

In carrying out the method, the cutter is selected for the inner, 






Fig. 17. — Surplus metal removed 
by filing. 



Fig. 18. — Blank set for 
cutting. 



Fig. ig. — Gear in various 
stages. 



For teeth coarser than five diametral pitch in cast iron, it is ad- 
visable to take a preliminary stocking cut, and in steel such a cut 
should be taken in nearly all cases. With this system of cutting 
the face of the gear should not exceed two and one-half times the 
outer pitch or one-third the apex distance — whichever is less. 

Fig. 18 shows a gear blank set up for cutting, the cut angle fog 
being the angle fog of Fig. 13, and equal to the center angle minus 
the angle increment. Fig. 19 shows a gear in various stages of 
progress. At X is a tooth at the completion of the stocking cut and 
too thick throughout; at L is a tooth as it leaves the miller, correct 
at the outer but still too thick at the inner end; at M are teeth as 
they should be after filing. The blanks should be accurately made 
in order that the depth of cut may be marked on them — this for the 
outer end being taken from a table of tooth parts for spur gears. 
Figs. 18 and 19 also are from the Brown and Sharpe publications. 

The Parallel Depth Method 

This method, due to A. D. Pentz {Amer. Mach., Sept. 10, 1891) 
has the following features: Both the adjustments are made posi- 
tively from calculations; spur gear cutters are used; filing the teeth 
is eliminated; the tooth proinles depart less from the correct forms 
than those cut by the offset method and such departure as there is 
is in the opposite direction where it does less harm, and the teeth at the 
outer diameter are of the stub type and hence stronger. Finally, 
the method may be used on any gear cutting or universal milling 
machine. The gears have an unusual appearance because of their 



or smaller, pitch circle. Just as when the cutter is selected for the 
outer circle, surplus metal is left at the inner ends of the teeth, so, 
when the selection is made for the inner circle, there is a deficiency 
at the outer ends because of which the filing is eliminated. More- 
over, the teeth being stubbed at the outer ends, the length of profile 
there is less, and the amount of departure from the correct profile 
is correspondingly reduced. 

The principle of the method is shown in Figs. 20, 21 and 22. The 
center, or pitch cone angle is determined as usual, but the face is 
turned parallel with the pitch cone at a distance from it equal to the 
addendum for the inner pitch, this addendum being taken from a 
table of tooth parts of spur gears, and this work should be accurately 
done, as the adjustments depend upon it. Spur gear cutters are 
used, the pitch of the cutter being selected with reference to the 
inner, or smaller, pitch circle, the number of the cutter being that 
for a spur gear of radius ab, Fig. 20, the number of teeth in this spur 
being equal to 2 Xaby. diametral pitch. 

The blank is adjusted in the miller at the pitch cone angle, the feed 
being parallel with this cone face and the teeth throughout their 
length of a constant depth, having the usual ratio with the inner 
pitch. Two cuts are taken for the two sides in the usual way. 
Above six pitch in cast iron and for all pitches in steel, a central 
stocking cut is advisable. 

Consulting Fig. 20, it is clear that, the feed being parallel with the 
face of the pitch cone, the pitch line ac of the tooth is traced by the 
same point of the cutter, this point a', Fig. 21, being found as before 
by measuring up from the end of the cutter the distance ad, Fig. 20. 



122 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 





Fig. 



Fig. 21. 
Figs. 20 to 22.— The parallel depth method. 



Fig. 22. 



Hence, if the cutter be offset by one-half its pitch thickness, bringing 
the point a' to the center line of the blank, the pitch line ac, as shown 
in Fig. 21 will be radial. With the parts so set, one cut around is 




Fig. 23. — Parallel depth bevel gear. 

made. Next the cutter is moved to the right twice its pitch line 
thickness (that is the thickness of a tooth at the inner pitch circle) 
and the blank is turned to the right 
one-half an index spacing — both to the 
positions of Fig. 22 — and the second cut 
around is made, completing the gear. 

The outside diameter of the blank is 
determined by the formula: 

Outside diameter = inside pitch diameter 

-\-2f sin 4>-\-2e cos 4> 

the notation being as in Fig. 20. 

A pair of gears cut by this method is 
shown in Fig. 23, from which thes tub 
shaped teeth at the outer diameter will 
be apparent. Calculations for strength 
should be based on the outer pitch, re- 
membering, however, that the stub form 
gives an excess of strength over gears 
cut in the usual way. 

The only limitation of the face width 
of gears cut in this manner is that due 
to the increased stubbing of the teeth 
that goes with increase of face which, 
obviously, may be carried too far. A 
safe guide for the permissible amount 
of stubbing may be found in the propor- 
tions that have been worked out for 
stub toothed spur gears. 



Modified Addendum 

The undercut of low nutnbered pinions is an outgrowth of the feature 
of interchangeable sets of spur gears. This feature does not enter 
the case of bevel gears and it is desirable to eliminate the weakening 
effect of the undercut. In generated bevel gears this may be done 
by decreasing the addendum of the gear and increasing that of the 
pinion. Gears so modified are more noisy and less durable than those 
of standard proportions, but the advantage of increased strength 
leads to their frequent use, especially in automobiles. 

The addendum angle of the wheel teeth of a right-angle pair of 
gears, if undercutting of the pinion teeth is to be avoided, is given 
by the equation: 

tan /3= "vtan^ a+sin-S cot-a+2 sin^S — tan a 
in which a = pitch cone angle of large gear, 

/3 = addendum angle or angle increment of large gear teeth, 
S = pressure angle. 



Atiigle of CorrecK'oa 



oNo.c 
2 JO 


f Teefh of Wheel 2 30 
15 1 


2 20 

20 1 


2 1C 

1 


25 


1 




150 
|30 




140 

1 


35 




130 


40 




120 


45 








50,1°3 


Iv 


1 


^^^^ 


s 


=H=- 






_- 


-^ 




.^ 




■-' 




^ 




' 














-- 










-- 












^ 


-' 






On 


-_l i 








'__ 





— 




' 




^- 








—- 










, — 


,— 












^ 


-- 












-1° 




^^^ 


^^D 


~ 


r. 




— 


— 


— 








— 


■-- 


























-- 














, 




a 




P 


8 


i 


g 


^ 


^ 




Z^ 


— 


— 


= 


= 


— 




— 




















— ■ 


IJ 








. 


= 


=: 


1 















-50' 


■3 15 


— 


- 






— 


^ 


^ 


^^ 


^ 


=:^ 


^ 


— 


— 







= 




= 




= 








= 


^r 


— 






— 


- 


_ 


z::. 


= 




— 


-= 


==: 



























— 






— 













































_ 


. 




.40' 


























t^ 


<; 


;;:^ 


--, 


■^ 




^ 














~ 




— 


— 




























H 




























"^ 


^ 


~~ 




■ — ■ 


^1- 


■ — 














— 


— 






— 1 


cz 


[_ 










— 


— 


— 




-30' 


O-20 

































— 


_ 


J 


" 




== 








"■■^ 










= 


=. 








-_ 


— 























^ 


^ 


— 




— 


— 


— 




— 






— 


— 


— 


— 


— 


— 




— 


















— 


== 








^ 


— 


— 


— 




— 


— 


— 




-20 


























































~~ 






















— 




-10' 






































































— 


■— 




__ 








o 


2.i 
































_ 




1 


























_ 


















-0 



Angls of Correctioa 

No.of Teeth of Wheel 1 20' 1 lO' 1 50' 

a 
.2 10 

C4 



40' 



20' 



= 40 





20 


30 


40 1 


50 


60 


70| 


60 


90 


100 


uo 


12( 


i 


130 


140 


150 


160 


170 


IS 


lo 


190 


20 














^— ■ 




-- 


■^ 


'- 






^ 




— 






— 






"^ 

















— 


— 




— " 












— 




— 












= 


^ 


— 






^ 




— 






=: 






— 




_ 


_ 






— 




-^ 


_ 











_ 


^ 


^ 


— 




— 














-= 


= 




= 


~ 




^ 






— 






— 




— 


- 






— 




— 


~ 


— 


— 


— 


— 


- 


— 


— 



























































































































































.10 



Fig. 24. — Angle of correction to avoid undercut of bevel gear teeth. 



BEVEL GEARS 



123 



With low numbered pinions this angle j3 is less than the standard 
addendum angle, the difference being the angle of correction. Prac- 
tically, if this correction is made equal to Y^, of the calculated amount 
the resulting undercut is so small as not to be noticeable. Fig. 24, 
by the Bilgram Machine Works, gives, without calculation, this 
y^ of the theoretical angle of correction. The chart applies to gears 
of 15 deg. pressure angle. 

The Gleason Works give a minimum number of teeth for the pinion 
to be used with the standard addendum as shown in the following 
table: 

Minimum No. of 
teeth in pinion — 
std. teeth 
I to I 14 



Ratio of 
diams. 



}4 to I 

2 to I 

3 to I 

4 to I 

5 to I 

6 to I 



18 

19 
21 
21 
21 
21 



If the pinion has fewer teeth than those given in the table the 
following proportions are used : 

Gear addendum = .^y,working depth 
Pinion addendum = .'/Xworking depth 
These proportions are used for a pressure angle of 14H deg. 

Table 5. — Axial Thrust of Bevel Gears 
By L. S. Cope 



SECTION ON LINE SS* 




a = pressure angle of the gear teeth, 
P = tooth pressure at middle of tooth face, 
iV = a normal through the point of contact, 
F=P tan q; = pressure resolved along line 55\ 
^^'^ = a normal section through the gears, 
& = pinion angle, 

r = thrust on pinion =P tan a sin b, 
r^ = thrust on gear=P tan a cos b. 

Following table gives factors by which the tooth pressure is multi- 
plied to find the thrust. 



Gear 


Pressure angle (a) 


I 


4H° 


15° 


20° 


22° 




Gear 


Pinion 


Gear 


Pinion 


Gear 


Pinion 


Gear 


Pinion 


I — I 


.183 


• 183 


.189 


.189 


• 257 


■ 257 


• 286 


.286 


iH—i 


• 215 


• 143 


,223 


.148 




303 


.202 


• 336 


.224 


2 — I 


.232 


.116 


.239 


. 120 




325 


• 163 


• 361 


.181 


2K2— I 


.240 


.096 


.249 


.100 




338 


• 13s 


• 375 


• 150 


3 — I 


.246 


.082 


.254 


• 085 




34S 


• 115 


• 383 


.128 


3^—1 


.249 


• 071 


• 258 


.074 




350 


.100 


• 389 


• III 


3H~l 


.250 


.067 


.259 


.069 




352 


.094 


• 390 


. 104 


4 —I 


• 251 


.062 


.260 


.065 




353 


.088 


• 392 


.097 


4K2— I 


.253 


• 056 


.262 


.058 




355 


.079 


■ 394 


.087 


S —I 


.254 


• 051 


.263 


.053 




357 


.072 


.396 


.080 


SH—l 


.255 


.046 


.264 


.048 




358 


.065 


.398 


.072 



Skew Bevel Gears 

The calculation and construction of strictly correct skew bevel gears 
are much involved. Certain approximations which result in gears 



that are entirely satisfactory in use lead also to great simplification 
of the work. The following explanation of this method is due to 
Reginald Trautschold {Amer. Mach., Oct. 7, 1915). 

These gears are of two types: (a) those in which the pinion is of 
the ordinary bevel gear type, the oblique teeth being applied to the 
gear only. (6) Those in which the teeth of both gear and pinion 
are cut askew. Of these types (a) is the more common and is illus- 
trated in Fig. 25 in which dimension a is the offset of the pinion 
shaft. The pinion differs in no way from a regular bevel gear and 
its proportions are easily calculated from ordinary bevel gear 
formulas, once its center angle is ascertained. The apex point of the 
pinion must lie at point i in the perpendicular axis plane of the gear. 




Fig. 25. — Skew bevel gears of type a. 

The teeth of the gear which are actually in mesh with those of the 
pinion converge toward the same apex, the others converging to 
points which lie on a circle having a radius equal to the offset of the 
pinion shaft to which circle the tooth profiles prolonged are tangent, 
all as shown in Fig. 25. 

If the pinion shaft was not offset and the gear combination simply 
a set of ordinary bevels, the pitch diameter would be FG and the 
number of teeth required, etc., could easily be calculated. The fact 
that the pinion shaft is offset can in no way affect the number of 
teeth in the gear and the tooth proportions. The number of teeth 
in the skew bevel gear is therefore the same as would be required 
for an ordinary bevel gear having a pitch diameter equal to FG. 
It is the failure to recognize this simple relationship that has led to 
the common belief that skew bevel gears are hard to lay out. The 
actual pitch diameter of the skew bevel gear is considerably greater 
than this "equivalent pitch diameter," FG, depending upon the 
amount of offset to the pinion shaft. 

The normal pitch of the gear BC must conform to the circular 
pitch of the pinion, but the circular pitch DE of the gear depends 
upon its actual pitch diameter, the number of teeth being fixed by 
the pinion and equal to the number of teeth required for a common 
bevel gear of a pitch diameter equal to FG. 

The sliding action of the teeth upon one another also depends upon 
the amount of offset to the pinion shaft. In the combination illus- 
trated in Fig. 25 it is evident that the sliding of the pinion tooth on 
the gear must take place from / to D. This sliding action is increased 
by any increase in the amount of pinion shaft offset. 

The various angles and the tooth proportions of the pinion are 
the same as those for any plain bevel gear of the same number of 
teeth, pitch, etc., and of similar center angle. The calculations for 
the gear require the somewhat different formulas that follow: 



124 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



d = 



D' 



tan « = 



2A 



D'e 

2A 

sin a 
3.1416 J>' 

N 
tan Ei=jy- 



D' 



P'i = 



tan / = 



2 sin El 



tan /C = 



2.314 sin El 



Fi = Ei+J 
Ci=Ei-K 
d' 
n 
Vi = s cosEi 
D = D'+2Vi 



or, for greater accuracy, 



D = D'+ 



2ViD'e 

D' 



£2 = (go-El) 
C2 = {qo-Ei)-K 
{go-Ei)+J\D' 



F2 = 



D'e 



(a) 
(6) 
(c) 
(d) 
ie) 
(/) 

(s) 

W 
a) 

0-) 

ih) 
Q) 

M 
(o) 

ip) 



in which p = diametral pitch, 

n = number of teeth in pinion, 
N = number of teeth in gear, 
A = offset of pinion shaft, 
a = angle of offset, 
«f' = pitch diameter of pinion, 
D'e = equivalent pitch diameter of gear, 
Z)' = pitch diameter of gear, 
(f = outside diameter of pinion, 
D = outside diameter of gear, 
p' = circular pitch of pinion, 
/>i" = normal pitch of gear, 
/»'i = circular pitch of gear. 
El = center angle of pinion, 
Fi = face angle of pinion, 
Ci = cutting angle of pinion, 
£2 = center angle of gear, 
F^ = face angle of gear, 
C2 = cutting angle of gear, 
£'2 = contact angle of gear and pinion = £1, 
/ = angle increment, 
i2' = angle decrement, 
Vi = diameter increment of pinion, 
5 = addendum. 

The formulas for ascertaining the various dimensions and angles 
of the pinion are similar to those for any ordinary bevel gear, once 
the center angle of the pinion is obtained. 

The equivalent pitch diameter of the gear is the same as the pitch 
diameter of the regular bevel gear that would give the required speed 
ratio, and is obtained by dividing the number of teeth in the gear by 
the diametral pitch. 

The angle of offset is the angle between the axis plane of the gear 
and a plane passing through the axis of the gear and the common 
contact point of the pitch circumferences (outer) of the pinion and 
gear. Its tangent is obtained by dividing the offset of the pinion 
shaft by half the equivalent pitch diameter of the gear, or twice the 
offset of the pinion shaft divided by the equivalent pitch diameter 
of the gear. 



The pitch diameter of the gear is then obtained by dividing twice 
the pinion shaft offset by the sine of the angle of offset. 

The circular pitch of the gear is equal to the quotient of the pitch 
circumference by the number of teeth. 

The tangent of the center angle of the pinion is found by dividing 
the pitch diameter of the pinion by the equivalent pitch diameter 
of the gear. 

The outside diameter of the gear is usually found by adding to its 
pitch diameter twice the diameter increment of the pinion. This 
method is not quite accurate, however, as the diameter increments 
of the gear and the pinion are only the same when there is no offset 
to the pinion shaft. The greater this offset is, the smaller propor- 
tionally does the diameter increment of the gear become. A more 
accurate way of ascertaining the outside diameter of the gear then 
is by the use of the second formula. This more accurate method is 
not absolutely correct, for it is based on the assumption that the 
decrease in diameter increment of the gear is proportional to the ratio 
of the equivalent pitch diameter to the pitch diameter of the gear, 
which relationship is not quite true. The possible error for any 
ordinary gear is so small, however, as to be quite immaterial. 

The center angle of each individual gear tooth is equal to the 
complement of the center angle of the pinion, so that the center angle 
of a skew bevel gear may be taken as the complement of the center 
angle of the pinion with which it is to mesh. 

The cutting angle of the skew bevel is likewise the same for each 
individual tooth and is equal to the difference between the center 
angle of the gear and the angle decrement. 

The face angle of the skew bevel gear as obtained by the formula 
given is not absolutely accurate, but the error is sufficiently trivial 
to be overlooked in practice with safety, unless the face of the pinion 
is usually wide and the pitch equally small. In such a case, the cut- 
and-try method of fitting the gear to the pinion is advisable, as the 
calculations involved for an accurate mathematical solution are 
extremely complex. 

The face angle of a skew bevel gear would not be the same as that 
of a bevel gear matched to mate with the skew gear pinion unless 
the offset of the pinion shaft was zero. Such a condition, which 
would be that existing between a set of bevels of proper proportions, 
would fix the minimum face angle for the skew bevel gear. The 
maximum face angle would occur when the pinion shaft's offset was 
equal to half the pitch diameter of the gear and would be one of 
90 deg. Between these limits the face angle of a skew bevel gear may 
be anything, depending upon the difference in the pitch and equiva- 
lent pitch diameters of the gear. Formula (/>) is derived on the 
assumption that the increase in face angle of the skew bevel gear, 
from that of a set of plain bevel gears of similar number of teeth, 
etc., to the condition where there would be no rolling action, is 
governed by the ratio of the pitch diameter of the gear to its equiva- 
lent pitch diameter This relationship is only approximately accu- 
rate, for the actual increase in face angle is not constant between 
its minimum and maximum values. For all practical shop require- 
ments, however, formula {p) may be considered correct. Any 
possible error that might arise would be slight and would affect 
only the total depth of tooth at the smaller end of the gear where 
it would be least noticeable and least harmful. 

The following example shows the application of the formulas: 

Required, a pair of skew bevel gears, 10 diametral pitch, 85 teeth 
in gear, 13 teeth in pinion; pinion shaft offset ij^ in. 
Pitch diameter of pinion, 

'^' = ^^{0 = 1-30 in. (a) 

Equivalent pitch diameter of gear, 

■D'e= 8^0=8.50 in. 
Angle of offset, 



2X1.5 
tan a= ^ =0.3529 

a =19 deg. 26 min. 



(6) 
(c) 



BEVEL GEARS 



125 



Pitch diameter of gear, 



^, 2X1-5 

Z)'= = o.oi in. 

0.33271 



Say, 9 in. 



Circular pitch of gear. 



^, 3-1416 X9 

pi = g^ = 0.33 "»• 



Center angle of pinion, 
£1 = 
£1 = 8 deg. 42 min. 



1.3 
tan£i = g-- = o.iS29 



Angle increment, 

2X0.15126 
tan / = = 0.02327 

/ = I deg. 20 min. 

Angle decrement, 

2.314X0.15126 

tan K = = 0.02692 

13 

K = i deg. 33 min. 
Face angle of pinion, 

i?i = (8°42') + (i° 20') = 10 deg. 2 min. 
Cutting angle of pinion, 

Ci = (8° 42') - (1° 33') = 7 deg. 9 min. 

Addendum, 

1-3 
s = — =0.10 in. 
13 

Diameter increment of pinion, 

71 = 0.1X0.1513=0.01513 in. 
Outside diameter of gear, 

Z> = 9 + (2Xo.oi5i3)=9.03 in. 
2X0.01513X8.S 



or, 



D = g+- 



Center angle of gear, 

£2=(9o°-8°42')=8i deg- 18 min. 
Cutting angle of gear, 

C2 = (90° -8° 42')-!° 33' = 79 deg. 45 min 

Face angle of gear, 

[(90°-8°42')+i°2o']9 



{e) 
if) 

(«) 

W 

(»•) 
0') 
ik) 

H) 
(w) 

(«) 
io) 



/?2 = - 



8-S 



87 deg. 29 min. {p) 



Skew bevel gears can be machined on any of the machines used 
for cutting the ordinary type of bevel gear, if simple adjustments or 
modifications are made. The carrying spindle of the machine must 
be offset from the plane of the cutting tool a distance equal to the 
ofiset of the pinion shaft, and the path of the cutting tool must con- 
form to the axis plane of the pinion shaft reinlation to its position 
in respect to the center of the carrying spindle. All subsequent 
operations are then similar to those employed in cutting plain bevel 
gears, except that the path of the cutting tool is always tangent to 
the circle of apexes instead of toward a common apex point. The 
rotary adjustment of the gear blank is governed by the circular 
pitch of the gear, not by its normal pitch, which corresponds to the 
circular pitch of the pinion. The adjustments are slightly more 
complicated than when cutting the simpler plain bevel gears and must 
be performed with great care, as there is no common apex toward 
which to work. This adds to the diflSculties of accurate workman- 
ship and explains the machinist's dislike of this type of gear. 

When extreme accuracy is required, the gear blank can be faced 
off when mounted on the ofifset spindle, the cutting edge of the facing 
tool being in a plane corresponding to the axis plane of the pinion 



shaft and its advance along a line conforming to the supplement of 
the cutting angle of the pinion. This is not ordinarily necessary, 
as sufScient accuracy may be obtained by facing the gear blank in 
a lathe to conform to the face angle as ascertained from formula (/>). 

Second Type of Skew Bevel Gears 

The second type of skew bevel gear is illustrated in Fig. 26. It 
requires slightly more complicated calculations, as both gear and 
pinion have their teeth cut askew. The manufacture of such gears 
is usually simplified by making the obliquity of the teeth the same 
in both pinion and gear. 

The gears are turned up according to the dimensions for plain 
bevel gears of the same number of teeth, pitch and ratio, and no 
alteration in the diameters is ordinarily made nor are any alterations 
in the angles necessary — the angles being made the same as for plain 
bevel gears. This decided advantage is possible because of the fact 
that, although the apex points of the two gears do not coincide, the 
converging conical surfaces are parallel to those of bevel gears with 
a common apex point. That is, angles e and e' are complements. 




2 

2 + 1'' 



Fig. 26. — Skew bevel gears of type b. 

Both the gear and pinion are machined with the plane of the cutting 
tool offset from the carrying spindle of the machine, but the offset 
is different for the two gears unless they are of the same size. For 
gears of similar dimensions, the total offset of the shafts is divided 
by two and the offset between the spindle of the machine and the 
cutting tool planed for both gears is the same and equal to half the 
total offset. For any other combination of gears the total offset 
is divided proportionally to the speed ratio, the smaller value being 
employed as the machine offset for cutting the pinion and the larger 
for machining the gear. For instance, in cutting a pair of skew bevel 
gears of this type having a shaft offset of 2 in. and a speed ratio of 

2 to I, the machine drop, or offset for the pinion, would be 

0.666 in., and for the gear, 0.666X2 = 1.333 in. 

Skew bevel gears cut to this method have proved very satisfactory 
and the only criticism that can be advanced is on account of the 
decreased strength of the teeth as they are commonly cut. The 
teeth not radiating from the center of the gear nor being normal to 
the pitch circumference, the circular pitch must necessarily be 
greater than that of common bevel gears, if standard tooth thick- 
nesses are to be retained. The circular pitch of the common bevels 
corresponds to the normal pitch of skew bevel gears, which is of 
necessity less than the circular pitch of such gears. To retain the 
proper thickness of tooth for a given pitch, an increase in the diam- 
eters of skew bevel gears is necessary, the amount of increase depend- 
ing upon the angularity of the teeth. If this increase in diameter 
is attended to, however, the full strength of the standard tooth will 
be developed in a skew bevel gear as well as in a plain bevel gear. 



FRICTION GEARS 



The working loads on friction gearing formed the subject of a 
series of experiments by Prof. W. F. M. Goss {Trans. A. S. 
M. E., Vol. 29). Various materials were tested for both the fibrous 
and the metal wheels. The materials of the fibrous wheels were 
straw fiber, straw fiber with belt dressing, leather fiber, leather, 
leather-faced iron, sulphite fiber, and tarred fiber. 

The straw-fiber wheels were worked out of blocks built up of 
square sheets of straw board laid one upon another with a suitable 
cementing material between them and compacted under heavy hy- 
draulic pressure. In the finished wheel the sheets appear as disks, 
the edges of which form the face of the wheel. 

The wheel of straw fiber with belt dressing was similar to that of 
straw fiber, except that the individual sheets of straw board from 
which it was made had been treated, prior to their being converted 
into a block, with a belt dressing, the composition of which is 
unknown. 

The leather-fiber wheel was made up of cemented layers of board, 
as were those already described; but in this case the board, instead 
of being of straw fiber, was composed of ground sole-leather cuttings, 
imported flax and a small percentage of wood pulp. The material 
is very dense and heavy. 

The leather wheel was composed of layers or disks of sole leather. 

The leather-faced iron wheel consisted of an iron wheel having a 
leather strip cemented to its face. After less than 300 revolutions 
the bond holding the leather face failed and the leather separated 
itself from the metal of the wheel. This wheel proved entirely in- 
capable of transmitting power and no tests of it are recorded. 

The wheel of sulphite fiber was made up of sheets of board com- 
posed of wood pulp. The sulphite board is said to have been made 
on a steam-drying continuous-process machine in the same way as 
is the straw board. 

The tarred-fiber wheel was made up of board composed principally 
of tarred rope stock, imported French flax and a small percentage 
of ground sole-leather cuttings. 

Each of the fibrous driving wheels was tested in combination 
with driven wheels of the following materials: Iron, aluminum, 
type metal. 

Regarding the metallic wheels the conclusions are that those driv- 
ing wheels which are the more dense work more efiiciently with the 
iron follower than with either the aluminum or type-metal followers; 
but in the case of the softer and less dense driving wheels, and espe- 
cially in the case of those in which an oily substance is incorporated, 
driven wheels of aluminum and type metal are superior to those of 
iron. Finely powdered metal which is given off from the surface 
of the softer metal wheels seems to account for this effect, and the 
character of the driving wheels is perhaps the only factor necessary 
to determine whether its presence will be beneficial or detrimental. 
Finally, with reference to the use of soft-metal driven wheels, it 
should be noted that no combination of such wheels with a fibrous 
driver appears to have given high frictional results. Except when 
used under very light pressures, the wear of the type metal was 
too rapid to make a wheel of its material serviceable in practice. 

Regarding the fibrous wheels the conclusions are that the addi- 
tion of belt dressing to the composition of a straw-fiber wheel is 
fatal to its frictional qualities. The highest frictional qualities are 
possessed by the sulphite-fiber wheel which, on the other hand, is 
the weakest of all wheels tested. The leather fiber and tarred fiber 
are exceptionally strong; and the former possesses frictional qualities 
of a superior order. The plain straw fiber, which in a commercial 



sense is the most available of all materials dealt with, when worked 
upon an iron follower possesses frictional qualities which are far 
superior to leather, and strength which is second only to the leather 
fiber and the tarred fiber. 

A review of the data discloses the fact that several of the friction 
wheels tested developed a coefficient of friction which in some cases 
exceeded .5. That is, such wheels rolling in contact have trans- 
mitted from driver to driven wheels a tangential force equal to 50 
per cent, of the force maintaining their contact. These wheels, 
also, were successfully worked under pressures of contact approach- 
ing 500 lbs. per in. in width. Employing these facts as a basis 
from which to calculate power, it can readily be shown that a fric- 
tion wheel a foot in diameter, if run at 100 r,p.m., can be made to 
deliver in excess of 25 h.p. for each inch in width. It is certainly 
true that any of the wheels tested may be employed to transmit 
for a limited time an amount of power which, when gaged by ordinary 
measures, seems to be enormously high; but obviously, performance 
under limiting conditions should not be made the basis from which 
to determine the commercial capacity of such de\dces. In view of 
this fact, it is important that there be drawn from the data such 
general conclusions with reference to pressures of contact and fric- 
tional qualities as will constitute a safe guide to practice. 

The recommended contact pressures, which are one-fifth of the 
ultimate resistance established by tests under destructive pressures, 
are given in Table i. 

Table i. — Working Contact Pressures per Inch of Face 

Pressure, lbs. 

Straw fiber 150 

Leather fiber 240 

Tarred fiber 240 

Sulphite fiber 140 

Leather 150 

The recommended values of the coefificient of friction, which are 
60 per cent, of the laboratory results, are given in Table 2. 

Table 2. — Working Values of the Coefficient of Friction 

Coefficient 
of friction 

Straw fiber and iron . 255 

Straw fiber and aluminum .273 

Straw fiber and type metal . 186 

Leather fiber and iron . 309 

Leather fiber and aluminum . 297 

Leather fiber and type metal . 183 

Tarred fiber and iron . 150 

Tarred fiber and aluminum .183 

Tarred fiber and type metal . 165 

Sulphite fiber and iron .330 

Sulphite fiber and aluminum .318 

Sulphite fiber and type metal .309 

Leather and iron . 135 

Leather and aluminum .216 

Leather and type metal . 246 

The recommended formulas for the working loads in h.p. are 

given in Table 3 

in which (^ = diameter of friction wheel, ins., 
I-F = width of face, ins. , 
N = T. p. ra. 
126 



FRICTION GEARS 



127 



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4375 
3500 

2625 



1750 



1300 



875 a 
700 ?: 



to 



518 



340 



263 



175 



20 



30 



50 60 70 80 90 100 150 

Speed, Kevolutions per Minute 



200 



300 



400 600 600 700 800 900 



To find peripheral speed, locate the intersection of the vertical line representing the given speed in r.p.m., with the diagonal 
one representing the given diameter. The horizontal line passing through this point will give the surface speed in ft. per min. on 
the vertical scale to the right of the chart. 

To find the horse power for a given wheel, locate the intersection of the vertical line representing the given speed in r.p.m. 
with the diagonal line representing the given diameter. Follow the horizontal line passing through this point to the right or left 
until the intersection between it and the vertical line representing the given width, as shown on the scale at the top of the chart, 
is reached. The diagonal line passing through this point marked Total Horse Power will represent the required horse power. 

To find the face width of a given wheel necessary to transmit a given horse power, the speed being known, locate the inter- 
section of the vertical line representing the given speed in r p.m. with the diagonal line representing the given diameter. Follow 
the horizontal line passing through this point to the right or left until the intersection between it and the diagonal line representing 
the required horse power is reached. The vertical line passing through this point will give the width of face in ins. on the scale at 
the top of the chart. 

For other material than straw fiber and iron, multiply the horse power by the following factors : 



Sulphite fiber and iron 1.23 

Leather fiber and iron 1.97 



Leather and iron .53 

Tarred fiber and iron 97 



Fig. I. — Dimensions of friction wheels of straw fiber and iron. 



128 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 3. — Formulas tor Working Loads 

Horse-power 

Straw fiber and iron . 00030 dWN 

Straw fiber and aluminum . 00033 dWN 

Straw fiber and type metal .00022 dWN 

Leather fiber and iron . 00059 dWN 

Leather fiber and aluminum. . . . .00057 dWN 

Leather fiber and type metal ... . 00035 dWN 

Tarred fiber and iron . 00029 dWN 

Tarred fiber and aluminum .... .00035 dWN 

Tarred fiber and type metal. . . . .00031 dWN 

Sulphite fiber and iron .00037 dWN 

Sulphite fiber and aluminum . . . .00035 dWN 

Sulphite fiber and type metal. . . .00034 dWN 

Leather and iron .00016 dWN 

Leather and aluminum .00026 dWN 

Leather and type metal .00029 dWN 

All usual problems connected with the dimensions and power 
capacity of friction wheels may be solved by the use of Fig. i, with 
which the necessary explanations are given. The chart repre- 
sents the formiila for straw fiber and iron: 

h.p. = .0003 dWN 

In the application of friction gearing, the fibrous wheel must 
always be the driver; the roUing surfaces should be kept clean or, 
if this is impossible, the wheels should be increased in size to provide 
for a lower coefiicient of friction due to the presence of dirt; and the 
pressure should be by positive, inflexible mechanism — springs are 
not admissable. 

The formulas and chart are equally applicable to face friction 
gearing, with the proviso that it is advisable to make the width of 
face of the driver and the distance between the driver and the 
center of the follower such that the variation in the velocity of the 
two edges of the driver shall not exceed 4 per cent. This may be 
secured by making the minimum distance between the driver and 
the center of the follower twelve times the width of the face of 
the driver. If this distance is made smaller, as it frequently must 
be, the gearing wiU work successfully but its power capacity will 
be decreased because of the fact that the coefl&cient of friction dimin- 
ishes if the slip exceeds 4 per cent. 

In making friction wheels, one |-inch bolt should be provided for 
every 20 sq. in. of disk. 

Bevel friction wheels, unless supported at the outer angle, give 
trouble by failure under the pressure at that point. E. R. Plaisted 
{Amer. Mach., Sept. 18, 1902) states that a disk of soft wood about 
f in. thick as a backing for the paper at that point obviates the 
difficulty. 

Practice with Friction Drives 

In the practice of the Rockwood Mfg. Co. (1916) the materials 
having the best combination of properties — high coefficient of fric- 
tion and physical ability to withstand the conditions of service — are 
specially prepared straw, leather or tarred fibers. Cork composition 
has an application for driving light, fluctuating loads. The coeffi- 
cient of friction varies with the slip, being at a maximum when the 
slip lies between 2 and 6 per cent., beyond which it decreases until, 
at 100 per cent, slip (the condition of starting), the coefficient has only 
about one-third of its value at 2 per cent. slip. Table 4 gives the 
working values of the coefficient of friction recommended by the 
Rockwood Mfg. Co. and Table 5 the recommended pressures per 
inch of face — these latter values being approximately one-fifth of the 
ultimate crushing resistances of the materials. The horse-power 
formulas used with these materials are given in Table 6 while Table 



7, computed from the appropriate formula of Table 6, gives the 
horse-power of wheels of straw fiber and cast iron, multipliers for 
other materials being included. The reduction of the coefficient of 
friction under the high slip due to starting the load, makes necessary 
the initial application of pressures equal to about three times those 
given in Table 5. After the load is started, the pressure should be 
reduced to prevent needless wear. 

Referring to the formulas for horse-power, it should be noted that, 
while leather fiber gives the maximum transmitting capacity, this 
material is of a hard dense composition and, when subjected to fre- 
quent or prolonged slippage, its face is liable to become glazed or 
burnished, resulting in a decided lowering of its value of the co- 
efficient of friction and a corresponding drop in transmitting capacity. 
It is therefore adaptable only to drives operating continuously and 
under steady loads. For drives operating under conditions of high 
slip or frequent starting and stopping, tarred fiber or straw fiber 
should be used. 

In all cases positive action thrust boxes should be used to apply 
the pressure. Springs are inadmissible. 

Table 4. — Working Values oe the CoEFFicrENT or Friction 

Leather and cast iron 135 

Wood and cast iron 150 

Cork composition and cast iron 210 

Straw fiber and cast iron 255 

Tarred fiber and cast iron 277 

Leather fiber and cast iron 300 

Table 5. — Pressures of Contact per Inch of Face 

Lbs. 

Leather 150 

Wood 150 

Cork composition 50 

Straw fiber 150 

Tarred fiber 250 

Leather fiber 300 

Table 6. — Horse-power Formulas for Spur and Bevel Friction 

Drives 

Leather and cast iron 00016 dWN 

Wood and cast iron 00018 dWN 

Cork composition and cast iron. . .00008 dWN 

Straw fiber and cast iron 00030 dWN 

Tarred fiber and cast iron 00055 dWN 

Leather fiber and cast iron 00071 dWN 

In which J = mean diameter of wheel, ins., 

TF = total efi^ective width of face, ins., 
iV = revolutions per minute. 

Table 7. — Horse-powers Transmitted by Friction Gears 
Having Straw-fiber Driving and Cast-iron driven Members 



For other speeds the powers are in direct proportion, 
diameters add or multiply the figures for those given. 



For other 



Diara. 
of 


Revolutions per minute 


wheel 


100 


120 


140 


160 


180 


200 


220 


240 


260 1 280 


300 


3 


0.09 


0. n 


0. 13 


0. 14 


0. 16 


0. 18 


0. 20 


0.22 


0.23 


0.25 


0.27 


4 


0. 12 


0. 14 


0. 17 


0. 19 


0.22 


0.24 


0.26 


0.29 


0.31 


0.34 


0.36 


S 


0. IS 


0. 18 


0.2 1 


0.24 


0.27 


0.30 


0.33 


0.36 


0.39 


0.42 


0.4s 


6 


0. 18 


0.22 


0.25 


0.29 


0.32 


0.36 


0.40 


0-43 


0.47 


o.so 


0.54 


7 


0.2 I 


0.25 


0.29 


0.34 


0.38 


0.42 


0.46 


0.50 


0.S5 


0.59 


0.63 


8 


0.24 


0.29 


0.34 


0.38 


0.43 


0.48 


0.53 


0.58 


0.62 


0.67 


0.72 


9 


0.27 


0.32 


0.38 


0.43 


0.49 


O.S4 


0.59 


0.6s 


0.70 


0.76 


0.81 


10 


0.30 


0.36 


0.42 


0.48 


O.S4 


0.60 


0.66 


0.72 


0.78 


0.84 


0.90 






FRICTION GEARS 



129 



Multipliers for Other Materials 
Spur and Bevel Friction Drives 

Combination Pressure l-in. face Multiplier 

lbs. 

Leather and cast iron 150 . S3 

Wood and cast iron 150 .60 

Cork composition and cast iron < . . . . 50 .27 

Straw fiber and cast iron 150 1. 00 

Tarred fiber and cast iron 250 1. 83 

Leather fiber and cast iron 300 2.37 

Disk Friction Drives 

Combination Pressure l-in. face Multiplier 

lbs. 

Tarred fiber and cast iron 250 2 . 10 

Tarred fiber and copper alloy 250 2.30 

Tarred fiber and zinc alloy 250 2.37 

Tarred fiber and aluminum alloy 250 2.53 

Variable Speed Disk Friction Drives 

In this form of drive there is a differential slippage between the two 
edges of the fiber disk which may even be negative at the inner edge. 
This condition leads to a loss of effective pull and should be guarded 
against by using a small ratio of the face width of the fiber wheel to 
its distance from the center of the iron disk. The width of the 
fiber wheel should be from 3ii2 to J^g of the diameter of the iron 
disk. 

For a ratio of i to 1 2 and average working values of the coefficient 
of friction, the slip between the driving and driven members ranges 
from 6 per cent, to 8 per cent. For a ratio of i to 16 the slip ranges 
from 4 per cent, to 6 per cent. 

The diameter of the fiber friction wheel relative to that of the disk 
is not important as regards transmitting capacity. However, it 
should be noted that as the diameter of the wheel is decreased its 
driving torque decreases proportionately; also the relatively smaller 
its diameter, the faster it revolves, resulting in a somewhat more 
rapid wear. When the friction wheel is made larger than the disk, 
the objection is offered that its face width is relatively small for its 
diameter and too great a space of installation is required for the drive. 
Preferably the diameter of both should be the same. 

The value of the coefficient of friction in disk drives is practically 
independent of the pressure of contact and, excluding positions at the 
extreme center, independent of the position of the fiber wheel on the 
disk. Thus the driving torque of the driven wheel varies directly 
with the pressure of contact but is independent of relative speed 
positions. Below are given the Rockwood Mfg. Co.'s recommended 
working values of the coefficient of friction for tarred fiber friction 
wheels as generally used in these drives, running in combination 
with different kinds of commercial disk materials: 

CoErnciENT OF Friction^ — Working Values for Disk Drives 

Tarred fiber and cast iron 323 

Tarred fiber and copper alloy 352 

Tarred fiber and zinc alloy 364 

Tarred fiber and aluminum alloy 390 

These values represent approximately 60 per cent, of the safe maxi- 
mum values regardless of slip, as obtained from tests. The slight 
variation from values heretofore given for like combinations of mate- 
rials is due in part to the changed conditions of operation and in part 



to the difference between the maximum values of the coefficient of 
friction regardless of slip as here used and values at 2 per cent, 
slip as used in those drives. 

In addition to combinations of the tarred fiber as given, numerous 
tests have been made to determine the transmitting capacity and 
efficiency of other materials as straw fiber, leather fiber, cork com- 
position, vulcanized fiber and leather, but in the main, none of these 
has approached the generally satisfactory results secured through 
use of the tarred fiber. 

Tests made to determine the effect of fillings or dressings for the 
frictions have shown conclusively the disastrous results following 
use of any mixture containing free oil or grease. Preparations con- 
taining rosin and pine tar are good, but to obtain satisfactory results, 
the mixture must be incorporated in tlje make-up of the fiber filler 
before being placed in service. 

The maximum normal working pressure recommended for frictions 
of tarred fiber is 250 lbs. for each i in. of effective face width. 
To satisfactorily meet the exacting conditions of service of this form 
of drive, with its strict limitations as to face width of the fiber wheel, 
only such material can be used as is capable of withstanding great 
pressure of contact and without danger of crushing or breaking down. 
At the same time it must not be of a sufficient hardness to be subject 
to glazing or burnishing under high slip, as encountered at times of 
starting or when operating at low speeds. In this respect the tarred 
fiber shows a marked superiority over the others. 

A common source of trouble in disk drives is a tendency on the part 
of the fiber filler projecting beyond the supporting flanges to break 
down over the flanges, causing often a softening of the entire width 
of face and resulting in greatly decreased life. To guard against such 
action, wheels should be designed to have a fiber projection of not 
to exceed % in. for the larger diameters and face widths and ranging 
down to H to % 6 in. for the sinaller sizes. As a further precaution 
it has been found highly beneficial to make the total thickness of 
the fiber filler somewhat in excess of the effective face width desired 
and bevel off the edges on either side to form a backing or retaining 
wall to support the outer edges of the effective face. Wherever con- 
ditions wUl permit this construction should be used. The edges are 
usually beveled on a 30° angle from the inner edges of the supporting 
flange to the face. 

From recommended values of the coefficient of friction and safe 
working pressures as given above, horse-power formulas for the differ- 
ent combinations of frictions may be computed as follows: 

Tarred fiber and cast iron h.p. = . 00063 dWN 

Tarred fiber and copper alloy h.p. = .00069 dWN 

Tarred fiber and zinc alloy h.p. = . 00071 dWN 

Tarred fiber and aluminum alloy h.p. = .00076 dWN 

In which (f = mean diameter at which friction wheel operates on 
disk, ins., 
PF = width of face of friction wheel, ins., 
N = revolutions per minute of disk. 

By means of these formulas the transmitting capacity of drives 
may be figured for any position of the friction wheel on the disk. 
The maximum transmitting capacity occurs of course when the fric- 
tion wheel operates at outer positions or highest speeds. Drives 
designed in accordance with these formulas are capable of starting 
their full rated loads from rest, though use must momentarily be 
made of pressures of contact approximately three times that of 
normal running. 



WORM GEARS 



For the distinction between lead and pitch, see Lead and Pitch. 

The thread profile of worms is most commonly made to the Brown 
and Sharpe standard which is a direct outgrowth of their gear-tooth 
standard, the section of a worm and wheel through the axis of the 
worm being the same as that of a rack and gear in mesh. When 
this standard is used the following formulas (from the Brown & 
Sharpe Mfg. Co's. Formulas in Gearing) apply, reference being made 
to Fig. I. 

Table 6 of spur-tooth parts by circular pitch (page 90) contains, in 
the 2d, 9th and loth columns, a list of worm-tooth parts. 

Formulas por Brown and Sharpe Standard Worm Gearing 

I, = lead of worm. 

iV = number of teeth in gear. 

m = turns per inch of worm. 

<i = diameter of worm. 
d' = pitch diameter of worm. 
d" = diameter of hob. 
Z> = throat diameter. 
D' = pitch diameter of worm wheel. 
B= blank diameter (to sharp corners). 

C = distance between centers. 

p = diametral pitch. 

P = circular pitch for worm wheels or axial pitch for worms. 

-r, I See Fig. i. 

5 = addendum. 

/ = thickness of tooth at pitch line. 
/« = normal thickness of tooth. 
j = clearance at bottom of tooth. 
D" = working depth of tooth. 
D"+ /= whole depth of tooth. 

6 = pitch circumference of worm. 

V = width of worm thread tool at end. 
w = width of worm thread at top and width of hob tool at 

end. 
^ = angle of tooth of worm wheel with its axis, or the angle 
of thread of worm with a line at right angles to its axis. 
If the lead is for single, double, triple, etc., thread, then 

L=P, 2P, 3P, etc. 
In multiple-threaded worms and their mating wheels, if the angle 
8 is more than 15° the tooth parts should be figured on the normal 
as for spiral gears. In using the formulas for spiral gears, it should 
be borne in mind that while P is the axial pitch for worms it is the 
circular pitch for spiral gears. 



a 


= 60° to 


90 


Z,= 


I 
' m 




P = 


nD 

N+2 






NP 


N 


D' = 


71 


P 


D-- 


^ 1 
= X + " 





h=Tt {d—2s) = Tt d' 
Practi 
circle is not more than f pitch diameter of worm. 



/'» = ^ cos 5 

r = 25 

2 

r" = r'+D"^-f 
^ D'+d D'+d' 



B=D-\-2 I r'~r' cos 

d"=d+2j 
v = .3i P 
"' = .335^ 



!) 



measurement of sketch is 
generally sufficient. 




, Li Practical only when width of wheel on wheel-pitch 
tano = -j| 



Fig. I. — Notation of formulas for worm gearing. 

The profiles of worm teeth being the same as those of rack teeth, 
the same interference takes place if the wheel has less than 30 teeth. 
If the wheel be finished with a hob, the interference will be overcome 
but at the expense of undercut teeth. Both interference and under- 
cut may be prevented by increasing the throat diameter of the wheel, 
making the diameter in accordance with the formula: 

iV 
D = cos^ 141° —+45 

The increase of throat diameter increases also the center distance, 
the amount of increase being shown by comparing this value of D 
with the one previously given. To keep the original center distance, 



130 



WORM GEARS 



131 



the outside diameter of the worm must be reduced by the amount 
the throat diameter of the wheel is increased. 

The pitch diameters of circular-pilch worms, Brown and Sharpe 
standard, may be obtained from Table i. For larger or smaller 
worms than those given, add or subtract the required number of 
inches thus: 

Given a worm 4x1 ins. outside diameter, xi-in. pitch. From 
the table itf ins. outside diameter, y^ pitch = 1.375 ins. pitch 
diameter. Therefore, 4xf ins. outside diameter, H-in. pitch = 
1-375+3 =4-375 ii^s. pitch diameter. Given a worm f -in. outside 
diameter,!]^ pitch. From the table i| ins. outside diameter, 
Ys in. pitch = 1.676 pitch diameter. Therefore | in. outside diam- 
eter, ]^ ins. pitch must = i.676 — i =.676 in. pitch diameter. 

The pitch diameters of ciradar pitch worm wheels may conviently 
be found from Table 5 of the section on spur gears which also ap- 
plies to worm wheels. 

Cutting Diametral Pitch Worms 

The culling of diametral pitch worms requires the introduction in 
the change gear train of the lathe of gears having to one another 

22 . 
the ratio of tt, for which, for ordinary purposes, the value — is a 

sufficiently close approximation. It gives rise to an error of less 
than half a thousandth per inch of length of the worm. The 
formula is: 

teeth in screw gear_22X threads per in. of lead screw 
teeth in stud gear 7 X diametral pitch 

Table 2, by E. J. Rantsch {Amer. Mack., April 11, 1907) gives the 
ratio of the gears for ordinary cases on the above basis together 
with other dimensions of worms of 145 deg. obliquity. The numer- 
ators of the fractions represent the screw gear and the denomi- 
nators the stud gear. When necessary, both numerator and denomi- 
nator are to be multiplied by a (the same) number to give actual 
gears. If gears to satisfy the ratio -t are not at hand, less accurate 
ratios are often sufficient, useful ratios in the order of accuracy 
being as follows: 



22 
7 ' 



= 3.1429 



69 



= 3-1364 



47. 
is' 



-3-'^2>i3 



22 . . . . 355 

If the ratio — is considered not sufficiently accurate the ratio ■ — 

7 "3 

is adequate to any possible requirement, its decimal value being 

3.1415929. The relations of circular and normal pitches are given 

in Table 3. 



Table 


2. — Change Gears for Diametral Pitch Worms 




Diame- 


Single 
depth 


Width 


Width 


Pitch of lead screw 


tral 


of tool 


of top of 
















pitch 


point 


thread 


2 3 


4 5 


6 


7 


8 


10 


2 


1.078 in. 


.487 in. 


.526 in. 


22 


3 3 

"7" 


¥- 


¥ 


-V 


-'J- 


8 8. 


ip 


2I 


.862 


•390 


.421 


^»- 


13 2 

"3 S' 


,Vk 


¥ 


w 


w 


¥.^ 


-«,» 


3 


.719 


• 325 


-350 


tt 


"V" 


-H 


1 1 
21 


-M 


¥ 


Vi*"- 


W 


32 


.616 


.278 


.300 


H 


w 


W- 


_2 2 


2 64 
"T"9' 


*f 


m- 


440 

^T9" 


4 


• S40 


-243 


.263 


1 1 


3 3 


¥ 


m 


-¥ 


¥ 


"¥ 


¥ 


s 


• 431 


• 19s 


.210 




fl 


8 8 
3.f 


22 


1 3 2 
"35 


22 


V¥ 


¥ 


6 


.360 


.162 


• 17s 


¥r 


-V 


tt 


*? 


-¥ 


¥ 


If 


¥« 


7 


.308 


-139 


.150 




n 


14 


w 


321 

'19' 


"¥- 


Vi? 


¥if 


8 


.270 


.122 


■131 


1 1 


3 3 

2 8 


1 1 


t* 


¥i 


1 1 


-¥ 


H 


9 


.240 


.108 


.117 


44 
53 


ii 


1* 


110 
"63" 


M 


V/" 


¥3" 


10 


.216 


.097 


-105 


22 

3 7t 


3 3 
3 5 


44 

7*77 


"¥ 


If 


¥ 


1* 


"¥ 


II 


.196 


.088 


.096 


f 


f 


f 


-V 


1_2. 


-¥" 


¥ 


¥- 


I2 


.180 


.081 


.088 


11 
2 1 


H 


M 


5 5 

42 


-V- 


1 1 
"fi" 


4 4 

2T 


55 


14 


• 154 


.069 


■ 075 


22 


3 3 

¥9 


44 


5 5 

T9 


6 6 
?9 


-U 




¥"" 


16 


■'^iS 


.061 


.066 


M 


fl 


If 


il 


II 


U 


¥ 


4-^ 


18 


.120 


.054 


.058 


22 

6T< 


H 


44 
B3 


6 5 
TTTT 


2 2 
2T 


¥ 


If 


¥" 


20 


.108 


.048 


-053 


H 


M 


M 


u 


If 


n 


It 


¥ 


24 


.090 


.040 


-044 


i* 


-n 


i\ 




f- 


7 7 


2 2 

"5T 


^^ 


28 


.077 


■ 034 


.038 


1 1 

¥3 


33 


2 2 


5 5 


H 


H 


n 


4* 


32 


.067 


.030 


-033 


ii 


t¥^ 


i\ 


t¥2 


*l 


t't'^ 


H 


H 


40 


-054 


.024 


.026 


1 1 


t¥o 

t¥f 


. 1 1 
3? 


1 1 

2¥ 




t¥o 


If 


W 


48 


-04s 


0.20 


.022 


S 5 



Durability and Efficiency of Worm Gearing 

The durability of worm gearing is largely dependent on the angle 
of the helix with the tangent to the pitch line. In order that a 
worm gear may be durable, the helix angle should be large — that is, 
the worm should be a steep pitch screw. This fact is established 
by theory, by experiment and by experience. The unfortunate 
experience that many have had with worm gearing is due to bad 
design and not to any inherent defect of the construction. 

An analysis of the worm-gear problem with examples collected 
from practice by the author {Amer. Mack, Jan. 13, 20, 1898, repub- 
lished as No. 116 of Van Nostrand's Science Series) is the source of 
much that follows. 

The reason why an increase of pitch, other things being equal, or, 
in other words, an increase of the angle of the thread, gives in- 
creased efficiency reduced wear and longer life, will be understood 



Table 1. — Pitch Diameters for Circular Pitch Worms 



Worm, outside 


Pitch in inches 


diameter 


i 


A 


1 


* . 


4 


A 


i \ a \ i \ 


if 


i 


1-1 


I 1 


li 


li 


Ij 


ins. 


Pitch diameters 


I 


.8408 


.8011 


.7613 


.7215 


.6817 


.6419 


.6021 


.5623 


■ 5225 


.4827 


.4430 


.4032 


■ 3634 


.2838 


.2042 


.0451 


I. '8 


.903 


.864 


.824 


.784 


■ 744 


.704 


.665 


-62s 


■ 585 


■ 545 


.505 


.466 


.426 


■ 346 


.267 


.108 


Ij 


.966 


.926 


.886 


.846 


.807 


■ 767 


.727 


.687 


■ 647 


.608 


.568 


.528 


.488 


.409 


■ 329 


.170 


lr% 


1.028 


.989 


.949 


.909 


.869 


.829 


.790 


.750 


.710 


.670 


■ 630 


■ 591 


-551 


■ 471 


■ 392 


■233 


II 


1.091 


i.osi 


I. Oil 


.971 


.932 


.892 


.852 


.812 


.772 


■ 733 


.693 


.653 


.613 


■ 534 


■ 454 


• 295 


I A 


1.IS3 


1. 114 


1.074 


1.034 


■ 994 


■ 954 


■915 


.875 


-835 


.795 


.755 


.716 


.676 


■ 596 


■ 517 


■ 358 


If 


I. 216 


1. 176 


1. 136 


1.096 


I^0S7 


1. 017 


■ 977 


.937 


.897 


.858 


.818 


.778 


-738 


.659 


■ 579 


.420 


I ['a 


1.278 


1.239 


1.199 


1. 159 


1. 119 


1.079 


1.040 


I. 000 


.960 


.920 


.880 


.841 


.801 


.721 


.642 


.483 


H 


1-341 


1. 301 


1. 261 


1.221 


1.182 


1. 142 


1. 102 


1.062 


1.022 


-983 


■943 


■ 903 


■ 863 


.784 


.704 


■ 545 


Ii°s 


1.403 


1.364 


1.324 


1.284 


1.244 


1.204 


1. 16s 


1.12S 


1.08s 


1.045 


1.005 


.966 


.926 


.846 


■ 767 


.608 


If 


1.466 


1.426 


1.386 


1-346 


1-307 


I. 267 


1. 227 


1.187 


I. 147 


1. 108 


1.068 


1.028 


.988 


.909 


.829 


.670 


lU 


1.528 


1.489 


1.449 


1.409 


1.369 


1-329 


1.290 


1.250 


1.210 


1. 170 


1.130 


I. 091 


1.051 


■ 971 


.892 


.733 


li , 


I.59I 


I.5SI 


1. 511 


I -47 1 


1-432 


1-392 


1-352 


1.312 


1.272 


1-233 


I -193 


I- 153 


I. 113 


1-034 


■ 954 


-795 


I'n 


1.653 


1.614 


1-574 


1-534 


1-494 


1-454 


1.415 


I. 375 


1-335 


1-295 


1.255 


1 . 216 


1 . 176 


I. 096 


I. 017 


-858 


Is 


I. 716 


1.676 


1.636 


1-596 


1-557 


I-517 


1-477 


1.437 


1-397 


1-358 


1. 318 


1.278 


1.238 


I-IS9 


1.079 


.920 


I 


1.778 


1.739 


1.699 


1-659 


1. 619 


I-S79 


I 540 


1. 500 


I -460 


1 .420 


1.380 


I-341 


I-301 


I. 221 


I. 142 


.983 



132 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



■ Table 3. — ^Relation of Circular and Normal Pitches of Worm Wheels 
By Wm. Haughton {Amer. Mach., June i, 1911) 



Pitch diameter of 


Number of threads 


worm 


i normal pitch 


/b normal pitch 


1 


normal pitch 


15 normal pitch 


5 normal pitch 


ins. 


I 


2 1 3 


4 


I 


2 


3 


4 


I 


2 


3 


4 


I 


2 


3 


4 


1 


2 


3 


4 




.25075 


• 2530 


.2572 


.2623 


.3143 


.3185 


.3262 


• 3363 


•3777 


.3854 


.3983 


.4155 


.4419 


■ 4S40 


•4740 


.500 


• 5063 


.5246 


• 5539 


.5928 


n 


.2506 


■ 2525 


. 2556 


.2598 


• 3137 


.3174 


■323s 


.3315 


.3770 


■ 3834 


.3935 


.4073 


.4409 


.4509 


.4683 


.4881 .5050 


.5196 


.5431 


■5743 


li 


.25055 


. 2520 


. 254s 


.2580 


.3136 


•3164 


.3213 


• 3279 


• 3767 


.3817 


.3907 


.4012 


.4402 


• 4484 


•4613 


.4789 


• S040 


.5160 


.5352 


• 5611 


If 


.25042 


.2517 


■ 2537 


.2566 


.31334 


.31572 


.3197 


.3251 


• 3764 


.3805 


• 3875 


.3966 


• 4397 


.4464 


• 4571 


.4724 


• 5033 


.5132 


.5293 


.5509 


li 


.25037 


.2515 


• 2531 


• 2555 


.31318 


.3152 


•3184 


• 3233 


.3761 


■ 3797 


.3856 


.3935 


.4396 


.4449 


.4540 


.4666 


• 5028 


.5111 


• 5247 


•5431 


If 


.2503 


.2512 


.2527 


.2547 


.3131 


.3148 


.3176 


.3217 


.3760 


• 3791 


• 3840 


.39C8 


.4391 


.4438 


•4519 


.4625 


• S024 


.5094 


• 52II 


.5370 


I- 


.25025 


.25103 


.25236 


.2541 


.31303 


• 314s 


.3170 


.3205 


.37586 


•3785 


• 3828 


• 3887 


.4389 


.4429 


• 4498 


• 4591 


.502c 


.5082 


.5183 


• 5322 


li 


. 25022 


.2509 


.2520 


.2535 


.31298 


.3142 


• 3165 


• 3194 


.3758 


.3780 


• 3818 


• 3869 


.4387 


• 4424 


.4482 


• 4564 


.5018 


• 5071 


• Si6c 


• 5280 


2 


.25017 


.2508 


■ 2517 


.2531 


.3129 


.3140 


.3159 


• 3185 


.3756 


•3776 


• 3809 


• 3855 


.4385 


• 4417 


.4469 


•4541 


.5016 


.5062 


• 5140 


• 5247 


2j 


.25017 


• 2507 


•251S 


.2528 


.3128 


.3138 


.3155 


■ 3179 


•3756 


.3773 


.3802 


• 3842 


.4384 


.4412 


• 4459 


• 4523 


.5014 


.5055 


• 5125 


.5220 


2i - 


■25015 


.2506 


■ 2514 


.2524 


.3128 


• 3136 


.3152 


.3172 


• 3753 


.3771 


• 3796 


• 3833 


.4383 


.4409 


•4449 


.4507 


.5012 


.5048 


• 5111 


• 5 196 


2t 


.25014 


.25057 


.2512 


.2522 


.3127 


.3136 


.3150 


.3168 


.3754 


.3768 


.3792 


• 3825 


.4382 


• 4405 


.4442 


• 4493 


.5011 


.5045 


• 5099 


• 5176 


2 J 


.25012 


.2505 


.2511 


.2520 


• 3127 


■ 3135 


•3147 


.3164 


.3754 


.3767 


• 3788 


.3817 


.4382 


.4402 


• 4435 


• 4482 


.5010 


• 5040 


• SO90 


.5160 


2i 


.2501 


. 25047 


.25102 


.2518 


.3127 


.3134 


.3144 


.3161 


.3753 


.3765 


•3784 


.3811 


.4381 


• 4399 


• 4430 


.4473 


.5009 


.5037 


• 5081 


■ S144 


2| 


.2501 


.25042 


.2509 


.2516 


• 3127 


• 3133 


■3143 


• 3157 


• 3753 


.3764 


• 3781 


.3805 


.4380 


.4398 


• 4425 


.4464 


.5008 


• S033 


• 5074 


.5131 


2j 


.25007 


.25037 


.25085 


.2514 


.3126 


• 3132 


.3142 


.315s 


.3753 


.3762 


• 3779 


.3801 


.4379 


.4395 


.4420 


.4456 


.5008 


.5030 


.5068 


.5121 


3 


.25007 


.25035 


• 2507 


.2514 


.3126 


■ 3132 


•3139 


.3152 


• 3753 


.3762 


• 3776 


.3796 


.4379 


• 4393 


• 4417 


.4449 


.5007 


.5028 


.5063 


.5111 


3i 


.25007 


.2503 


. 2506 


.2512 


.3126 


• 3130 


•3138 


.3148 


• 3752 


• 3760 


• 3772 


.3790 


•4379 


.4391 


.4411 


.4439 


.5006 


.5024 


• 5054 


• 5095 


3J 


.25007 


. 2502 


. 2506 


.2510 


.3126 


.3130 


• 3136 


.3145 


.3751 


.3758 


• 3769 


.3785 


.4378 


.4389 


.4405 


■ 4430 


.5005 


.5020 


.5046 


.5081 


3i 


.25005 


.2502 


.2505 


■ 2509 


.3126 


.3129 


.3135 


.3142 


.3751 


•3757 


• 3767 


• 3780 


.4378 


■4387 


.4401 


.4423 


• 5004 


.5018 


• 5 040 


.5072 


4 


. 25005 


.2502 


.2504 


.2508 


.3126 


.3129 


■ 3133 


.3140 


.3751 


• 3757 


.3765 


• 3776 


.4377 


.4385 


• 4399 


• 4417 


.5004 


.5016 


• 5036 


• 5063 



Pitch diameter of 














Number of threads 














worm 


1 normal pitch 


i normal pitch 


1 normal pitch 


I in. normal pitch 


ins. 


I 


2 


3 


4 


I 


2 


3 


4 


I 


2 


3 


4 


I 


2 


3 


4 


I 


.6371 


.6727 


.7278 


.7988 


.7710 


.8310 


.9224 


1.037 


.9083 


1. 001 


1. 140 


1.310 


1.049 


1.18s 


1.382 


1. 619 


i\ 


.6346 


.6629 


.7079 


.7658 


.7667 


.8147 


.8891 


.9836 


.9014 


.9764 


1.089 


I. 231 


1.0393 


1.1488 


1.3115 


1 . 5 096 


li 


.6328 


.6558 


.6925 


• 7398 


• 7635 


.8028 


.8642 


.9443 


.8965 


.9579 


1.0524 


1.1719 


1.032 


•1 . 1 1 23 


1.258 


1.427 


If 


• 6315 


.6507 


.6816 


.7229 


.7611 


.7939 


• 8456 


.9131 


.8927 


.9440 


1.0238 


1. 113 


1. 0263 


1. 1019 


1.2175 


1.3630 


li 


• 6305 


.6465 


.6725 


.7078 


.7594 


.7870 


.8310 


.8892 


.8901 


.9333 


1.0148 


I . 0903 


1.0233 


I . 0863 


1.1855 


I^31IS 


i| 


• 6296 


.6435 


.6658 


• 6959 


.7581 


.781S 


.8195 


.8698 


.8879 


• 9249 


.9839 


1.0608 


I. 0188 


1.0739 


1.1597 


1^2705 


i| 


.6289 


.6410 


.6603 


.6866 


.7571 


.7773 


.8102 


.8542 


.8859 


.9182 


.9695 


1.0373 


I. 0164 


1.0641 


I. 1390 


I. 2371 


n 


.6286 


.6389 


.6559 


• 6789 


• 7561 


.7739 


.8028 


.8417 


• 8845 


.9128 


.9579 


I. 0176 


I. 0142 


1.0560 


I. 1223 


1.2087 


2 


.6280 


.6372 


.6522 


.6727 


.7553 


.7710 


.7966 


.8309 


• 8835 


.9083 


.9484 


I. 01 14 


1.0125 


I . 0494 


I. 1079 


1.1855 


2i 


.6277 


.6358 


.6492 


.6673 


■ 7546 


.7687 


.7914 


.8222 


• 8825 


.9046 


.9401 


.9879 


I. 0112 


I . 0440 


1.0963 


1.1569 


2i 


.6274 


.6348 


.6466 


.6629 


.7542 


.7668 


.7870 


.8147 


.8816 


.9015 


.9333 


.9763 


l.OIO 


1.0392 


1 . 0863 


I. 1489 


2| 


.6272 


.6336 


.6444 


.6591 


.7537 


.7649 


• 7833 


.8083 


.8809 


.8987 


.9277 


.9665 


1. 009 


1.0354 


1.0778 


1.1334 


2i 


.6270 


.6328 


• 6425 


.6559 


.7533 


.7634 


.7802 


.8028 


.8803 


.8963 


.9226 


.9578 


1.008 


1.0318 


1.0704 


1 . 120 


2| 


.6267 


.6321 


.6409 


.6533 


.7531 


.7623 


.7773 


.7981 


.8799 


.894s 


.9182 


.9504 


1.007 


1.0289 


I. 064 


I. Ill 


2l 


.6266 


.631S 


.6394 


.6506 


• 7528 


.7612 


.7750 


.7939 


• 8794 


.8928 


.9145 


.9440 


1.006 


1.0263 


1.0586 


1. 102 


2i 


.6265 


.6309 


.6385 


.6484 


.7526 


.7602 


.7729 


.7901 


.8791 


.8912 


.9111 


.9386 


1.006 


1.024 


1-537 


1.0936 


3 


.6261 


.630s 


.6372 


.6466 


.7524 


• 7593 


.7710 


.7870 


.S788 


.8895 


.9081 


.9333 


1.0057 


1.0222 


I . 0494 


1.0863 


3i 


.6261 


.6296 


■ 6354 


.6435 


.7520 


.7579 


.7679 


.7816 


• 8782 


.8878 


.9034 


• 9249 


I . 0048 


I. 0190 


1.0422 


1.074 


3\ 


.6260 


.6289 


.6340 


.6409 


.7517 


• 7568 


.7655 


.7773 


.8778 


.8859 


.8995 


.9182 


1.004 


I. 0163 


I . 0366 


I. 0641 


3i 


.6258 


.6285 


.6328 


• 6389 


■ 7515 


• 7560 


.7636 


.7740 


.8773 


.884s 


.8965 


.9128 


1.003 


I . 0144 


1.032 


1.056 


4 


• 6257 


.6281 


• 6319 


.6372 


• 7513 


• 7553 


• 7619 


.7710 


.8771 


.8834 


.8938 


.9082 


1.003 


I. 0125 


1.028 


1.049 



from Fig. 2. If ah be the axis'of the worm and cd a line represent- 
ing a thread, against which a tooth of the wheel bears, it will be seen 
that if the tooth bears upon the thread by a pressure P, that pres- 
sure may be resolved into two components, one of which, ef, is per- 
pendicular, while the other, eg, is parallel to the thread surface. The 
perpendicular component produces friction between the tooth and 
the thread. The useful work done during a revolution of the thread 
is the product of the load P and the lead of the worm, while the 
work lost in friction is the product of the perpendicular pressure ej, 
the coefficient of friction and the distance traversed in a revolution, 
which is the length of one turn of the thread. Now, if the angle of 
the thread be doubled, as indicated, the load P remaining the same, 
the new perpendicular component fh of P will be slightly reduced 
from the old value ef, while the length of a turn of the thread will 



be slightly increased. Consequently their product and the lost 
work of friction per revolution will not be mtich changed. The 
useful work per revolution will, however, be doubled, because, the 
pitch being doubled, the distance traveled by P in one revolution 
will be doubled. For a given amount of useful work the amount 
of work lost is therefore reduced by the increase in the thread angle, 
and, since the tendency to heat and wear is the immediate result 
of the lost work, it follows that that tendency is reduced. For small 
angles of thread the change is very rapid, and continues, though in 
diminishing degree, until the angle reaches a value not far from 
45 deg., when the conditions change and the lost work increases 
faster than the useful work, an increase of the angle of the thread 
beyond that point reducing the efficiency. 

These general principles have been given mathematical expres- 



WORM GEARS 



133 



sion by Prof. J. H. Baer {Amer. Mach., Jan. 13, 1898). Assuming 

frictionless bearings — a condition that is nearly fulfilled by ball 

bearings — the formula for the efi&ciency of a worm gear is : 

_tan a (i —/tan a) 

tan a + f 

in which 

e = efficiency, 

a = angle of thread, being the angle dfi of Fig. 2, 

/ = coefficient of friction. 

Assuming a plain step bearing of the collar type having the same 

mean friction radius as the worm, the formula becomes: ^ 

tan a (i— /tan a) . 

e = — — T i 7 — (approximately). 

tan a + 2/ ^ ^'^ ^ 

Notation as before. 

Note that a worm being essentially a screw these formulas and the 
following chart, Fig. 3, apply also to the efficiency of screws. 

In order to present to the eye a picture of the meaning of 
these formulas, Fig. 3 has been plotted from them. 

The scale at the bottom gives the angles of the thread from o to 
90 deg., while the vertical scale gives the calculated efficiencies, the 
values of which have been obtained from the equations and plotted 
on the diagram. The upper curve is from the first equation, and 
gives the efficiencies of the worm thread with a frictionless step; 
while the lower curve, from the 
second equation, gives the combined 
efficiency of the worm and step. In 
the calculations for the diagram it is 
necessary to assume a value for /, 
and this has been taken at .05, 
which is probably a fair mean value, 



worms are used the efficiency of the transmission, as such, is of very 
little account. What the designer concerns himself with is the 
question of durability and satisfactory working, and the results to 
be expected in this respect are best shown by the upper curve, in 
which high efficiency means a durable worm. 

The chief significance of efficiency in this connection is that since 
lost power is expended in friction and wear, low efficiency means 
rapid wear and high efficiency the reverse. 

The above conclusions are confirmed by the well-known experi- 
ments of Wilfred Lewis for Wm. Sellers & Co. {Trans. A.S.M . E., 
Vol. 7). The small crosses plotted on Fig. 3 represent the results of 
such experiments as developed the same coefficient of friction as 
that used in plotting the curves from the above equations. The 
experimental results will be seen to have a very satisfactory agree- 
ment with the lower curve with which they should be compared, as 
the step bearings used by Mr. Lewis were of the plain pattern without 
balls. 

Similar high efficiencies have been obtained repeatedly and most 
recently by Prof. Wm. K. Kennerson for the Brown & Sharpe 
Mfg. Co. {Trans. A. S. M. E., Vol. 34). Using worms of 45° and 
38° 16' helix angle with ball step bearings on both worms and wheels, 
Professor Kennerson obtained efficiencies as high as 97! per cent. 

The articles by the author, above referred to, include particulars 




100 : :::-:::::-:::::-::-:::::::-::::-::---:::-::::-::::■"-::::::::::::-::::::::: 












^==-- __---._-=_-,__;^ =--,._ ^^__ 






80 -?'- -j;»' ""=:;: :; ::: 


"" ;'"- — ,-' " "" " — — ^U - 


m * ^Kl r^b 


t ^ - %\ - - - 




'0: ::"' r--7 : ..V . _ 


— I ■' ,' IIS- I 




a ? " ;' — " " _ _ - __ -A -- 


Gee -• / _ _ _ - -- - -- -- ;.;_.H__ .;:;i. 


„bU---7--,r,' --■ --.-\- ------ 


j< : 2:::_:.:::: : :: : _ _::r__ :i:_ ' 


«< ., y___i _ ___ 


0< 1 ..' _ -- _- -- _ _ __ _ - 




ffoO 1 S 


c : I" " " " " I __: _" " hi:; ^ a" ^ :: :: : : : _ ? ::i :: 


at a 


u f - _ _ -^_ - - ^ -' _--_ - 


9 ,in 1 , ?. , ■f', \ 


iE *" -. ./_ -._ --M - -- - 


tt^ _X_L_ _g g„ 


" :i :..iiii_.i_ii _: " 11 i" "" a S~ " P i "' i- -ii: -__: i i i ---:- r :: : 


J : _;; .Sc ".S3 — _' -- 


in u >A V U \ 


*' — I — ::::::::::::: — S S 3 3 :i : : : : :: ^ _ _ 


__:: — ::__;_ --__bo 5!o-__:_ :__ ___ C_ 


_ ----si^ '^ -^ _ 


_ _.ii 0-1 t ■" 5- 


20. P tt I ° EI II II 


iu J — ... _ ^^ _ .. 


_;:_ -----s^-^^f^ ' \ 


1/ M U to >l ■ \ 


/ " " f S \ 1 


10 t III I 11: . "" ^ £ ""■" -S kL " ~ " " ' . -i.. 


1 1^ ■ n \ 


C- __ __2;3 o>i3 - -\ 


c _ __±t"g 3" a J 




t x::~i:::::::::--+i::-i:z:±i.^:i::iM^.:::i:ii::i:±:::i + :::±z.:^im:±: 



10 



20 



30 



40 50 60 

Angle of Thread- Degrees 



70 



80 



90 



Fig. 2. — The principle of worm 
gear efficiency. 



Fig. 3. — Relation of helix angle and efficiency of worm gearing. 



although, since it varies with the rubbing speed, no single curve 
can represent all conditions. 

The curves will be seen to rise to a maximum and then to drop. 
The values of the helix angle a for maximum efficiency as found 
from the foregoing equations by the methods of the calculus are: 

For a frictionless step bearing, a = 43° 34'. 

For a plain step bearing, a = 52° 49'. 

Of more importance than the angle of maximum efficiency is the 
general character of the curves, of which the most pronounced pecu- 
liarity is the extreme flatness, showing that for a wide range of angles 
the efficiency varies but little. Thus, for the upper curve there is 
scarcely any choice between 30 and 60 deg. of angle, and but 
little drop at 20 deg. 

At first sight the lower curve might be thought the more useful 
of the two, as it includes the effect of the step, but a little consider- 
ation will show that this is not the case. For most cases in which 

1 In both formulas the worm thread is assumed to be square in section. 
Thread sections in common use affect the results but little. 



of 18 worm drives of various helix angles doing heavy duty, some of 
which were successful while others failed from rapid wear. Sum- 
ming up the results of the investigation it was found that all worms 
having helix angles greater than 12° 30' were successful, that all 
having angles less than 9° were failures from rapid wear, while be- 
tween these angles some were successful and some failures. In 
several cases unsuccessful worms of low angle had been replaced 
with others of high angle with the result of changing failure to success. 
The prevailing materials for worm gearing are hardened steel for 
the worm and bronze for the worm wheel. Referring to the examples 
collected by the author and already cited, of eleven successful gears 
doing heavy duty, five had bronze and six cast-iron gears, and, 
moreover, numerous other cases of successful cast-iron wheels 
could be cited. The Sellers planer drive, which is essentially a worm 
drive and which has been successful through a long series of years, 
always has the rack of cast-iron. The conditions under which worm 
gearing operates, especially if an oil bath is used, as it should be, 



134 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



regardless of the materials, would seem to be especially favorable 
to the well-known glazing action of cast-iron bearings. 

It may be concluded that, given suitable dimensions for the work, 
worms having helix angles exceeding 15 deg. or better 20 deg. and 
running in oil baths should transmit continuous loads with good and 
even high efficiency and long life. 




ft78 1.41 
0.2G 



2.77 5.40 

Sliding Velocity at Pitch Circle 



8.61 



Sliding velocity at the pitch circle of the worm in meters per second ; 
differences of temperature in deg. Cent. Pressures in kilograms. 

Fig. 4. — Relation of pressure and velocity at observed differences 
of temperature in worm gearing. 

Load Capacity of Worm Gears 

The loads to be carried by worm gearing depend upon the dimensions 
of the gearing and the speed. 

The law connecting the speed, pressure and temperature was dis- 
closed in experiments by Profs. C. Bach and E. Roser 
{Zeitschrift des Vereines Deutscher Ingenieure, 1902, atid Amer. Mack., 
July 16, 23, 1903). This law is plotted in metric measures in Fig. 4 
from which unfortunately, the scale of pressure is omitted. For com- 
parative purposes it may be scaled. The two curves are for 
observed differences of temperature (centigrade) as noted, between 
the oil cellar and the surrounding air. 

Professors Bach and Roser deduced formulas for the perform- 
ance of worm gearing from their experiments, these formulas trans- 
lated into British units being as follows: 

P = i4.23s[cf {to-ta)+d]bp 
in which 

P = axial thrust on worm, lbs., 
to = temperature in oil cellar, Fahr., 
la = temperature of surrounding air, Fahr., 
b = breadth of wormwheel teeth measured on the arc at the 

roots of the teeth, ins., 
p = Pitch (not lead in the case of multiple thread worms) ins., 

13-17 , 

--f.4192, 



d = 



V 

21,476 



-24.92, 



= circumferential velocity at pitch line of worm, ft. 
min.i 



per 



In the experiments the included profile angle of the worm thread 
was 29 deg., to which angle the application of the formula is prop- 
erly limited. Flooded lubrication was used in the experiments and 
is assumed in the formula. 

^ In the original papers v was given as the sliding velocity at the pitch 
line, but a checking of the calculations shows that the quantity actually used 
was the circumferential velocity. 



Fig. 5 by Prof. J. B. Peddle (Amer. Mach., Jan. 23, 1913) has 
been constructed to give the same results as the Bach and Roser 
formula. The use of the chart is shown by the example below it. 
Several trials will usually be required to find suitable proportions of 
pitch to face. 

In the use of the formula or chart, the temperature rise to be 
permitted must be determined by the designer. There does not seem 
to be any cause for alarm at a considerable rise if suitable oil be used. 
In Professors Bach and Roser's experiments the extreme rise was 
95 deg. C. (171 deg. Fahr.), while in experiments by Professor 
Kennerson {Trans. A. S. M. E., Vol. 34) a rise of 225 deg. Fahr. 
was experienced repeatedly, and in one case a rise of 322 deg. Fahr., 
the room temperature ranging between 66 and 88 deg. 

In the former experiments the oil used was "an extremely ^viscous 
steam cylinder oil" while Professor Kenerson used an oil intended for 
use with superheated steam. In both sets of experiments the worms 
were flooded, as they always should be. 

There is no doubt that bearings frequently operate at higher tem- 
peratures than is commonly believed and without giving trouble. 
Ordinary bearing oil loses its lubricating qualities at a temperature 
of about 250 deg. Fahr. See Index. 

Professor Kenerson's experiments show that, when subjected to 
varying loads, worm gearing need not be designed for the greatest 
load. The final temperature of the oil cellar was not reached until 
the lapse of two, and in some cases three, hours, while abnormally 
high loads were carried for an hour and more without failure. The 
uniform experience was that, while the rise of oil temperature was 
rapid at the start, it became more gradual as the rim continued. 

The materials used in Professors Bach and Roser's experiments 
were unhardened steel for the worm, and bronze for the wheel. In 
Professor Kenerson's experiments the worms were of case-hardened 
machinery steel, and the wheels of various grades of bronze, among 
which no great differences were observed so far as the rise of tem- 
perature was concerned. 

Good reason exists for doubting the correctness of Professors 
Bach and Roser's fundamental analysis and hence of the formula 
and chart based upon it. They are, however, included here be- 
cause they embody the best existing information at the time of 
writing. 

As a contribution to this chapter, the Hindley Gear Co. (successors 
to Morse Williams Co.) have placed at my disposal their method 
of determining the dimensions of their well-known Hindley worms 
which are based on the most extended experience with this gear 
extant. 

The method is based on the use of a constant for the product of 
the velocity and unit pressure, this constant being chosen to suit 
the conditions and being thus largely a matter of experience and 
only partially capable of reduction to a formula. The initial as- 
sumption is that the load is carried by two teeth and that 70 per 
cent, of the area of these teeth is effective bearing area. The area 
of the teeth is determined from the drawing and multiplied by .7 
when the chosen constant is substituted in the formula 

in which 

^ = pressure on effective area, lbs. per sq. in., 

D = circumferential speed of worm at the throat diameter 

pitch line, ft. per min., 
C = constant. 

For the most unfavorable condition, that is, for a steady load 
acting continuously and for an indefinite period of time, a conserva- 
tive value is C= 250,000. For intermittent loads much larger values 
may be taken for C — an experience that is directly in line with Pro- 
fessor Kenerson's experimental determination that abnormally high 
loads may be carried for considerable periods. In general, for inter- 
mittent loads C may be increased in the inverse proportion which 



WORM GEARS 



135 



0- 


r 


1000-^ 


r 100 


1T)00-^ 


^ 200 


-. 


r 300 


2000 -: 


E- 400 


■i 


E- 500 


3000^ 


- 600 


-. 


r 700 


4000 - 


- 800 



5000- 



COOO- 



7000- 



8000- 



9000- 



10,000. 




1000-3_iooo 

fOntsoo .3 
'OOf-yoo 

4ooJ: '^ 




300-1-300 



250 — 250 



z 



200- 






100^,^ 100 



5 f^ 50-$- 50 



O :: 



V 



150 



g, 100 100 

s :: 



50- 



-200 



50 



Join oil cellar and air temperatures and note intersection with axis X; connect this intersection with the velocity and note intersection 
with axis F. Any two lines from this point and from the thrust which intersect on axis Z will give the required pitch and breadth of teeth. 
For light loads use light faced figures on both thrust and breadth scales ; for heavy loads use heavy faced figxures on both scales. The ex- 
ample is for oil -cellar temperature = 190 deg., air temperature = 60 deg., velocity = 200 ft. per min., thrust = 11 14 and 5570 lbs., giving 
pitch = 2 ins. and breadth = i and 5 ins. 

Fig. 5. — Dimensions of worm gears. 



the load time bears to the total time. Thus should the load time be 
one-half the total time, C becomes 250,000 X 2 =500,000, while should 
the load time be one-quarter of the total, C becomes 250,000X4 = 
1,000,000. In extreme cases of intermittent loads applied for short 
periods and at large intervals, C may reach as large a value as 
3,500,000. 

In the matter of materials the timid are disposed to favor the 
general use of bronze for the gear. The experience of the Hindley 
Gear Co. has led chem to the following choice of materials. 

For pressures per sq. in. of effective area not exceeding 350 lbs. 
and velocities not exceeding 500 ft. per min., cast-iron for both 
worm and gear. 

For velocities of 1000 ft. per min. and over, the pressure per sq. 
in. of effective area being 350 lbs. for steady or 500 lbs. for inter- 
mittent service, openhearth steel of about 35 points carbon for the 
worm and bronze for the wheel. 

For very heavy thrusts (say 10,000 lbs.) and low velocities (300 
ft. per min. or less) openhearth steel as above for the worm and 
Cramp's special gear bronze for the wheel. 

Worm-gear cases and, for that matter, all gear cases, should be 
provided with vents; if this is not done the expansion of the air by 
the heat will drive the oil out through the bearings. The action 
repeats itself every time the gearing is started from the cold state 
and ultimately empties the case of most of its oil. 



Tooth Parts of Worms and Hobs 

The tooth parts of circular pitch worms and hobs of the Brown and 
Sharpe standard may be taken, for most cases, from Table 4 by 
Geo. W. Jager {Amer. Mach., June 18, 1914). The table is based 
on the formulas given above it which may be used for cases not 
included in the table. 

Milled Worm Wheels 

Considerable use has been made of worm wheels cut with spur gear 
cutters, the teeth being identical with those of spur gears except that 




Fig. 6. — Milled worm wheel. 

they are cut at an angle with the axis equal to the helix angle of the 
worm as shown in Fig. 6. Very little published information regard- 



136 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



K--P— H 



W-._P_^ 




WORM 



HOB 



Table 4.- — Tooth Parts of Worms and Hobs 

A =thickness of tooth on pitch line = .50 P., 

B = width at top of thread = .3354 P., 

C = clearance = .05 P., 

D = working depth of tooth = .6366 P., 

£ = total depth of tooth = .6866 P., 

F = width of thread tool at end = .3095 P.. 

C = depth of space below pitch line = .3683 P., 

i? = tooth above pitch line = .3i83 P., 

P = circular pitch (not lead in multiple thread worms). 



p 


A 


B 


C 


D 


E 


F 


G 


- 


Threads 
, per in. 


iH 


.8750 


.5869 


• 0875 


I. 1141 


I. 2016 


.5416 


• 6445 


•5570 


^-i 


iH 


.8125 


■ 5450 


.0812 


I . 0444 


I.II57 


• 5029 


■ 5984 


.5172 


M3 


1 1/2 


• 75 


■ 5031 


■ 075 


• 9549 


I .0299 


• 4643 


• 5525 


■4775 


% 


iH 


.6875 


.4611 


• 0687 


• 8753 


■ 9440 


• 4255 


.5063 


.4376 


Ml 


iH 


.62s 


■ 4193 


■ 0625 


• 7958 


■ 8583 


.3869 


• 4604 


■3979 


^5 


iH 


.5625 


• 3773 


.0562 


.7162 


• 7724 


.3482 


• 4143 


• 3581 


% 


I 


• SO 


• 3354 


• 050 


.6366 


.6866 


.3095 


• 3683 


.3183 


I 


1^6 


.4687 


■ 3144 


.0468 


.5968 


.6436 


.2901 


•3452 


.2984 


I Ms 


% 


■ 4375 


.2934 


.0437 


• 5570 


.6007 


.2708 


• 3222 


■ 2785 


iW 


'Me 


.4062 


.2724 


.0406 


• S172 


.5578 


• 2514 


.2992 


.2586 


iMa 


% 


.375 


• 2Si6 


• 0375 


.4775 


• S150 


• 2321 


.2762 


.2387 


iH 


'He 


■ 3437 


■ 230s 


.0343 


• 4376 


• 4720 


• 2127 


• 2533 


.2188 


iMx 


5i 


■ 3125 


.2096 


.0312 


• 3979 


.4291 


• 1934 


.2302 


. 1989 


iH 


H 


.3333 


.2236 


0333 


• 4244 


• 4577 


• 2063 


• 2455 


.2122 


iH 


Me 


.2812 


.1886 


.0281 


• 3581 


• 3862 


.1741 


.2071 


.1790 


iH 


1/2 


■25 


.1677 


.025 


• 3183 


• 3433 


• 1548 


.1842 


.1592 


2 


Vie 


.2187 


■ 1467 


.0218 


• 278s 


• 3003 


■ 1354 


. 1611 


.1392 


2^ 


H 


.2000 


■ 1341 


.0200 


• 2546 


.2746 


.1238 


■1473 


■1273 


2H 


H 


• 187s 


• 1258 


.0187 


• 2387 


• 2575 


. 1161 


.1381 


■I 194 


2H 


Me 


. 1362 


• 1048 


• 0156 


.1989 


• 2145 


.0967 


.1151 


■0994 


3 'A 


H 


.1666 


.1118 


.0166 


• 2122 


.2289 


.1032 


. 1228 


. I06I 


3 


?^ 


.1429 


.0958 


.0143 


■ I819 


. 1962 


.0884 


. 1052 


.0909 


3H 


H 


■125 


.0839 


.0125 


.1592 


• 1717 


.0774 


.0921 


.0796 


4 


H 


.1111 


• 074s 


.0111 


• 1415 


• 1526 


.0687 


.0818 


■ 0707 


4H 


Me 


■ 0937 


.0629 


.0094 


• I 193 


. 1287 


• 0580 


.0690 


•0597 


5]A 


Vs 


. 10 


.0671 


.010 


.1273 


• 1373 


.0619 


• 0737 


■0637 


5 


H 


.0833 


.0559 


.0083 


.1061 


.1144 


.0516 


.0614 


■ 0S3I 


6 


M 


.0714 


.0479 


.0071 


.0909 


.0981 


.0442 


.0526 


■0455 


7 


H 


.0625 


.0419 


.0062 


.0796 


• 0858 


.0387 


.0460 


■ 0398 


8 


'A 


.0556 


■ 0373 


.0056 


.0707 


.0763 


• 0344 


.0409 


•0354 


9 


Mo 


■ 050 


• 0335 


.0050 


• 0637 


• 0687 


• 0310 


.0368 


■ 0318 


10 


M2 


■ 04I6 


.0279 


.0041 


• 0530 


• 0573 


.0258 


• 0307 


.0265 


12 


Hi 


■ 0357 


.0239 


• 0035 


• 0454 


.0490 


.0221 


.0263 


.0227 


14 


Me 


.0312 


.0209 


.0031 


.0398 


.0429 


.0193 


.0230 


.0198 


16 


Ms 


.0278 


.0186 


.0027 


• 0353 


.0381 


.0172 


• 0204 


.0177 


18 



ing this construction is available, but there is excellent reason for 
believing that such gears are as satisfactory as those cut with hobs, 
a conspicuous example of long-continued success being found in the 
Sellers planer drive. For occasional constructions this type of 
wheel offers the advantage that the cost of the hob is saved. 

An examination of Fig. 6 wiU show that the depth of the spaces 
diminishes as the cut approaches the edges of the wheel. This effect 
increases with increase of the angle of the cut and of width of wheel 
face and with decrease of the diameter of the gear. With high helix 
angles it may lead to interference and, in such cases, a careful lay out 
should be made before deciding on this construction. 




Table 5. — Constants for Use with the Wire System of Meas- 
uring THE Brown and Sharpe 29 Deg. Worm Thread 

For particulars of this method see index 
Wire diam. for given pitches 



Pitch, 


Wire diam., 


m. 


m. 


2 


1.0298 


iM 


0.9010 


iH 


0.7723 


^H 


0.6436 


I 


0.5149 


M 


0.3862 


M 


0.2574 


H 


0. 1716 


H 


0. 1287 


H 


0.1030 


H 


0.0858 


y, 


o^o73S 


M 


0.0643 



WORM GEARS 



137 



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On 



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ri 


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ft 


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to 


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to 
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oil 


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^'O 



HELICAL (COMMONLY MISCALLED SPIRAL) GEARS 



0} the load-carrying capacity of helical gears the author has no data. 

Note that in what follows the angle of the helix is taken as the angle 
between the teeth and the tangent to the pitch circle — that is, as the 
angle kal, Fig. 4. There is no uniformity of practice in this notation, 
some writers using the complement of this angle — that is, the angle 
lar, Fig. 4. This difference of practice should be kept in mind when 
comparing formulas from different sources. 

There is no geometrical difference between worm and helical gear- 
ing. This fact is illustrated in Fig. i of which the example in the 
immediate foreground is plainly a case of worm gearing, while that 
in the far background is as plainly a case of helical gearing. The 
difference between them is, however, one of degree only, as shown by 
the intermediate constructions. 

Such difference as there is relates to the method of production. 
Worms are commonly cut with threading tools in a lathe, the pitch 
with which we are chiefly concerned being that parallel with the axis 
— the axial pitch — which, in the case of the worm wheels, becomes the 
circular pitch, precisely as in spur gears. The section through the 
worm center line of a worm and gear in mesh is the same as the section 
of a rack and gear in mesh. Helical gears, on the other hand, are 
cut with cutters in a milling machine, the pitch with which we are 
chiefly concerned being that perpendicular to the teeth — the normal 
pitch. 



Helical Gears of 45 deg. Helix Angle on Shafts at Right Angles 
by Calculation 



Calculations of helical gears of 4.$ deg. helix angleonshafts alright angles 
may be made for most cases by the use of Table i by E. J. Kearney 
{Amer. Mach., June 29, 191 1). 

The law connecting the durability and efficiencywith the helix angle 
of worm gearing, which see, applies also to helical gearing. For all 
practical purposes the angle of maximum durability and efficiency is 
45 deg. Helical gears having this heUx angle are also the easiest of 
all to calculate and to make. They are, hence, deservedly popular. 
The speed ratio of such gears is the same as that of spur gears — that is, 
the speeds are inversely as the diameters and as the number of teeth, 
the nimibers of teeth being proportional to the diameters, and the 
pitch-line speeds of mating gears are equal. Such gears should be 
used whenever possible, there being, in fact, little reason for using 
other angles except in cases when the required speed ratio cannot be 
obtained with 45 deg. gears on the given center distances. 

The calcidation of helical gears not found in Table i begins with 
findingthelengthof the normal helix, that is, the length of a line equal 
to the normal pitch multiplied by the number of teeth. The normal 
pitch of helical gears is determined by the cutter used and is the same 
as the circular pitch of spur gears. In spur gears the circular pitch 
multiplied by the number of teeth equals the circimiference, from 
which it is plain that, in the calculations, the normal helix of helical 
gears is analogous to and takes the place of the circumference in spur 

circumference 
gears. Similarly, in spur gears, = diameter, and in 



helical gears, 



length of normal helix 



= a quantity which has no name 



but which is analogous to, and, in the calculations, takes the place of, 
the diameter in spur gears. 



The author has shown {Amer. Mach., Nov. 28, 1901. Van Nost- 
rand's Science Series No. 116) that for45-deg. heUcal gears: 
li 1.4142C 

{a) 



1 + 



r\ 
ri 



in which /i = length of normal heUx of driver, ins., 
C = distance between centers, ins., 
ri = r.p.m. of driver. 
r2 = r.p.m. of follower. 

—being the quantity which takes the place of the diameter in spur- 
gear calculations as already noted. 




Fig. I. — :Helical and worm gears. 
The author has also shown (same references) that for 45 deg. gears: 

7t (6) 



di- 



.7071 



in which (ii = diameter of driver, ins. 

The use of these formulas is best shown by an example: 



Let 



and 



15 , 
C=4 — = 4.468 ms. 
32 
ri 
ri 



Inserting these values in (u) we have: 

^1 1.4142X4.468 

= 1.2637 ins. 

Assuming next that a 6 diametral pitch cutter is to be used, we 
multiply this quantity by the diametral pitch to find the number of 
teeth, precisely as we multiply the diameter of a spur gear by the 
diametral pitch and obtain : 

number of teeth in driver = 1.2637X6 
= 7-58 

(Contirtued on page 140, first column) 

138 



HELICAL GEARS 



139 



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3 



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01 


+ 






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2 A 



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o ft < ft 2: 1^ < <; 



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pitch 
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ickness 
ooth at 
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140 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



o o o 



8075 70 65 60 55 



50" 



Helix Angle of the Helical Gear 
45 40 35 30 



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Locate the intersection of the lines for the helix angle and the number of teeth. The number in the area in which the inter- 
section falls is the cutter number required. 

Fig. 2. — Cutters for helical gears. 



This number is fractional as it almost invariably is. A fractional 

niunber of teeth is, of course, impossible and the obtaining of such a 

number shows that the assumed conditions are impossible and that 

they must be changed. More specifically, the center distance must 

be increased to admit 8 teeth or reduced to provide for 7. Adopting 

8 teeth, the result is to change the length of the normal helix from the 

trial value first foimd. As in spur gears: 

number of teeth 

diameter = —r. .^ , 

diameter pitch 

So in helical gears: 

true normal helix number of teeth. 



diametral pitch 



= 5 = 1-333 ms. 



instead of the trial value 1.2637 ins., as first found. Inserting this 
true value in (6) we have : 

di = -^^ =1.885 ms. 

.7071 

as the diameter of the driver. Since the numbers of teeth and the 

diameters of the two gears are inversely as the speeds we have: 

Number of teeth in follower = 8X4 = 32 

Diameter of follower = tf 2 =1.885X4 = 7.540 ins. 

T- ,1 ^ di+di 1.885 + 7.540 

Finally C= = ^ ' ^ -^ = 4.7125 ins. 

= corrected center distance that must be used. 



To find the outside diameter of the blank add -r: — -, — r— rto the 

■' ' diametral pitch 

pitch diameter. 

To find the Brown and Slmrpe cutter to be used, multiply the number 

of teeth in the gear to be cut by 2.83 ( = sec.' 45°) and select a cutter 

for the resulting niunber of teeth or use the chart. Fig. 2, by A. E. 

Larsson {Amer. Mack., July 2, 1908), the use of which is explained 

below it. 
Helical Gears of 45 Deg. on Shafts at Right Angles by Graphics 
The results may be checked by a graphical construction. Fig. 3, 

the use of which is explained below it. 

Helical Gears of any HeUx Angle on Shafts at Right 
Angles by Calculation 

The solutioti of helical-gear problems for helix angles other than 45 
deg. depends upon the same principle — finding the length of the nor- 
mal helix to contain the required number of teeth- — the process 
always involving a trial and a final solution. A clear understanding 
of the methods involves a knowledge of fundamental principles. 

Fig. 4 is a conventional representation of a helical gear with its 
pitch cylinder prolonged. One of the teeth is also prolonged to 
make a complete turn around the cylinder, the resulting curve, abcdef, 
being the tooth helix. For purposes of calculation this helix is defined 
by the angle kal between a tooth and the tangent to the pitch cylinder. 
For shop purposes it is more commonly defined by the length af of 



HELICAL GEARS 



141 





Lay down ab = twice the assumed center distance. 

= 2 X 4.468 = 8.936 ins. 
Divide ab at c into two parts, the lengths of which are in the ratio 
of the speeds, that is 

cb : ca : : I : 4 
or cb = 1/5X8.936 = 1.7872 ins. 

and ca = 4/5X8.936=7.1488 ins. 

Draw de at an angle of 45 deg. with ab and draw ad and be at 
right angles with de, giving cd and ce which equal the 
trial lengths of normal values 

Scale cd and ce and multiply them by the diametral pitch, that 
is: 

Scale length 1.26x6= 7.56= Trial number of teeth in driver. 
Scale length 5.04X6 =30.24 = Trial number of teeth in follower. 
The results being fractional, increase (or decrease) them to 
the nearest whole numbers having the given ratio of I to 4, that 
is, increase them to 8 and 32. Divide these numbers by the 
diametral pitch 

8/6 = 1.333 ins. 
32/6=5-333 ins. 
and lay down cd' =5.333 ins. and ce' = 1.333 ins. Draw d'a' 
and e'b' perpendicular to de and we have : 

ca' = diameter of gear = 7.540 ins. 

c'b' = diameter of pinion = 1.885 ins. 

a'b' = twice the center distance = 9.425 X ins. 

Fig. 3. — Graphical construction for helical gears of 45 deg. helix 

angle. 



nearly a right angle. It is apparent from this illustration that the 
length of the normal heUx from atod takes in all the teeth and that 
ao, multiplied by the number of teeth, must equal ahpd and not 
ahpq. This length ahpd is always less than ahpq, and usually much 
less. Fig. 6 ^ is a development of the gear end of Fig. 5 on a reduced 
scale, ad being the developed length of the normal helix. Fig. 6B 
and Fig. 6 C show how with the same circumferential pitch and the 
same niunber of teeth but a reduced value of the angle kal, the 
length of the normal helix, which cuts all the teeth, grows shorter until 
it may make but a small part of a complete turn around the cylinder. 
It is clear that in all cases the line ad cuts all the teeth precisely as 
does the circumference aa, which goes completely around the cylinder. 
It is also clear that if the normal pitch is decided upon at the start, a 
diameter of cylinder and a helix angle must be found such that the 
normal pitch, multiplied by the number of teeth, shall equal the length 
of the normal helix between two intersections with the tooth helix. 

It is natural to ask: Why not employ the circumferential pitch and 
so deal directly with the circumference instead of the normal helix? 
Because we do not know what it is. The normal pitch is determined 
by the cutter used, while the circumferential pitch depends also upon 
the helix angle, and until this angle is known the circumferential 
pitch is not known. 

In the extreme case of a helical gear in which the helix angle is so 
small that the gear becomes a single thread worm, as in Fig. 7, points 
and d of Fig. 5 coincide and the length of the helix between a and d 
becomes the normal pitch. It is, however, true as before that the 
normal pitch, multiplied by the number of teeth, which is now one, 
is still equal to the length of the normal helix betweed two intersec- 
tions with the tooth helix. 

A glance at Fig. 6 will show that in gears of the same diameter the 
length of the normal helix^ grows shorter as the angle kal grows less, 
and hence that it and its gear will contain successively fewer and fewer 
teeth of the same normal pitch. That is to say, the number of teeth 
in a gear varies with the helix angle as well as with the diameter and 
the number of teeth in two gears of the same normal pitch is not necessarily 
proportional to the diameters. In fact, it is never so proportional, 
except when the angle kal is equal to 45 deg. The diametral pitch of 
the cutters and the diameter of the gear thus do 7iot determine the num- 
ber of teeth. 

Fig. 8 illustrates the simplest possible case of a pair of helical gears. 
The shafts are at right angles, the gears are of equal size and the tooth 
helix has an angle kal of 45 deg. Such a pair of gears will obviously 
run at the same speed — that is, have a speed ratio of i — and as 
obviously both will have the same number of teeth. Unlike spur 




Fig. 4. 
Figs. 4 and 5. — Tooth and normal helices of helical gears. 



Fig. 5. 



a complete turn around the cylinder — that is, by giving the pitch of 
the helix. 

The normal helix, aghdipq, is also drawn in. The normal pitch is 
the distance ao, an being the circtdar pitch. The normal pitch mul- 
tiplied by the number of teeth must equal the length aghd of the nor- 
mal helix between its intersections with the tooth helix — not the 
length aghipq of a complete turn around the cylinder. That this is 
true may be seen by reference to Fig. 5 in which the angle kal is 



gears, there are two ways in which the speed ratio of such a pair of 
helical gears may be varied. First, the diameters of the gears may be 
changed, as with spur gears, the angle of the tooth helix remaining 
unchanged, as in Fig. 9; and second, the angle of the helix may be 
changed, the diameters of the gears remaining unchanged, as in Fig. 
10. These methods act in very different ways. The first method is 

'"Length of normal helix" is to be understood as meaning the length of 
that helix between two intersections with the same tooth helix. 



142 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



analogous to the procedure with spur gears. As with spur gears the 
circumferential or pitch-line speed of the two gears remains, as before 
the change, equal, but the length of the circumference of the two gears 
is unequal and the larger one thus makes a less number of revolutions 
than the smaller one. The second method is entirely unlike anything 
seen in connection with spur gears. Byit the pitch4ine speeds of the 
two gears are made imequal, and hence, while their diameters are 
equal, the lower one revolves the more slowly. This points out 
another fundamental difference between helical and spur gears: 
With hehcal gears, unless the helix angle is 45 deg., the pitch-line 
speeds of two mating gears are not the same. 




Fig. 6. — Developed helical gears. 




Fig. 7. — Normal helix of a worm. 

The two methods of changing the speed ratio shown in Figs. 8 and 
9 may be combined. That is, part of the desired change in speed may 
be obtained by changing the diameters of the gears and the remainder 
by changing the angle of the hehx. Given the speed ratio and the 
diameter of one of the gears, we may assume a helix angle and find a 
diameter for the second gear to go with it which shall give the desired 
speed ratio and, having done this, a second angle may be assumed and 
a second diameter be found. There are thus an indefinite number of 
combinations of angles and diameters which will give the required 
speed ratio. Note, however, that with the diameter of one gear 
fixed, every change in the diameter of the other changes the distance 
between centers, and that the lengths of both normal helixes must 
be exact multiples of the normal pitch of the teeth. The problem 
of designing spiral gears thus consists of finding a pair which shall 
have a given center distance, helix angles which can be cut with 
' the means at hand, and such a normal pitch that stock cutters can 
be used. 

Geometrically speaking, there is a wide range of choice in the helix 
angle. As regards the desirability of different angles from the stand- 
point of durability, the conditions are essentially the same as in worm 



gearing, in which the most favorable angle for durability is not far 
from 45 deg. There is, howevet, but a trifling increase in wear down 
to 30 deg., no serious increase down to 20 deg., and no destructive 
increase down to about 12 deg. Where gears are to transmit consid- 
erable power, the best results should attend the use of angles between 
30 and 45 deg., while angles as low as 20 deg. may be used in case of 
need, and as low as 1 2 deg. if the gears are to run in an oil bath or do 
light work only. The angle may also be increased above 45 deg. by 
similar amounts and with similar results. 




Follower 'L^ 




Fig. 10. 



Figs. 8 to 10.- 



FiG. 9. 
-The speed ratio of helical gears. 



The author has shown (^we^. If ac/j.,iVoz). 21,1901; Van Nostrand's 
Science Series No. 116) that for helical gears having shafts at right 
angles: 



ri di 

— = 'T tan a 

ri di 



ic) 



in which ri=r.p.m. of driver, 
;-2 = r.p.m. of follower, 
di = diameter of driver, ins., 
^2 = diameter of follower, ins., 
a = helix angle of driver, deg., 
= angle kal of Figs. 5, 6 and 7 measured on the driver. 
Note that formula (c) differs from the corresponding formula for 
spur gears only by the introduction of the factor tan a. 

In any actual case the center distance and the speeds are given 
and the diameters and hehx angle must be foimd. We may assume 
a ratio for the diameters and find the angle, or we may assume an 
angle and find the ratio of diameters. It is desirable to assume the 
angle first, as on it depends, largely, the durability of the gears. To 
do this the above formula may be more conveniently written: 



di r-i 

^ = — tan a 

di Ti 

The author has also shown (same references) that 

2C 



d,- 



— tan a -F I 
r2 



id) 



W 



in which C = distance between centers, ins. 

Having assumed a value for « and substituted its tangent and the 
ratio of the desired speeds in (e), we find a value for di, and, having 
found di, di may obviously be found by subtracting di. from 2C. 

Such a solution is complete in a geometrical sense, and if it were 
feasible to make a cutter to suit each case, it would be complete in a 



HELICAL GEARS 



143 



practical sense also. When, however, we go a step further and find 
the length of the normal heUxes, the probabiUties are all against their 
being exact multiples of the pitch of any stock cutter. The solution 
so obtained must therefore be considered as provisional and be modi- 
fied to suit the cutters to be used. 

The author has also shown (same references) that 
/i=Ci sin a 
= 71(^1 sin a. 



and that 



^1 , ■ 
-=ai sm oi 

h = C2 cos oc 
= Ttdi cos a 

- =di cos a 

7t 



(/) 



(g) 



in which Ci = circumference of driver, ins., 

C2 = circmnference of follower, ins., 
rfi = diameter of driver, ins., 
^2 = diameter of follower, ins., 

^1 = length of normal helix of driver between intersec- 
tions with tooth helix, ins., 
^2= length of normal helix of follower between intersec- 
tions with tooth helix, ins., 
a = tooth helix angle of driver, deg. 
Note that (/) and (g) give the lengths of the normal helixes divided 
by T and not their actual lengths. This is done because, in dealing 
with diametral pitch cutters the calculations are made less laborious 
as has been explained in connection with 45-deg. gears. Dividing 

(/) by (g) gives: 

h di sin a 



h di cos a 
= 3- tan a 

0,2 



ih) 



Comparing (c) with (h) proves what is almost self-evident, that the 
lengths of the normal heUxes are to each other inversely as the number 
of revolutions, and hence that a pitch which will exactly divide the 
short helix will also divide the long one and that the numbers ofteethin 
the gears are inversely as the speeds. 

The use of the formulas is best shown by an example: 
r.p.m. of driver ri 



Assume that 
and 



r.p.m. of follower r^ 



= ,~ = 4 



center distance = C =4 Jf 

= 4.468 ins. 
We are, at the start, entirely at sea regarding the whole matter; 
but as an angle of 30 deg. is favorable to durability we may use it as a 
trial angle and see what it wiH lead to. Finding the tangent of 30 

deg. in a table and substituting it and the value of — in (e) we obtain : 

ri 



di = - 



2X4-468 



4X.S7735 + I 
= 2.7 ins."^ 

Obviously tii-|-(i2 = 2C or 

d2 — 2C — di 
that is, (^2 = 2X4.468 — 2.7 

= 6.236 ins. 
h 



From (/) we find 



I 

^^B,nd from (g) 



: = 2-7X.S 



= i.3Sms. 



- = 6.236X.866 
= 5.4 ins. 



■ For a method of greatly abbreviating the calculations from this point 
on, in most cases, see Table 2 of Real Diametral Pitches of Helical Gears. 



These values oidi, d^, - , and - are the provisional values belong- 

ing'with 30 deg. for a. We must next find if these lengths of the nor- 
mal heUxes will contain exact whole numbers of teeth. Assume that 
6 diametral pitch cutters are to be used. With spur gears, 

circumference 
diameter X diametral pitch = X diametral pitch 



and so with heUcal gears, 

length of normal helix 



= No. of teeth 



X diametral pitch = No. of teeth 



Performing the multiplications we have: 
^'X6 = I.3SX6 



and 



= 8.1 teeth 
:X6 = s.4X6 



= 32.4 teeth 

The provisional normal heUxes thus contain 8.1 and 32.4 teeth of 
the desired pitch, and as these numbers are impossible, we take the 
nearest whole numbers having the desired ratio of i to 4, namely, 
8 and 32. That is, we decide to make the gears smaller and so shorten 
the normal helixes until they contain exactly 8 and 32 teeth — 
the result being also to reduce the center distance from the 
assumed value. 

To determine how much to reduce the diameters we must first 
find the reduced lengths of the normal helixes, which must be such 
that: 

^X6=8 

It 

h 8 
or . — = 7 

n 6 



and 



= 1.333 ms. 

h . 
-X6 = 32 

Tt 



: = 5-333 ms. 



h 



Knowing these corrected values of - and -, it is easy to find the 

new diameters thus: The ratio between the provisional and final 
diameters is the same as that between the lengths of the provisional 
and final helixes, which latter is 8.1 to 8 or, its equal, 32.4 to 32. 
That is: 

final diameter 8 

provisional diameter 8. i 



or final diameter = provisional diameter X 
or final (ii = 2.7 X^ 



8.1 



and 



= 2.667 ins. 

a 

final (^2 = 6.236 X'^ 

o. I 



= 6.159 ins. • 

and (fi-|-<f 2 = 2. 667 -f 6.159=8.825 ins. = twice the new center distance. 
These calculations may he greatly abbreviated in most cases by the use 
of Table 2 by Wm. Haughton (Amer. Mach., Sept. 7, 1911). In this 
table Mr. Haughton has given a series of numbers (which he calls 
real diametral pitches) having the same relation to the circular pitches 
of helical gears that the usual diametral pitches have to the circular 
pitches of spur gears. We have the well known relation: 
No. of teeth in a spur gear 



diametral pitch 
And so with this table: 

No. of teeth in a helical gear 
real diametral pitch 



= pitch diameter 



= pitch diameter 



144 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



The use of the table is best shown by appl3ang it to the example 
already worked out, in which 

revolutions of driver 
revolutions of follower 
Trial center distance =4.468 ins. 

Helix angle =30 deg. 

Diametral pitch of cutter = 6 
We first apply formula (2), for which a column of tangents will be 
found in Table 2, the result being as before: 

trial diameter of driver = 2.7 ins-. 
We now consult the Table 2 and, opposite 30 deg. helix angle and 
below 6 pitch, we find 3 as the real diametral pitch. Now: 
No. of teeth = pitch diameter Xreal diametral pitch 
= 2.7X3=8.1 
This result being fractional and hence impossible, we reduce it to 
8 and then find 

r , ,. r ^ ■ No. of teCth 

final diameter of driven = — tj^ : — i — rrr 

real diametral-pitch 

8 
=- = 1.667 ms. 
3 ' 

The speed ratio being i to 4 the mating gear must have 
8X4=32 teeth 
Looking in the table again for the real diametral pitch of the mating 
gear, we find it to be 5.196 and, as before, 

£ 1 ,- r r ,. No. of teeth 

final diameter of follower = — t-t- : — i — v~i: 

real diametral pitch 

32 

= ; =6.150 ms. 

5.196 ^^ 

Note that for depth of tooth and diameter of blank the diametral 
pitch as given in the top line of the table is to be used. 

The special values of helix angles 26 deg. 34 min. and 63 deg. 
26 min. are for gears of equal diameter with a speed ratio of 2. 

By an adjustment of the angle a it is possible to solve the prob- 
lem for the assumed center distance. In this, helical gears possess 
a property not shared by spurs of diametral pitch, which can only be 
made of such diameters as will contain an exact whole number of 
teeth. The lack of this property has not been found of moment 
with spur gears and it would seem that its possession is of correspond- 
ingly small value with helical gears. For this reason the author has 
omitted its consideration here. Those who desire to learn its use 
are referred to Worm and Spiral Gearing — (Van Nostrand's Science 
Series No. 116) by the author. 

Helical Gears of any Helix Angle by Graphics 

The above determinations may be made or checked by a graphical 
construction. Fig. 11, the use of which is explained below it. 

To find the outside diameter of the blank, add -r- t — j — 7--^ 

•^ "^ diametral pitch 

to the pitch diameter. 

To find the Brown and Sharpe cutter to be used divide the number 

of teeth in the gear to be used by sin^ a. and select a cutter for the 

resulting number of teeth or use the chart. Fig. 2. 

The pitches of the tooth helixes are found by the formulas: 

pitch of tooth hehx of driver = 7:di 1 tan a 

pitch of tooth helix of follower = Tcd^ cot a 

a being taken from the driver in both cases. 

Helical Gears of any Helix Angle on Shafts at any Angle 

The calculation of helical gears on shafts at other angles than 90 deg. 
brings in the angle between the shafts. 
Let ?-i = r.p.m. of driver. 
r2 = r.p.m. of follower. 
(fi = diameter of driver, ins. 
£f2 = diameter of follower, ins. 
« = helix angle of driver, deg. 
S6= angle between the shafts, that is, the lesser of the 

supplementary angles, deg. 
C = trial distance between centers, ins. 



b- 


v--^ 


^:>\ ^ 


71"^ 


\ 


^~^^^ 






"' \ 


^ /' 






/\ 




/ 

/ 




h 






'^ 




l'^ 


0^ 


.^^^^^ 



Lay off ab = 2C = 8lf inches. At any convenient distance 
lay off the indefinite line cd parallel to ab. At c lay off the pro- 
visional angle a = 30 degrees. Draw ef at any convenient 
point perpendicular to cd. Take ef in the dividers and step it 
off from e toward d as many times as will represent the ratio of 
the desired speed of the driver divided by that of the follower. 
That is, in the present case, lay off ef 4 times above e and thus 
obtain d. Draw ca and db and extend them till they meet at 
g.i Draw ge, giving ah and bh, which are provisional diameters 
of driver and follower respectively. Draw hp perpendicular 
to ab, at h lay down hk and hi to repeat a, and from h strike 
arcs ak and bl. Draw ko and nl perpendicular to ab and we 
have the provisional values. 



ah 


= 


di 


bh 


= 


d2 


ho 


= 


li 

1. 



hn = 

t: 

Scale ho and hn and multiply them by the diametral pitch 
number — 6. If the results are not whole numbers, as they 
usually are not, select the nearest whole numbers having the 
desired speed ratio, and they are the final numbers of teeth. 
Divide these numbers by the diametral pitch number to obtain 

the final values of — and- and lay them down as ho' andhn'. 

Draw o'k' and I'n' and k'a' and I'b', giving : 
a'h = the final dj, 
b'h = the final d2, 
a'b' = twice the new center distance. 

Fig. II. — Graphical solution of helical gears of any helix angle on 
shafts at right angles. 



Formula (e) becomes for this case: 



di = - 



2C 



As before d2 = 2 C—di. 



r\ sin a 
r-i sin («+^) 



+ 1 



(0 



1 Had cd been taken shorter than ab, g would have fallen to the left of 
the diagram, but the construction would otherwise have been unchanged. 



HELICAL GEARS 



145 



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10 



146 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 




Fig. 12. Fig. 13. Fig. 14. 

Figs. 12 to 14. — Helices of gears on shafts at other than righi angles. 

As before these values of di and d2 are to be treated as trial values 
and tested and adjusted for exact whole numbers of teeth. For the 
driver, formula (/) applies directly, but not so with (g) for the fol- 
lower. If the shafts are at right angles the helix angles of mating gears 
are compliments and the cosine of one may be used for the sine of 
the other, as in {g). With the shafts at other angles, this is no 



longer true, and it is necessary to find the angle of the follower by 
the relation : 

Sum of helix angles=i 80° — shaft angle 
Calling the helix angle of the follower i?, we have in place of (g) : 

I2 

- = d2 sin ^. (i) 

So also, in finding the pitch of the tooth helix, the cotangent of 
the angle of the driver cannot be used for the tangent of the angle 
of the follower, the formulas becoming: 

pitch of tooth helix of driver = n di tan a 
pitch of tooth helix of follower = 7id2 tan /? 

The helixes of gears on shafts at right angles are always of the 
same hand but, with shafts at other angles, the relation given above 
for the sum of the helix angles makes it possible that one of the gears 
may be a spur, or its helix may be of opposite hand to its mate. 
This is more clearly shown in Figs. 12, 13 and 14, by H. B. McCabe 
(Amer. Mach., Oct. 11, 1906). In Fig. 12 the driver has a left-hand 
helix with an angle equal to the shaft angle and the follower in a 
plain spur. In Fig. 13 the angle of the driver has been reduced and 
the helix of the follower is right hand, while in Fig. 14 the angle of 
the driver has been increased and the helix of the follower is left 
hand. 



PLANETARY (EPICYCLIC) GEARS 



The action of epicyclic or planetary trains of gearing may be deter- 
mined from the following collection of formulas by F. J. Bostock 
{Amer. Mach., May i6, 1907). 

An epicyclic gear is one that revolves around the center of another 
with which it is in mesh. The formulas that follow begin with simple 
and lead up to more complex arrangements. 

Example i. — If in Fig. i R and N are two gears in mesh, ^-and n 
being their respective numbers of teeth, their bearings being fixed, 
then: 

velocity of driven gear N _f_ 
velocity of driver gear R n' 

r 

or Ws velocity = 22'5velocityX- 

n 

If, however, R revolve in a positive direction, N must revolve in the 

opposite, that is, in a negative direction. 



,'.N's velocity =— jR'i velocityX" 

71 



(a) 



In all these calculations it is essential that great care be taken in 
order to obtain the correct sign of the resulting velocity. 

Example 2. — An intermediate gear / is placed in contact with 
both N and R, Fig. 2. The effect will be that of giving N motion 
in the same direction as R. 

f 
.*. iV'5 velocity = i?'5 velocity X- (b) 



Simple Epicyclic Train 

Example 3. — Two gears, F and N, are in mesh, the centers of 
which are on the arm R, which is capable of revolving aroimd the 
center of F. It is required to find the velocity ratio between R and N 
when R revolves around the fixed gear F; Fig. 3 shows the arrange- 
ment. The gear N is subject to two motions due to the following two 
conditions: 

a. The fact of its being fixed to the arm R. 

b. The fact that it is in contact with the gear F. 

We will therefore in the first place suppose that they are not in 
gear, and that N cannot rotate on the arm R. Then if R makes one 
revolution around F it is obvious that N must also make one revolu- 
tion aroimd F, as in Fig. 4. 

.'.N's velocity due to condition a, = R's velocity, 
the direction being the same as R's. 

Secondly, if instead of R making one revolution around F'lna, ■}- 
direction, we cause F to make one in the opposite, that is, negative 
direction, we shall have exactly the same effect. Therefore place 
F and N in mesh, and fix the arm R, as in Fig. 5. 

/ 

Then if F makes— i revolution. A'" will make -|-- revolutions. 

(According to equation (a). 
But -I oiF=+i of R. 

f 
.'. I revolution oi R = - revolutions of N, 
n 



or. 



/ 
N's velocity, due to condition b = - X R's velocity 



By addition we obtain the total impulses given to N, that is: 

f 
N's velocity = iJ'5 velocity -^-Xi?' J velocity 



=R's velocity X 



HY 



ic) 



Epicyclic Train with an Idler 

Example 4. — If an intermediate gear / be inserted between F and 
N, as in Fig. 6, we have a similar case to the above, but the inter- 
mediate gear has the effect of changing the direction of revolution of 
N (equation b) , due to its contact with F through /. 



N's velocity=jR'5 velocity X (i— ~) ■ 



(d) 



It will be seen that if / = «, N will not have any motion of rotation at 
all; and it will have a positive one if /<w and negative if />«. Thus 
by the adjustment of / and n one can obtain great reduction in speed 
by means of few moving parts. 

Simple Epicyclic Train with Internal Gear 

Example 5. — Instead of the driven gear N being external, it might 
have been internal, as shown in Fig. 7. The effect will be the same 
as inserting an intermediate gear in Example 3, giving the same 
result as case 4, namely: 



N's velocity = i?':? velocity X 

In this case n >/. 

.*. The final direction is always -J-. 



(-i) 



(e) 



Internal Gear Epicyclic Train with Intermediate Gear 

Example 6. — Fig. 8 shows a still further modification of this condi 
tion, / being an intermediate gear. The result is: 

/ 



N's velocity =2?'s velocityX I i 



(-)■ 

nf 



if) 



The Same Train with the Internal Gear Driving 



Example 7. — With the above type, one often arranges the outer 
internal gear to be the driver, imparting motion to the arm carrying 
the intermediate gear. See Fig. 9. 

We have seen by equation 6 that: 

N's velocity (driven) _ i 
R's velocity (driver) f_ 

r 

•'. N's velocity=i?V velocity -i- ( i -| — J 

= 2?V velocityX (;q~/) (s) 

The latter two examples constitute what is known as the " Sun and 
Planet" gear, which is largely used in many mechanisms. All the 
above examples show "simple" gearing, but they can be compounded 
with great advantage. 

Compound Gears in Fixed Bearings 

Example 8. — Gears compounded together are shown in Figs. 
10 and II, II being a diagram of 10. One repeats the well-known 
rule that: 

velocity of driven gear product of number of te e th of driver gears 
velocity of driver gear product of number of teeth of driven gears 

or, N's velocity = R's velocity X -r^ — W 

iX n 

The direction is the same as N's namely, -{-. 

{Continued on page 149, first column) 



147 



148 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 

+ 




Fig. I. — Simple pair of 
gears in fixed bearings. 

-Eq. (a) N's V.= -R's V.X^- 



FiG. 2. — Gears in fixed bear- 
ings with an idler. 

Eq.{b) N'sV. =R'sV.X- 



\ J 


y 


^ h 


p'^TTT^ 


N 


'". 


Fig. 3. — Simple 
train. 

Eq. (c) N's V.= 


epicyclic 
R's V.X 






■^^-'' Fig. 4. — Illustrating rota- 
tion of N when it is revolved 
about the center of F. 

+ 




Fig. 5. — Second stage in deriving 
equation 3 ; arm assumed to be fixed, 
F turned backward. 




Fig. 6. — Epicyclic 
train with an idler. 
Eq. (d) N's velocity = 
R's velocity X 

-A 



Fig. 7. — Simple epicy- 
clic train with internal 
gear. 
Eq. (e) N's velocity = R's 

velocity X [1 - -j 
+ 



Fig. 8. — Internal gear train with 
intermediate gear; the arm driving 
Eq. (/) N's velocity =R's velocity X 





Fig. 9. — Same train as Fig. 8 but 
with the internal gear driving. 

Eq. {g)N's V. = R'sV.x{^^ 



Fig. 10. — Compounded 
gears in fixed bearings 
r 
Eq.{h)N'sV.=R's VX- 



FiG. II. — Compounded 
gears in fixed bearings. 
See equation (h). Fig. 10. 



Fig. 12. — Compounded epicyclic 
train. 



Eq. (i) N's V. = R's V.X 






Fig. 13. — Second stage in deriving equa- 
tion (i); arm assumed to be fixed, F turned 
backward. 



Fig. 14. — Compound epicyclic train 
with one internal gear. 

Eq. (./) N's V.=R's V.X (i + j^) 




Fig. 15. — Compound epicyclic train 

with two internal gears. 

See Eq. (i), same as Fig. 12. 



NOTATION 

R = Denotes Driving Gear, or in some 

cases, Arm. 
r = Number of Teeth in Driving Gear. 
N= Denotes Driven Gear or Arm. 
n = Number of Teeth in Driven Gear. 
I, S and M denote Intermediate Gears 
F= Denotes Fixed Gear, 
f = Number of Teeth in it. 
V ^ Angular Velocity. 
''*"•''"" Denotes part which is Fixed. 



Fig. 16. — An epicyclic train consisting of 
two central gears, one arm carrying two 
planetary gears, and two internal gears, one 



Fig. 17. — Diagram of the train of Fig. 16. 

Eq.ik) N's V.=^R's V.X {^^)- 



of which is fixed. 



Figs, i to 17. — Planetary gear trains with corresponding velocity ratio formulas. 



PLANETARY GEARS 



149 



Compound Epicyclic Train without Internal Gear 

Example 9. — We will now arrange to fix one of the gears F, and 
by means of the arm R revolve the others aroimd it, thereby 
causing N to revolve as shown in Figs. 12 and 13. As before, 
we will assume the gears M and 5 to be out of mesh, so that 
when the arm R, carrying with it the gear N, makes one revolu- 
tion aroimd F, N must also make one revolution relatively to F. 
Also when they are in mesh, the arm R being fixed and F makes one 

fm 
revolution in a negative direction (see Fig. 13), N will make 

revolutions (equation h). 

Now the total motion imparted to N must be the sum of these two, 
namely: 

fm 



I revolution of i?=i revolutions of N, 

sn 



N's velocity = 7? '5 velocity X 



/ _fXnA 
\ sXnJ 



Compound Epicyclic Train with One Internal Gear 

Example 10. — Fig. 14 shows a slight modification of the last case, 
N being an internal instead of an external gear. Obviously the only 
difference will be in the direction of N's motion, that is: 



(■+f:) 



N's velocity = i?'5 velocity X ( i + 



Compotmd Epicyclic Train with Two Internal Gears 



0') 



Example 11. — A further modification, however, is one in which 
both F and N are internal gears. Fig. 15, the effect of such being a 
change of sign in the equation. 



.'. N's velocity = i?'i velocityX 1 1 



_fm\ 
sn I 



(0 



The tjrpe shown in Figs. 12 and 15 is, perhaps, one of the best 
methods of obtaining a good reduction of speed in an easy and 
cheap manner. 

There are several combinations of the examples shown, but as they 
are all somewhat similar we will take another typical case as a guide 
for future calculations. 

An Epicyclic Train Consisting of Two Central Gears, One Ann 

Carrying Two Planetary Gears, and Two Internal Gears, 

One of Which Is Fixed 

Example 12. — The writer has successfully used the arrangement 
shown in Figs. 16 and 17, in which R and R' are two spur gears 
moimted on one shaft; / and 7' are two "planet" pinions, while F 
and TV are two internal gears, the former being fixed. R and R' are 
made to revolve, which has the effect of giving N a very slow speed. 

Finding the Velocity Ratio 

As this is somewhat complicated, we will work it out in stages: 

1. Obtain the revolutions of the arm A when R' makes one revolu- 
tion, F, of course, being fixed. 

2. Obtain N's revolutions when the arm A is fixed and R makes 
one revolution. 

3. Assume R fixed, and that the arm makes one revolution; obtain, 
then, N's revolutions. 

4. Then if N makes so many revolutions to one of the arm, as 
given by stage 3, we can by proportion obtain how many will be 
caused by the amoimt given by stage i. 



5. Add the results of 2 and 4 together, and obtain the motion 
given to N by one revolution of R, which is the desired result. 
Working the above out we obtain: 

I. When F is fixed and R' makes one revolution, the arm A must 
r' 



make-|- 



/+/■ 



(According to equation {g) 



2. R makes one revolution, arm A being fixed; then N must make 

T 

— -revolutions. (According to equation (6).) Negative sign used 
n 

because of the internal gear. 

3. When R is fixed and arm A makes one revolution, N will make 

(i-| — I revolutions. (According to equation (f)-) 

r 

4. With one revolution of arm, N makes i + - revolutions, from 

n 



+ 



stage 3 ; 
.'. with 
^^z will make 



r'+f 



revolutions of the arm, as derived in stage i, N 



(■+^)x(4:> 



5. The aggregate is the sum of the effects derived in stages 4 and 
2, namely, to one oi R, N makes: 



K)xC-4)-K) 



_(M+r)r' r 
~n{r'+f) n 
_rr'-\-r'n—rr' — rf 

= W+J) 

r'n—rf 

The final direction of revolution of N will depend upon the re- 
lation which r'n bears to rf; if the former be greater, then the direc- 
tion will be positive (-|-), and vice versa. The formula for this 
combination is, then: 



N's velocity = i?'i velocity X 



/r'n—rf\ 

\W+f)) 



ik) 



Some Numerical Examples in Epicyclic Gearing 



In order to illustrate the above examples we will take one or two 
cases. 

If in example and Fig. 3,/ = 30, 11 = 2$, then to one revolution of 

R, N will make ( i -t- r I = i -|- f f = 2^ revolutions. 



(-i) 



It will be obvious that with/ = M, N would revolve at twice the 
speed of R. 

In the type shown in Fig. 7,/ = 60, ^ = 65. 
then 

/ 



Velocity of N 



I— - 



n 



65 



= T^. 



Velocity of i? i i 

The arrangement of Fig. 12 is much used. Let «=6o, / = 6i„ 
5=40, m=4i. 

/ fm \ 
Then the velocity ratio between N and R is I i : i I 

_ 6iX4i_ 2501 

40X60 2400 

= say i: 24, in a minus direction. 

Illustrating example 12, Fig. i6,let r = go, r' = gi,f=i2Qfn = i2i, 

Velocity of iV r'w—r/_9rXi2i — 90X120 

Velocity of i?~w(r'-f/) 121(91-1-120) 

noil — io8co 211 I 

121X211 121X211 121 



ROPES 



Important differences exist between British and American prac- 
tice in rope driving. Thus British engineers prefer three strand 
cotton rope and the multiple-wrap system, while American engineers, 
with few exceptions, have adopted four strand manilla rope and the 
continuous-wrap system. This diversity of practice, based on ex- 
tended experience in both cases, is difficult to reconcile or explain. 

An obvious advantage of the multiple-wrap system is that the 
ropes give out one at a time and, as the failure of a single rope does 
not cripple the system, delays due to failure are lessened. This is 
at least partially offset by the fact that ropes never fail without 
givmg warning. An equally undoubted advantage of the continuous 
wrap system with its weighted idler is its flexibihty as regards 
center distance, incUnation from the horizontal, direction of rotation 
and the passage of obstructions by the use of guide pulleys. Ma- 
nilla rope is also most suitable for situations involving exposure to 
weather conditions or to dampness. 

The rival claims of cotton vs. manilla rope relate chiefly to cost 
and durabihty, regarding which no accurate data exist. Whatever 
the explanation, British engineers consider the superiority of cotton 
rope as proven beyond question. 

The Plymouth Cordage Co., who install both systems, consider 
the British system best when the driving and driven sheaves are of 
about equal diameters, the shafts enough out of the vertical to pre- 
vent the slack side from falling too much out of the sheave grooves, 
the shafts between 30 and 125 ft. apart, the load fairly uniform, the 
speed not excessive and the drive protected from the weather. In 
general, the multiple system incUnes to the larger and the continuous 
system to the smaller installations. Cotton rope is, however, better 
for small interior drives which take the place of belts. 

The continuous system should generally be used when shafts are 
nearer together than 30 ft., as, in the multiple system, a small 
amount of stretch will so decrease the initial tension that the centrif- 
ugal force carries the rope out of its groove and quickly diminishes 
its driving capacity. 

With shallow grooves to avoid chafing due to the necessary side 
lead of the continuous system, sheaves may be rim on 10 ft. centers 
while, without supporting idlers, the shafts may be placed as much as 
150 ft. apart. With idlers the distance may be increased almost 
indefinitely. 

Rope drives up to 4000 h.p. capacity are in operation, drives of 
between 1000 and 2000 h.p. being fairly numerous. 

The comparative first cost of rope and belt drives, according to the 
Plymouth Cordage Co., may be determined by assuming the belt 
to cost about two and one-half times as much as manilla rope of 
equal capacity and the rope sheaves to cost about one-third more than 
belt pulleys — the advantage of ropes increasing with the distance 
between shafts. The average life of manilla rope of good proportions 
may be taken as eight years, the Ufe of belts being materially longer. 
The most economical speed to run the rope, taking into consider- 
ation the first cost and durability, is given as about 4500 ft. per 
min. Lower speed increases the durabihty, while higher speed, 
within certain limits, reduces the first cost. 

The diameter of maniOa ropes used in the United States for heavy 
drives ranges between i and if ins., while the speeds used run up to 
5000 ft. per min. The following charts and tables are from a 
paper on Rope Driving by C. W. Hunt {Trans. A. S. M. E., Vol. 12). 
The relative first cost of manilla rope as related to the speed may be 
obtained from Fig. i, the cost of a rope running at 80 ft. per sec. 
being taken as 100. The first cost for other speeds is in proportion 



to the ordinates for those speeds. Thus if a rope is to rim at 60 ft. 
... , 112 

per sec, its cost will be of that for a speed of 80 ft. 

100 "^ 

The rule for the load to be carried by manilla ropes is that the 
stress on the tight side in lbs. shall be equal to 200 times the 
square of the diameter in- ins. Table i gives the horse-power 
suitable for usual sizes of rope on this basis. 

The horse powr including the effect of centrifugal force in relation 
to the speed is given in Fig. 2. 



2W 








"T 












~^ 


^^ 
























1 






































1 






































1 






























190 




























































































































































































'180 




























































































































































































170 




















































































1 












































































































































160 












































































a 










































































J 150 










































































•H 












































































•£ 






































^ 1-40 


























































































1 






































\ 






































\ 






















ISO 
















\ 




































\ 














































































































































































120 


























































V 






































\ 






































\ 








































s 
















110 






















\, 














1 


























s 












/ 


























\ 










/ 






























N 








/ 




inn 


























\ 


;:>_ 




y 







10 



20 



30 40 50 60 70 i 

Speed of the Rope, Ft. per Sec. 



90 



100 



Fig. I. — Relative first cost of rope at various speeds. 

The tension on the slack part of the rope, when transmitting the 
amount of power given in Table i , may be obtained from Table 2. - 
The use of this tension is in determining the weight to be applied 
to the idler in order to obtain the necessary adhesion. 

The sag of the rope (drive horizontal) when transmitting the 
amounts of power given in Table i may be obtained from Table 3. 
This sag is the same at all speeds for the driving part, but is variable 
for the slack part. 



150 



ROPES 



151 



- 


— 


— 


— 1 


— 




- 


^ 


- 




- 






y 










:is 


\ 








- 










- 


























^ 
















\ 






































/ 


/ 




















\ 
































c* • 
























































O^ 


/ 












--— 


— 1 


--. 










\ 




























..y 




loV-. 














s 








\ 
























ri-y 






V" 


V 


' 
















s 






s 
























;\ 


/ 




.^^ 


^ 






















^S- 
















_ 














-1^ 


A 


^. 


/ 


_ 


_ 


-^ 


Ko\^- 




— 


_ 




— 


— 


^ 


V 


— 




— 


— 




- 


- 


— 


— 




5 


m 


H 




»-e, 




g 






I 


z 


z 


~~ 





~ 


"n 




A 


^ 





\ 






I 


- 


— 


— 




y^ 


bo 


w 


n 


— 


— 










— 








X; 






X 


-\ 






~ 






■s 


/i / 


y 




h 


.oS 


6 


^^ 


























V- 








- 




^/ 


m\ 


>v 


b 


d! 






















"^ 








\ 








-?'/- 


'7^^^ 
































i^ 


s 










^ 


^/ 


/ 


y 




-^ 


p^ 


































N 








- 


-o 


-/ 


"7" 




p^ 












































\ 


"^ 




-^ 


" 


■ 












































\ 




— 


£■ 








_ 


L_ 














_ 










^ 












^ 









&24 

n 

12 



10 20 30 40 50 60 70 80 90 100 UO 120 130 140 
Speed of DjivXng Kope, Ft, per Sec. 

Fig. 2. — Horse-power of Manilla rope, including the effect of cen- 
trifugal force. 





Table i.- 


-Horse-power of Manilla Rope 




Diam. 

of 

rope 


Speed of the rope in ft. per min. 


Diam. of 
smallest 

pulley or 
idler in 


lUS. 


1500 


2000 


2S00 


3000 3500 


4000 


4500 


SOOO 


6000 


7000 


8400 


ins. 


\ 


1-43 


1-9 


2.3 


2.7 3-0 


3-2 


3-4 


3.4 


3.1 


2.2 





20 


\ 


2.3 


3.2 


3.6 


4.2 4.6 


5-0 


5.3 


5-3 


4-9 


3-4 





24 


\ 


3 3 


4-3 


5.2 


5.8 6.7 


7.2 


7.7 


7.7 


7.1 


4.9 





30 


\ 


4-5 


5.9 


7.0 


8.2 


9.1 


9.8 


10.8 


10.7 


9.3 


6.9 





36 


I 


S.8 


7.7 


9.2 


10.7 


ri.9 


12.8 


13.6 


13.7 


12. 5 


8.8 





42 


li 


9.2 


12. 1 


14.3 


16.8 


18.6 


20.0 


21.2 


21.4 


19.5 


13.8 





54 


li 


13. 1 


17.4 


20.7 


23.1 


26.8 


28.8 


30.6 


30.8 


28.2 


19.8 





60 


If 


18.0 


23.7 


28.2 


32.8 


36.4 


39-2 


41-5 


41.8 


37.4 


27.6 





72 


2 


23-2 


30.8 


36.8 


42.8 


47.6 


51.2 


54-4 


54-8 


50.0 


35.2 





84 



Table 2. — ^Tension on the 'Slack Part of the Rope 



Speed of rope in 
ft. per sec. 


Diameter of the rope and pounds tension on 
the slack rope 




\ 


1 


i 


\ I 


l\ 


li 


if 


2 


20 


10 


27 


40 


54 


71 


IIO 


162 


216 


283 


30 


14 


29 


42 


56 


74 


IIS 


170 


226 


296 


40 


IS 


31 


45 


60 


79 


123 


181 


240 


315 


SO 


16 


33 


49 


65 


85 


132 


195 


259 


339 


60 


18 


36 


S3 


71 


93 


14s 


214 


28s 


373 


70 


19 


39 


59 


78 


lOI 


IS8 


236 


310 


406 


80 


21 


43 


64 


85 


III 


173 


255 


340 


445 


90 


24 


48 


70 


93 


122 


190 


279 


372 


487 



Table 3. — Sag of the Rope Between Pulleys 









Slack side of rope 


Distance between 
pulleys in ft. 


All speeds 


80 ft. per 
sec. 


60 ft. per 
sec. 


40 ft. per 
sec. 




Ft. 


Ins. 


Ft. 


Ins. 


Ft. 


Ins. 


Ft. 


Ins. 


40 





4 





7 





9 





II 


60 





10 


I 


5 


I 


8 


I 


II 


80 


I 


5 


2 


4 


I 


10 


3 


3 


TOO 


2 





3 


8 


4 


5 


5 


2 


120 


2 


II 


5 


3 


6 


3 


7 


4 


140 


3 


10 


7 


2 


8 


9 


9 


9 


160 


S 


I 


9 


3 


II 


3 


14 






Ordinary manilla rope should not he used. American manilla 
transmission rope is always laid up with internal lubricant which is 
essential to long life. 

The cross-sections of rope sheaves used by the Pljmiouth Cordage 
Co. are shown in Figs. 3, 4 and 5. Fig. 3 shows the usual section, 
Fig. 4 being used to avoid side chafing when the sheaves are close 
together, xmder which circumstances the higher webs are not needed 
as there is no tendency for the rope to jump the grooves. Fig. 5 
shows the section used for idler sheaves. 



The diameter of sheaves for manilla rope should not be less than 40 
diameters of the rope. Cotton rope being more flexible, the sheaves 
for it may be smaller. The British rule for cotton-rope sheaves 
provides a minimum of 30 rope diameters. The same rule for di- 
ameters should be observed with idler as with driving sheaves, as it 
is the bending of the rope on the sheave that does the damage. 

For dimensions of rope sheave arms see Dimensions of Pulley 
Arms. 

The horse-power of cotton ropes, according to British practice, is 
given in Table 4 by Edward Kenyon {Trans. South Wales Ins. of 




Fig. s. 

Figs. 3 to 5. — Cross-sections of rope sheaves. 

Engrs., 1909). British practice with cotton ropes does not hesitate 
to adopt speeds of 7000 ft. per min., whereas American practice 
regards 5000 ft. as the economical limit with manilla ropes. Ac- 
cording to Mr. Kenyon, the angle between the driving faces of the 
groove should not exceed 40 deg. in order to prevent the rolling 
over of the ropes in their grooves which reduces their life at least one- 
third. 

For the efficiency of rope driving see below. 

Manilla Rope for Hoisting 

The proper working loads of hoisting rope, according to C. W. Hunt 
(Trans. A. S. M. E., Vol. 23) are well settled by extended experience. 
Table 5 gives Mr. Hunt's figures for the working loads and the sheave 
diameters under various conditions. The terms in the captions of 
the columns have the following meanings: 

Slow: Derrick, crane and quarry work; speed from 50 to 100 ft. 
per min. 

Medium: Wharf and cargo hoisting; 150 to 300 ft. per min. 

Rapid: 400 to 800 ft. per min. See also end of section. 

The efficiency to be expected from hoisting blocks is given in Table 
6 from some experiments made by Robert Grimshaw and quoted 
by Mr. Himt. The blocks experimented upon had a 6-fold pur- 
chase, the three upper sheaves having roller bearings and the three 



152 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 







Table 


4-- 


-Horse-power of 


Three Strand Cotton Ropes 


















Rope diameters 


i" 


Ik" 


ii"| 


ii"l 


li" li" 


If" 


li" 


2" 


Rope diameters 


r" 


li" 


Ij" 


ir 


li" 


If" 


li" 


II" 


2" 






Minimum diameter of small- 1 




w 
C 






i 




.5 




a 




Minimum diameter of \ 
smallest pulley J 


j 


.s 




1 

CO 


a 
to 


1 
0. 






g 

4J 
•* 


4i 








Velocity in ft. per min. 




















Velocity in ft. per min. 




















1000 


3-3 


4-1 


S-i 


6.1 


7.4 


8.6 


10 


II. 5 


13 


4100 


13-3 


16.9 


20.7 


25.1 


30.4 


35.3 


41 


47.1 


S3.S 


1 1 00 


3.6 


4-5 


S.6 


6.7 


8.1 


9.S 


II 


12.6 


14.3 


4200 


13.6 


17.3 


21.2 


25.7 


31.2 


36.2 


42 


48.3 


54.8 


1200 


3.9 


4-9 


6.1 


7.3 


8.9 


10.3 


12 


13.8 


IS. 6 


4300 


13-9 


17.7 


21.7 


26.3 


31.9 


37 


43 


49.4 


56. 1 


1300 


4.2 


53 


6.6 


8 


9.7 


II. 2 


13 


14.9 


16.9 


4400 


14-3 


18. I 


22.2 


26.9 


32.7 


37.9 


44 


50.6 


57.4 


1400 


4.6 


5.7 


7.1 


8.6 


10.4 


12 


14 


16 


18.3 


4500 


14.6 


18.5 


22.7 


27. 5 


33.4 


38.8 


45 


SI. 7 


58.7 


1500 


4-9 


6.1 


7.6 


9.2 


II. I 


12.9 


15 


17.2 


19.6 


4600 


14.9 


18.9 


23.2 


28.1 


34.2 


39.6 


46 


52.9 


60 


1600 


S.2 


6.6 


8.1 


9.8 


II. 9 


13.8 


16 


18.4 


20.9 


4700 


15.2 


19.3 


23.7 


28.7 


34.9 


40.5 


47 


54 


61.3 


1700 


S.5 


7 


8.6 


10.4 


12.6 


14.6 


17 


19. S 


22.2 


4800 


15.6 


19.8 


24.2 


29.4 


35.7 


41.4 


48 


55.2 


62.7 


1800 


S.8 


7.4 


9.1 


II 


13.4 


15. S 


18 


20.7 


23.5 


4900 


15-9 


20.2 


24.7 


30 


36.4 


42.2 


49 


S6.3 


64 


1900 


6.2 


7.8 


9.6 


II. 6 


14. 1 


16.3 


19 


21.8 


24.8 


5000 


16.2 


20.6 


25.3 


30.6 


37.1 


43.1 


SO 


S7.5 


6S.3 


2000 


6.5 


8.2 


10. 1 


12.2 


14.9 


17.2 


20 


22.9 


26.1 


5100 


16.5 


21 


25.8 


31.2 


37.9 


43.9 


SI 


S8.6 


66.6 


2100 


6.8 


8.6 


10.6 


12.8 


IS. 6 


18. 1 


21 


24.1 


27.4 


5200 


16.9 


21.4 


26.3 


31.8 


38.6 


44-8 


52 


59.8 


67.9 


2200 


7.1 


9. 


II. I 


13.4 


16.3 


18.9 


22 


25-3 


28.7 


S300 


17.2 


21.8 


26.8 


32.4 


39.4 


45.7 


53 


60.9 


69.2 


2300 


7-5 


95 


II. 6 


14 


17. 1 


19.8 


23 


26.4 


30 


S400 


17. S 


22.2 


27.3 


33 


40.1 


46. S 


54 


62.1 


70.5 


2400 


7.8 


9.9 


12. 1 


14-7 


17.8 


20.7 


24 


27.6 


31.3 


SSOO 


17.8 


22.6 


27.8 


33.6 


40.9 


47.4 


55 


63.2 


71.8 


2500 


8.1 


10.3 


12.6 


IS. 3 


18.5 


21. s 


2S 


28.7 


32.6 


5600 


18.2 


23.1 


28.3 


34.3 


41.6 


48.3 


56 


64.4 


73.1 


2600 


8.4 


10.7 


I3.I 


15.9 


19.2 


22 .4 


26 


29.9 


33.9 


S700 


18.5 


23-5 


28.8 


34.9 


42.3 


49.1 


57 


65.5 


74.4 


2700 


8.7 


II . I 


13.6 


16.5 


20 


23-3 


27 


31 


35.2 


5800 


18.8 


23.9 


29.3 


35.5 


43.1 


50 


58 


66.7 


75. 7 


2800 


9.1 


II. 5 


14. 1 


17. 1 


20.8 


24.1 


28 


32.2 


36.5 


5900 


19. 1 


24-3 


29.8 


36.1 


43.8 


50.8 


59 


67.8 


77 


2900 


9-4 


II. 9 


14.6 


17.7 


21. S 


25 


29 


33.3 


37.8 


6000 


19-5 


24.7 


30.3 


36.7 


44.6 


SI. 7 


60 


69 


78.3 


3000 


9-7 


12.3 


IS. I 


18.3 


22.3 


2S.8 


30 


34.5 


39.1 


6100 


19.8 


25.1 


30.8 


37.3 


45.3 


52.6 


61 


70.1 


79.6 


3100 


10 


12.7 


IS. 6 


18.9 


23 


26.7 


31 


35.6 


40.4 


6200 


20. I 


25.5 


31.3 


37.9 


46. 1 


53.4 


62 


71.3 


80.9 


3200 


10.4 


13.2 


16.2 


19.6 


23.8 


27.6 


32 


36.8 


41.8 


6300 


20.4 


25.9 


31.8 


38. 5 


46.8 


54-3 


63 


72.4 


8226 


3300 


10.7 


13.6 


16.7 


20.2 


24.6 


28.4 


33 


37-9 


43.1 


6400 


20.8 


26.4 


32.4 


39.2 


47.6 


55.2 


64 


73.6 


83.9 


3400 


II 


14 


17.2 


20.8 


25.3 


29.3 


34 


39.1 


44.4 


6500 


21. I 


26.8 


32.9 


39.8 


48.3 


S6 


65 


74.7 


84.. 


3500 


II. 3 


14.4 


17.7 


21.4 


26 


30.1 


35 


40.2 


45. 7 


6600 


21.4 


27.2 


33.4 


40.4 


49 


56.9 


66 


75.9 


86.2 


3600 . 


II. 7 


14.8 


18.2 


22 


26.7 


31 


36 


41.4 


47 


6700 


21.7 


27.6 


33-9 


41 


49.8 


57.7 


67 


77 


87. S 


3700 


12 


15.2 


18.7 


22.6 


27.5 


31.9 


37 


42. S 


48.3 


6800 


22. I 


28 


34-4 


41.6 


SO. 5 


58.5 


68 


78.2 


88.8 


3800 


12.3 


IS. 6 


19.2 


23.2 


28.2 


32.7 


38 


43-7 


49.6 


6900 


22.4 


28.4 


34-9 


42.2 


51.3 


59-4 


69 


79-3 


90.1 


3900 


12.6 


16 


19.7 


23.8 


29 


33.6 


39 


44.8 


50.9 


7000 


22.7 28.8 


35.4 


42.8 


52 


60.3 


70 


80. 5 


91.4 


4000 


13 


16,4 


20.2 


24-5 


29.7 


34-5 


40 


46 


52.2 





















Table 5. — ^Working Loads for Manilla Hoisting Rope 



Diameter 


Ultimate 


Working load, 


Minimum diameter of 


of rope, 


strength, 
lbs. 


lbs. 




sheaves, ins. 


ins. 


Rapid 


Medium 


Slow 


Rapid 


Medium 


Slow 




7,100 


200 


400 


1,000 


40 


12 


8 


i| 


9,000 


250 


500 


1,250 


45 


13 


9 


jl 


11,000 


300 


600 


1,500 


SO 


14 


10 


If 


13.400 


380 


750 


1,900 


55 


IS 


ir 


■ik 


15,800 


450 


900 


2,200 


60 


16 


12 


If 


18,800 


530 


1,100 


2,600 


6S 


17 


13 


1- 


21,800 


620 


1,250 


3,000 


70 


18 


14 



The splice in a transmission rope is not only the weakest part of the 
rope, but is the first to fail when the rope is worn out. If the splice is 
not strong, the rope will fail by breakage or pulling out of the spUce. 
If the rope is larger at the splice, the projecting parts will wear on the 
pulleys, and the rope fail from the cutting off of the strands. 

Do not put in a "short splice," or an ordinary "long splice," or 
get an old sailor to do the work, but have a handy man follow implic- 
itly the directions given herein for a splice in a four-strand rope. 

For spUcing, add to the net length the following amount for making 
a splice: 



Table 6 


— Efpiciency of Block and 


Fall 






Theoretical 


1 


Net load on tackle, 
weight raised 


amount required 
to raise the 
net weight 


Actual power 
required 


r-xtra power 
required over 
the theoretical 


600 lbs. 


100 lbs. 


158 lbs. 


58 lbs. 58 % 


800 lbs. 


133.3 lbs. 


198 lbs. 


64.3 lbs. 48 % 


1,000 lbs. 


166.7 lbs. 


243 lbs. 


76 lbs. 


45-8 % 


1,200 lbs. 


200 lbs. 


288 lbs. 


88 lbs. 


44 % 



lower ones plain, solid bushings. The rope was 3 strand of jf-ins. cir- 
cumference. The sheaves were of 8 ins. diameter. 

Splicing Manilla Rope 

The following particulars regarding the spUcing of rope and the 
various forms of knots are taken, by permission, from the publications 
of the C. W. Hunt Co. 



I in. diameter 12 ft. 

i\ ins. diameter 14 ft. 

1 5 ins. diameter 16 ft. 



if ins. diameter 18 ft. 

2 ins. diameter 20 ft. 



The splicing of a if-in. rope is shown in Figs. 6 to 9. Begin by 
tieing a piece of twine, 9 and 10, around the rope to be spliced, about 
six feet from each end. Then unlay the strands of each end back to 
the twine. Butt the ropes together, and twist each corresponding 
pair of strands loosely, to keep them from being tangled, as shown in 
Fig. 6. 

The twine 10 is now cut, and the strand 8 unlaid, and strand 7 care- 
fully laid in its place for a distance of four and a half feet from the 
jimction. The strand 6 is next imlaid about one and a half feet, and 
strand 5 laid in its place. The ends of the cores are now cut off so 
they just meet. Unlay strand i four and a half feet, laying strand 
2 in its place. Unlay strand 3 one and a half feet, laying in strand 4. 
Cut all the strands off to a length of about twenty inches, for conven- 



ROPES 



153 



ience in manipulation. The rope now assumes the form shown in 
Fig. 7, with the meeting points of the strands three feet apart. 

Each pair of strands is successively subjected to the following 
operation: 

From the point of meeting of the strands 8 and 7, unlay each one 
three turns; split both the strands 8 and the strand 7 in halves as 
far back as they are now unlaid, and the end of each half strand 




Fig. 6. 




Fig. 




Fig. 8. 




Fig. 9. 
Figs. 6 to 9. — Splicing Manilla rope. 

whipped with a small piece of twine. The half of the strand 7 is now 
laid in three turns, and the half of 8 also laid in three turns. The 
half strands now meet and are tied in a simple knot, 11, Fig. 8, 
making the rope at this point its original size. 

The rope is now opened with a marlin spike, and the half strand of 
7 worked around the half strand of 8 by passing the end of the half 



strand 7 through the rope, as shown in the engraving, drawn taut, and 
again worked around this half strand until it reaches the half strand 
13 that was laid not in. This half strand 13 is now split, and the 
half strand 7 drawn through the opening thus made, and then tucked 
under the two adjacent strands, as shown in Fig. g. The other half 
of the strand 8 is now woimd around the other half strand 7 in the 
same manner. After each pair of strands has been treated in this 
manner, the endsare cut off ati2, lea vingthemaboutfourinches long. 
After a few days' wear they will draw into the body of the rope or 
wear off, so that the locality of the splice can scarcely be detected. 

Knots 

A great number of knots have been devised, of which a few only 
are illustrated, but those selected are the most frequently used. In 
Fig. 10 they are shown open, or before being drawn taut, in order 
to show the position of the parts. The names usually given to them 



A . Bight of a rope. 

B. Simple or Overhand Knot. 

C. Figure 8 Knot. 

D. Double Knot. 

E. Boat Knot. 

F. Bowline, first step. 

G. Bowline, second step. 
H. Bowline completed. 

/. Square or Reef Knot. 

/. Sheet Bend or Weaver's 
Knot. 

K. Sheet Bend, with a toggle. 

L. Carrick Bend. 

M. Stevedore Knot completed. 

N. Stevedore Knot com- 
menced. 

0. Slip Knot. 



P. Flemish Loop. 
Q. Chain Knot, with toggle. 
R. Half-hitch. 
5. Timber-hitch. 
T. Clove-hitch. 
U. Rolling-hitch. 
V. Timber-hitch and Half- 
hitch. 
W. Blackwall-hitch. 
X. Fisherman's Bend. 
Y. Round Turn and Half- 
hitch. 
Z. Wall Knot commenced. 
A A. Wall Knot completed. 
BB. Wall Knot Crown com- 
menced. 
CC. Wall Knot Crown com- 
pleted. 



The principle of a knot is that no two parts, which would move in 
the same direction if the rope were to slip, should lie alongside of and 
touch each other. 

The bowline is one of the most useful knots, it will not sHp, and 
after being strained is easily untied. It should be tied with faciUty 
by every one who handles rope. Commence by making a bight in 
the rope, then put the end through the bight and under the standing 
part, as shown in G, then pass the end again through the bight, and 
haul tight. 

The square or reef knot must not be mistaken for the "granny" 
knot that slips under a strain. Knots H, K, and M are easily untied 
after being under strain. The knot M is useful when the rope passes 
through an eye and is held by the knot, as it will not slip, and is easily 
untied after being strained. 

The timber hitch, S, looks as though it would give way, but it will 
not; the greater the strain the tighter it will hold. The wall knot 
looks complicated, but is easily made by proceeding as foUows: 
Form a bight with strand i, and pass the strand 2 around the end of 
it, and the strand 3 round the end of 2, and then through the bight of 

1, as shown in the engraving Z. Haul the ends taut when the ap- 
pearance is as shown in the engraving ^^. The end of the strand i 
is now laid over the center of the knot, strand 2 laid over i, and 3 over 

2, when the end of 3 is passed through the bight of i, as shown in 
the engraving BB. Haul all the strands taut, as shown in the engrav- 
ing CC. 

The efficiency of knots, as determined at the Massachusetts Insti- 
tute of Technology, is given in Table 7. The efficiency compares 
the strength of the knots with the full strength of the rope. 



154 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Wire Rope 

Standard wire rope for hoisting purposes is composed of 6 strands 
and a hemp center with 19 wires to the strand. Extra pliable 
hoisting rope is of two constructions, one composed of 8 strands and a 
hemp center with 19 wires to the strand and the other of 6 strands 
and a hemp center with 37 wires to the strand. Standard coarse laid 
rope for haulage is composed of 6 strands and a hemp center with 7 
wires to the strand. Ropes are made of Swedish iron, cast steel, 




Fig. 10. — Knots, hitches and bends. 

extra strong cast steel, plough steel and improved plough steel. 
Special tiller rope, hawsers, ship's rigging rope, etc., are also made of 
them. Particulars may be obtained of the makers. 

Sweedish iron rope is soft, tough and pUable and is especially 
adopted for passanger elevators and similar service where the ten- 
dency to abrasion is slight, the speed high, the loads moderate and the 
arrangement of the sheaves such as to produce severe bending stresses 
in the rope. Other materials are used when it is desired to obtain in- 
creased strength or security without increasing the diameter. Cast- 
steel rope is used for general hoisting. Coarse laid rope is much 
stiflfer than standard hoisting rope and requires larger sheaves. A 
higher factor of safety should be used as the breaking of one or two 
wires materially reduces the strength. Tables 8-1 1 give the par- 
ticulars of the leading brands as made by the John A. Roebling's 
Sons Co. and bring out clearly the progressive increase of strength. 
The diameter of a wire rope is that of a true circle enclosing the rope. 

See also the end of this section. 

Splicing Wire Rope 

Wire rope is susceptible of almost perfect splicing and the opera- 
tion is so simple that it may be learned in an hour by any mechanic 
who is at all skillful in the use of ordinary tools. For all kinds of 



transmission rope the long splice is used and should not be less than 
16 ft. in length for J in. rope and increasing to 30 ft. for the 
larger sizes. 

Where the splicing must be done in position, rope blocks are used 

Table 7. — Efficiency of Knots 



Eye- 




Timber 
hitch, 


Bowline 


Square 
knot, 

weavers' 
knot, 
sheet 
bend 


Flemish 


Rope dry, 
average of 


splice 
over an 


Short 
splice in 


round 
turn. 


slip 
knot. 


loop, 
over- 


four tests 
from the 


iron 


the 


and 


■ clove 


hand 


same coil 


thimble 


rope 


half- 


hitch 


knot 


as the 






hitch 






knots 


90 


80 


65 


60 


50 


45 


100 



to draw the wire rope taut, as in Fig. 11, care being taken to make 
fast far enough from the ends to leave plenty of room for the splice 
and the men who make it. If possible, it is better to hold the rope 
taut, mark the splice on both ends, by securely winding with No. 20 
annealed-iron wire, throw it off the sheaves and make the splice on 
the floor or staging, as may be most convenient. 

The strands of both ends are unlaid, back to the points wound 
with wire, the hemp core cut off and the ends brought together with 
the strands interlaced. Fig. 12. Any strand, as a, is imlaid and 
closely followed by the corresponding strand i of the other end of 
the rope which is pressed closely into the groove left by the unlaid 
strand. The unwinding of one strand and the inwinding of the other 
are continued until all but about 12 ins. of strand are inlaid, when 
a is cut off at the same length with a sharp chisel. See Fig. 13. 
Strands 4 and d are next treated in the same way and the process 
is repeated with each pair of strands until all are laid and cut as in 
Fig. 14. 

Around each point where the free strands cross, a few turns of 
stout twine are made and the length of the splice is bent and worked 
in all directions until the tension in all the strands is equal and the 
rope as flexible there as elsewhere. If this is not done and there is 
more tension in some of the strands than in others when a stress is 
put on the rope, these strands will pull into the rope, making a bad- 
looking and weak spUce. 

Next, the open or free ends of the 1 2 strands are carefully trimmed 
and served or wound with fine wire, and two rope and stick clamps. 
Fig. 15, are secured to the rope, one on each side of an end crossing, 
as in Fig. 18, for the purpose of aiding in tucking the strand ends 
into the middle of the rope. 

There are two ways of tucking in these ends. They are first 
straightened with a mallet. The long ends of the rope-clamp handles 
are twisted in opposite directions, separating the strands and exposing 
the hemp core, which is cut off and puUed out between the points 
to where the tucked-in strands will reach and the ends forced into 
the place formerly occupied by the core. 

This is most easily done with the aid of a marine spike, which is 
passed over the strand which is to be tucked and under two strands 
of the rope, Fig. 16, and moved along the rope spirally following the 
lay and forcing the free end into the core space. Fig. 17. 

In the other method the strands are more widely separated by 
untwisting the rope with the clamps. Fig. 19, slipping the free end 
in between the strands and correcting slight kinks by the use of a 
mallet. 

The order in which the ends are tucked in is immaterial. Some 
operators prefer to tuck aU the ends pointing in one direction before 
any of those pointing the opposite way, while others finish each pair 
of ends in series. 

If the foregoing directions are intelligently followed the splice will 
be uniform with the rest of the rope, of nearly equal strength through- 
out, and after a few hours' use it will be almost impossible to detect 
the splice. (F. L. Johnson Poicer, Jan. 30, 1912). 



ROPES 



155 




Figs, ii to 19. — Splicing wire rope. 



Hoisting Drums 

A superior construction of large hoisting drums, including the lead- 
ing dimensions, by the Nordberg Mfg. Co. {Am. Mach., Sept. 21, 1899) 
is shown in Fig. 20. The customary practice with such drums is to 
have a number of spiders, four or five, at different points of the shaft 
for supporting the drum. These drums are, however, of consider- 
able length and great stiffness, and the long shafts, instead of sup- 
porting the drum, should be supported by the drum. The drum 
shown rests on two spiders only, one at each end, the shaft being 
supported by tension rods from the shell of the drum. The detail 
drawing of the drum, Fig. 20, shows plainly the mode of construc- 
tion. In Fig. 21 the position of the spiders is represented before the 



longitudinal rods A, Fig. 22, are put in. The deflection of the shaft 
is considerable, making the distance x between spiders on top less 
than xi on bottom. This deflection also causes the weight to come 
entirely on the edges of the bearings nearest to the drum. In 
erecting the drum, rods A are first put in place, and their tension is 
so adjusted as to make the faces BB, Fig. 22, perfectly parallel and 
bearings e level. Then the drum shell is put in place, and lastly the 
diagonal tension rods are so adjusted as to take out the deflection 
in center of shaft, as shown by d, Fig. 22. The shell and spiders 
are of abundant strength to transmit the whole power of the engines. 
There is a reel on each end, mounted on shafts inside the drum, for 
taking up the slack rope. 



156 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 




Fig. 20. — Nordberg construction of large hoisting drums. 




Fig, 21. 



K 



\i 



Fig. 22. 



Figs. 21 and 22. — Method of erecting the hoisting drum 
shown in Fig. 20. 

Durability of Wire Rope 

The durability of wire ropes f»rmed the subject of a paper by Daniel 
Adamson {Proc. I. M. E. 1912) from which the following is taken. 

If the wires are too large they are stressed considerably when pas- 
ing over the pulleys, and accordingly the material is quickly fatigued 
and the wires break. Smaller wires, on the other hand, are more 
quickly worn through by rubbing against the pulleys and against 
their neighbors in the body of the rope. 



Investigations of the durability to be expected from consideration 
of working stresses lead to calculated results that are never experi- 
enced and that cannot reasonably be expected, and it must be taken 
for granted that abrasion is the principal factor in limiting the life 
of wire ropes. 

Comparing two ropes of equal size, one from wires half the diameter 
of the other, when the rope of finer wires is passing over the pulley, 
there being four times as many wires in it, the pressure at each point 
of contact between the rope and the pulley and beween the individual 
wires of the rope may be assumed to be one-quarter of what it is 
in the rope of larger wires. The wires being of half the diameter 
the damage done to them by contact, even under this lower pressure, 
will be at least half as much as occurs to the coarser wires in the other 
rope, and this half damage done to a wire of one-quarter the sectional 
area will result in the cutting through of the wire in half the time, so 
that the effect of abrasion upon the rope of finer wires will be twice as 
great. If a smaller pulley be used for the rope of finer wires, as sug- 
gested by some authorities, the pressure at the points of contact and 
the stress due to bending will be proportionately increased, so that 
it may reasonably be expected that with a pulley diameter bearing 
the same proportion to the diameter of the wires, the hfe of the rope 
with fine wires will be one-quarter of that of the rope of coarser wires 
working over a pulley of correspondingly increased diameter. 

Mr. Adamson quotes from experiments by A. S. Biggart (Proc. 
I. C. E., Vol. loi) on apparatus consisting of two pulleys around which 
the rope under trial was passed, the lower piilley being weighted to 
give the required tension on the rope which was passed, under a nor- 
mal working load, to and fro over the pulleys until breakage ensued. 
Experiments were repeated with different diameters of pulleys and 
difierent makes of rope. 

{Continued on page 158, first column) 



ROPES 



157 



Table 8. — Standard Hoisting Rope Composed or 6 Strands 
AND A Hemp Center with 19 Wires to the Strand 

Swedish Iron 



Extra Strong Cast Steel 





Approxi- 


Approxi- 


Approxi- 
mate 
strength, 
tons of 


Proper 


Diameter of 


Diameter, 


mate cir- 


mate 


working 


drum or 


ins. 


cumfer- 


weight 


load, tons 


sheave in ft. 




ence, ins. 


per ft., lbs. 


of 2000 lbs. 


advised 








2000 lbs. 






2| 


8f 


II -95 


Ill 


22.2 


17 


2i 


7i 


9.8s 


92 


18.4 


IS 


2i 


7i 


8 


72 


14.4 


14 


2 


6i 


6.30 


5S 


II 


12 


II 


Si 


S-5S 


50 


10 


12 


li 


Si 


4-8S 


44 


8.8 


II 


li 


s 


4. IS 


38 


7.S 


10 


li 


4f 


3.5s 


33 


6.S 


9 


li 


4i 


3 


28 


S.S 


8.S 


li 


4 


2.4s 


22.8 


4-S6 


7S 


li 


3i 


2 


18.6 


3-72 


7 


I 


3 


I.SS 


14. S 


2.90 


6 


i ' 


2j 


I .20 


II. 8 


2.36 


SS 


i 


2l 


.89 


8.S 


1.70 


4-S 


i 


2 


.62 


6 


1 .20 


4 


iV 


l| 


• SO 


4-7 


.94 


3-5 


i 


Ij 


.39 


3.9 


.78 


3 


A 


li 


• 30 


2.9 


.58 


2.7S 


f 


Ij 


.22 


2.4 


.48 


2.25 


i's 


I 


•IS 


I.S 


• 30 


2 


i 


i 


. 10 


I. I 


. 22 


I. SO 



Cast Steel 



Diameter, 
ins. 


Approxi- 
mate cir- 
cumfer- 
ence, ins. 


Approxi- 
mate 
weight 
per ft., lbs. 


Approxi- 
mate 
strength, 
tons of 
2000 lbs. 


Proper 

working 

load, tons 

of 2000 

lbs. 


Diameter 

drum or 

sheave in ft. 

advised 


2i 


8f 


II-9S 


211 


42.2 


II 


2\ 


7i 


9.8s 


170 


34 


10 


2i 


7i 


8 


133 


26.6 


9 


2 


6i 


6.30 


106 


21.2 


8 


n 


5f 


s.ss 


96 


19 


8 


li 


Si 


4-8s 


85 


17 


7 


li 


S 


4-IS 


72 


14.4 


6i 


a 


4i 


3-SS 


64 


12.8 


6 


ii 


4i 


3- 


S6 


II. 2 


Si 


li 


4 


2.4s 


47 


9.4 


5 


li 


3i 


2 


38 


7.6 


4i 


I 


3 


I.S8 


30 


6 


4 


\ 


2i 


1.20 


23 


4.6 


3i 


\ 


2i 


.89 


17. S 


3.S 


3 


\ 


2 


.62 


12. 5 


2.5 


2i 


A 


Ij 


• SO 


10 


2 


2i 


h 


li 


.39 


8.4 


1.68 


2 


A 


li 


.30 


6.S 


1.30 


I| 


\ 


li 


.22 


4.8 


.96 


li 


h 


I 


.IS 


3.1 


.62 


li 


i. 


i 


. 10 


2.2 


■ 44 


I 



Diameter, 
ins. 


Approxi- 
mate cir- 
cumfer- 
ence, ins. 


Approxi- 
mate 
weight 
per ft., lbs. 


Approxi- 
mate 
strength, 
tons cf 
2000 lbs. 


Proper 

working 

load, tons 

of 2000 

lbs. 


Diameter 

of drum or 

sheave in ft. 

advised 


2i 


8f 


II-9S 


243 


48.6 


II 


2i 


7i 


9.8s 


200 


40 


10 


2i 


7i 


8 


160 


32 


9 


2 


6i 


6.3 


123 


24.6 


8 


^i 


Si 


S-SS 


112 


22.4 


8 


li 


Si 


4.8s 


99 


19.8 


7 


li 


s 


41S 


83 


16.6 


6i 


li 


4i 


3.SS 


73 


14'. 6 


6 


i| 


4i 


3 


64 


12.8 


Si 


ij 


4 


2.4s 


S3 


10.6 


S 


Ii 


3i 


2 


43 


8.6 


4i 


I 


3 


I.S8 


34 


6.80 


4 


i 


2i 


I . 20 


26 


5-20 


3i 


\ 


2i 


.89 


20.2 


4.04 


3 


\ 


2 


.62 


14 


2.80 


2i 


A 


li 


• SO 


II . 2 


2.24 


2i 


i 


li 


•39 


9^2 


1.84 


2 


A 


li 


• 30 


7.25 


1.4s 


1} 


1 


Ii 


.22 


530 


1 . 06 


Ii 


A 


I 


•IS 


350 


• 70 


li 


i 


h 


• 10 


2^43 


• 49 


I 



Plough Steel 



Diameter, 
ins. 


Approxi- 
mate cir- 
cumfer- 
ence, ins. 


Approxi- 
mate 
weight 
per ft., lbs. 


Approxi- 
mate 
strength, 
tons 
of 2000 Ibs^ 


Proper 

working 

load, tons 

of 2000 

lbs. 


Diameter of 
drum or 

sheave in ft. 
advised 


2\ 


8f 


11.95 


275 


SS 


II 


2\ 


7l 


9.8s 


229 


46 


10 


2i 


7i 


8 


186 


37 


9 


2 


6i 


6.3 


140 


28 


8 


Ij 


si 


5SS 


127 


25 


8 


l] 


5i 


4.85 


112 


22 


7 


Ii 


5 


4^15 


94 


19 


6i 


Ii 


4i 


3^55 


82 


16 


6 


i| 


4i 


3 


72 


14 


• 5i 


Ii 


4 


2.4s 


S8 


12 


5 


li 


3i 


2 


47 


9.4 


4i 


I 


3 


I. 58 


38 


7^6 


4 


{ 


2i 


1.20 


29 


5^8 


3i 


3 


2i 


.89 


23 


4.6 


3 


1 


2 


.62 


15^5 


31 


2i 


A 


li 


• so 


12.3 


2^4 


2i 


i 


jl 


• 39 


10 


2 


2 


A 


Ii 


• 30 


8 


1^6 


li 


§ 


i| 


.22 


5-7S 


I^IS 


li 


A 




• 15 


3.8 


• 76 


li 


i 


i • 


. 10 


2.6s 


• 53 


I 



Improved Plough Steel 



Diameter, 
ins. 


Approxi- 
mate cir- 
cumfer- 
ence, ins. 


Approxi- 
mate 
weight 
per ft., lbs. 


Approxi- 
mate 
strength, 
tons of 
2000 lbs. 


Proper 

working 

load, tons 

of 2000 

lbs. 


Diameter 

of drum of 

sheave in ft. 

advised 


Diameter, 
ins. 


Approxi- 
mate cir- 
cumfer- 
ence, ins. 


Approxi- 
mate 
weight 
per ft., lbs. 


Approxi- 
mate 
strength, 
tons of 
2000 lbs. 


Proper 

working 

load, tons 

cf 2000 

lbs. 


Diameter 

of drum of 

sheave in ft. 

advised 


2} 


8! 


11-95 


31s 


63 


n 


li 


3i 


2 


56 


II 


4i 


2} 


7i 


9^85 


263 


S3 


10 


I 


3 


1-58 


45 


9 


4 


2i 


7i 


8 


210 


42 


9 


i 


2i 


1 . 20 


35 


7 


3i 


2 


6i. 


6.30 


166 


33 


8 


i 


2i 


.89 


26.3 


5-3 


3 


1} 


Si 


S-SS 


ISO 


30 


8 


i 


2 


.62 


19 


3-8 


2\ 


li 


si 


+•85 


133 


27 


7 


A 


li 


• SO 


14-5 


2.9 


2i 


ij 


5 


4.1s 


no 


22 


6i 


i 


Ii 


.39 


12. I 


2.4 


2 


Ii 


4i 


3-SS 


98 


20 


6 


A 


li 


■ 30 


9-4 


1-9 


1! 


If 


4i 


3 


84 


17 


Si 


f 


Ii 


.22 


6.75 


1-35 


li 


Ii 


4 


2.4s 


69 


14 


s 


A 


I 


-IS 


4-50 


.9 


li 














i 


i 


. 10 


3-15 


.63 


I 



158 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 9.— Extra Pliable Hoisting Rope Composed of 8 Strands and a Hemp Center with 19 Wires to the Strand 



Cast Steel 



Diameter, 
ins. 


Approxi- 
mate cir- 
cumfer- 
ence, ins. 


Approxi- 
mate 
weight 
per ft., lbs. 


Approxi- 
mate 
strength, 
tons of 
2000 lbs. 


}'roper 

working 

load, tons 

of 2000 

lbs. 


Diameter 

drum oi 

sheave in ft. 

advised 


li 


4i 


3.19 


58 


II. 6 


375 


i| 


4i 


2.70 


51 


10.2 


35 


li 


4 


2 .20 


42 


8.4 


3^2 


li 


3^ 


1.80 


34 


6.8 


2.83 


I 


3 


1.42 


26 


5-2 


2.5 


i 


2£ 


1.08 


20 


4 


2.16 


i 


2i 


.80 


15.3 


3.06 


1.83 


! 


2 


.56 


10. 9 


2.18 


1-75 


A 


I| 


■45 


8.7 


1.74 


I.S 


i 


1} 


■ 35 


7-3 


1.46 


1.33 


TS 


li . 


.27 


5-7 


1. 14 


1. 16 


t 


I J 


.20 


4.2 


.84 


I 


A 


I 


.13 


2. 75 


■ 55 


.83 


i 


J 


.09 


1.80 


.36 


.75 





Extra Strong 


Cast Steel 






Diameter, 
ins. 


Approxi- 
mate cir- 
cumfer- 
ence, ins. 


Approxi- 
mate 
weight 
per ft., lbs. 


Approxi- 
mate 
strength, 
tons of 
2000 lbs. 


Proper 

working 

load, tons 

of 2000 

lbs. 


Diameter of 
drum or 

sheave in ft. 
advised 


a 


4i 


3-19 


66 


13 


3-75 


If 


4i 


2.70 


57 


II 


3.5 


li 


4 


2. 20 


47 


9-4 


3.2 


U 


3i 


1.80 


38 


7.6 


2.83 


I 


3 


1.42 


29.7 


5-9 


2.5 


i 


2i 


I .08 


23 


4.6 


2. 16 


i 


2i 


.80 


17.6 


3-5 


1.83 


1 


2 


.56 


12.4 


2.5 


1.75 


TS- 


If 


.45 


10. I 


2 


1.5 


i 


li 


• 35 


8 


1.6 


1.33 


A 


li 


■ 27 


6.30 


1.26 


1. 16 


t 


■ li 


.20 


4.66 


■ 93 


I 


A 


I 


.13 


3.0s 


.61 


.83 


i 


3. 


.09 


2.02 


.40 


.75 



The eflect of oiling the ropes was found to be very beneficial, in- 
creasing the life of a given rope by two or three times. Experiments 
were also made to ascertain the effect on the life of a rope of running 
it over pulleys so arranged that the rope was subjected to reverse 
stresses, Fig. 23. The results obtained from this series of experiments 
showed that, generally, the life of a rope working under such conditions 
was only one-half as long as a similar rope bent in one direction only. 

The experiments show that when the first wire breaks, the rope may 
be assumed to have passed through one-half of its life, and as no one 
knowingly works a rope until it breaks entirely, then the breakage 
of even a few wires is a sign that a rope should be carefully watched 
and replaced by a new one at an early opportunity. 

The effect of varying the proportions of diameter of pulley to 
diameter of rope is one of the most important features to be noticed. 
Speaking generally, Mr. Biggart's experiments show that increasing 
the diameter of the pulleys by an amoiuit equal to two circumferences 
of the rope will double the life of the rope. This is approximately 
correct for all the varieties of rope and conditions experimented with, 
and may therefore be taken as equally correct for all the varying 
conditions imder which cranes are worked. It is very remarkable 
that so simple a rule should evolve from such numerous and varied 
experiemnts. 

These conclusions enable one to express a definite value for the 
effect upon the durability of ropes, of the various arrangements of pul- 
leys that are commonly adopted in overhead cranes, some of which are 
illustrated in Figs. 24 to 30. Assuming that Fig. 25, in which the ropes 
make three bends in working, namely, one at the upper drum and one 
on each side of the lower pulley, i.e., at entering and leaving, is the 







Plough Steel 






Diameter, 
ins. 


Approxi- 
mate cir- 
cumfer- 
ence, ins. 


Approxi- 
mate 
weight 
per ft., lbs. 


Approxi- 
mate 
strength, 
tons of 
2000 lbs. 


Proper 

working 

load, tons 

of 2000 

lbs. 


Diameter of 
drum or 

sheave in ft. 
advised 


li 


4i 


3.19 


74 


14.8 


3.75 


i| 


4i 


2.70 


64 


12.8 


3.S 


li 


4 


2.20 


52 


10.4 


3.2 


ij 


3j 


1 .80 


43 


8.6 


2.83 


I 


3 


1.42 


33 


6.6 


2.5 


i 


2f 


1 .08 


26 


5.2 


2. 16 


1 


2i 


.80 


20 


4 


1.83 


i 


2 


.56 


14 


2.8 


I. 75 


A 


li 


■ 45 


II. 6 


2.32 


I 50 


5 


a 


.35 


8.7 


1.74 


I 33 


A 


li 


.27 


6.90 


1.38 


1. 16 


f 


li 


.20 


5.12 


1.02 


I 


A 


I 


.13 


3.35 


.67 


.83 


i 


^ 


■ 09 


2.2s 


■ 45 


.75 



Improved Plough Steel 



Diameter, 
ins. 


Approxi- 
mate cir- 
cumfer- 
erence, ins. 


Approxi- 
mate 
weight 
per ft., lbs. 


Approxi- 
mate 
strength, 
tons of 
2000 lbs. 


Proper 

working 

loads, tons 

of 2000 

lbs. 


Diameterof 
drum or 

sheave in ft. 
advised 


li 


4i 


3.19 


80 


16 


3.75 


i| 


4i 


2.70 


68 


13 


3.5 


li 


4 


2.20 


56 


II 


3.2 


li 


3i 


1.80 


46 


9.2 


2.83 


I 


3 


1.42 


36 


7.2 


2.5 


i 


2i 


1.08 


28 


5.6 


2. IS 


i 


2} 


.80 


22 


4.4 


1.83 


i 


2 


.56 


15 


3 


1.75 


A 


il 


.45 


12 


2.4 


I. 5 


i 


15 


.35 


9-5 


1.9 


1-33 



Table 12. — Comparison of Anticipated Length of Life of 
Ropes Arranged as Shown in Figs. 5 to ii 



Fig. No. 


Number of bends 


Relative life of rope 


24 


I 


300 


25 


3 


100 


26 


3^ 


75 


27 


7 


43 


28 


II 


27 


29 


7' 


37l 


30 


III 


25 



' Including one reverse bend which is twice as effective in wearing out 
the rope. 

Table 13. — Required Increase in Diameters of Rope Drums 
(Measured in Terms of Circumference of Rope) Re- 
quired to Give Equal Durability 



Fig. No. 


Increase over diameter called for by Fig. 25 


26 

27 
28 

29 
30 


I circumference of rope 
2I circumferences of rope 
4 circumferences of rope 

3 circumferences of rope 

4 circumferences of rope 



arrangement most frequently adopted in practice, and representing 
the anticipated life of the rope under these conditions by 100, then 
the relative lives of the ropes in each of the other arrangements in- 
dicated will be shown in Table 12. 

If it be desired to design each of the above arrangements of pulleys 
so that the ropes shall have equal durability, then the ratio of the 
drum diameters to rope circumferance (if the law mentioned above 
is to be relied upon) must be increased, as shown in Table 13. 

(Continued on page i6o, second column) 



ROPES 



159 



Table io. Extra Pliable hoisting Ropes Composed of 6 Strands and a Hemp Center with 37 Wires to the Strand 

Cast Steel Plough Steel 



Diameter, 
ins. 



If 

li 
I !- 



Approxi- 
mate cir- 
cumfer- 
ence, ins. 



7i 
7h 
6i 



4i 

4i 

4 

3-i 



Approxi- 
mate 
weight 
per ft., lbs. 



11.95 
9.8s 
8 

6.30 
4-85 

4-iS 
3-55 
3 

2.4s 



1.58 



.89 
.62 



■ 39 
• 30 



Approxi- 
mate 
strength, 
tons of 
2000 lbs. 



200 
160 

125 

los 

84 

71 
63 
55 
45 
34 



29 
23 

17.5 

II .2 

9-5 



7-25 

5-5 

4.2 



Proper 

working 

load, tons 

of 2000 

lbs. 



40 
32 
25 



9 

7 

6 
5 
3.5 

2 . 2 
1-9 

1-45 
I . I 
.84 



Diameter of 
drum or 

sheave in ft. 
advised 



3-75 
3-5 
3.2 
2.83 

2.S 

2. 16 
1.83 

1-75 
IS 

1-33 
1. 16 





Extra Strong Cast Steel 






Diameter, 
ins. 


Approxi- 
mate cir- 
cumfer- 
ence, ins. 


Approxi- 

mate 

weight 

per ft, lbs. 


Approxi- 
mate 
strength, 
tons of 
2000 lbs. 


Proper 

working 

load, tons 

of 2000 

lbs. 


Diameter of 
drum or 

sheave in ft. 
advised 


2i 


8| 


II .95 


233 


47 
37 




2j 


7s 


9.85 


187 




2i 


7s 


8 








2 


6i 




117 






li 


55 


4.85 


95 


19 




if 


5 


4.15 


79 


16 




li 


4i 


3-55 


71 


14 


3.7S 


I| 


4l 


3 


61 


12 


35 


li 


4 


2.45 


50 


10 


3-20 


ij 


3l 


2 


39 


8 


2.83 


I 


3 


1.58 


32 


6.4 


2.S 


i 


2i 


1 .20 


25 


5 


2.16 


i 


2i 


.89 


19 


3.8 


1.83 


L 


2 


.62 


12.6 


2.5 


I.7S 


ft 


II 


• SO 


10. 5 


2 . 1 


1-5 


i 


Ij 


• 39 


8.2s 


1.65 


1.33 


A 


li 


• 30 


6.35 


1.27 


1. 16 


"B 


li 


. 22 


4.65 


.93 


I 



Diameter, 
ins. 



2i 

2i 

2i 



I| 

If 
li 



Approxi- 
mate cir- 
cumfer- 
ence, ins. 



Approxi- 
mate 
weight 
per ft., lbs. 



8| 
(>i 
7i 
6i 
5^ 



4j 
4i 
4 
3^ 



li 
li 



11.95 
9.8s 
8 

6.30 
4-85 

4-15 
3-55 
3 

2.45 



1.58 



.62 
• SO 



• 39 

• 30 



Approxi- 
mate 
strength, 
tons of 
2000 lbs. 



265 
214 
175 
130 
108 

90 
80 
68 
55 

44 

35 
27 

21 
14 

"•5 

9-25 

7^2 
S. I 



Proper 

working 

load, tons 

of 2000 

lbs. 



S3 
43 
3S 
26 



18 
16 
14 



1.85 
1.4 



Diameter of 
drum or 

sheave in ft. 
advised 



375 
35 
3. 2 
2.83 

2.5 
2.16 
1.83 
1-75 
I- 5 

1-33 
I. 16 



Improved Plough Steel 



Diameter, 
ins. 


Approxi- 
mate cir- 
cumfer- 
ence, ins. 


Approxi- 
mate 
weight 
per ft., lbs. 


Approxi- 
mate 
strength, 
tons of 
2000 lbs. 


Proper 

working 

load, tons 

of 2000 

lbs. 


Diameter of 

drum or 

sheave in ft. 

advised 


2i 


8f 


11-95 


278 


55 




2i 


7i 


9. 85 


225 


45 
37 





2i 


7j 
(>i 


8 


184 




2 


6.30 






li 


5^ 


4-8s 


113 


23 




ih 


5 


4. 15 


95 






4i 


3.55 


84 


J7 


3.7S 


If 


4i 


3 


71 


14 


3 50 


li 


4 


2.45 


58 


II 


3-20 


I* 


3i 


2 


46 


9.2 


2.83 


I 


3 


1.58 


37 


7.4 


2.50 


i 


2i 


I . 20 


29 


5.8 


2. 16 


i 


2i 


•89 


23 


4^6 


1.83 


i 


2 


.62 


16 


3^2 


1-75 


ft 


li 


• SO 


12.5 


2^5 


I .50 


i 


i| 


• 39 


9.75 


1.9 


1.33 


T% 


1} 


•30 


7^50 


I^S 


I^IS 


t 


il 


.22 


S^30 


1 . 06 


I 




Eleven Bends 

Fig. 28 



Seven Bends, 
One Reverse 



Eleven Bends, 
One Reverse 



Three Bends, 
Large Bottom Pulleya 



Fig. 29 Fig. 30 Fig.'SI 

Figs. 23 to 31. — Various arrangements of wire ropes on cranes. 



160 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table h. — Standard Coarse Laid Rope tor Haulage and Transmission Composed of 6 Strands and a Hemp Center with 7 Wires 

TO THE Strand 



Swedish Iron 



Diameter, ins. 


Approxi- 
mate cir- 
cumfer- 
ence, ins. 


Approxi- 
mate 
weight, 
per ft., lbs. 


Approxi- 
mate 
strength, 
tons of 
2000 lbs. 


Proper 

working 

load, tons 

of 2000 

lbs. 


Diameter of 
drum or 

sheave in ft. 
advised 


li 


4l 


3.55 


32 


6.4 


16 


li 


4i 


3 


28 


5.6 


IS 


li 


4 


2.45 


23 


4.6 


13 


li 


3J 


2 


19 


3.8 


12 


I 


3 


1.58 


15 


3 


10} 


i 


2j 


I, 20 


12 


2.4 


9 


1 


2i 


.89 


8.8 


1.7 


Ti 


a . 


2i 


.75 


7.3 


i.S 


7i 


1 


2 


.62 


6 


1 . 2 


7 


A 


li 


.50 


4.8 


.96 


6 


i 


li 


• 39 


3.7 


■74 


Si 


A 


li 


.30 


2.6 


• 52 


4i 


f 


li 


. 22 


2. 2 


•44 


4 


A 


I 


•IS 


1.7 


•34 


3J 


A 


J 


.I2S 


1.2 


■24 


3 



Extra Strong Cast Steel 



Diameter, ins. 


Approxi- 
mate cir- 
cumfer- 
ence, ins. 


Approxi- 
mate 
weight 
per ft. lbs. 


Approxi- 
mate 
strength, 
tons of 
2000 lbs. 


Proper 
working 
load, tons 
tons of 
2000 lbs. 


Diameter 

of drum or 

sheave in ft. 

advised 


li 


4f 


3.SS 


73 


14.6 


II 


If 


4i 


3 


63 


12.6 


10 


I4 


4 


2.45 


54 


10.8 


9 


l| 


3} 


2 


43 


8.6 


8 




3 


1.58 


35 


7 


7 


1 


2i 


1.20 


28 


5^6 


6 


5 


2i 


.89 


21 


4^2 


S 


ii 


2i 


• 7S 


16.7 


3.3 


4f 


1 


2 


.62 


14-5 


2^9 


4^ 


A 


if 


■50 


II 


2.2 


4 


J 


I^ 


.39 


8.8s 


1^8 


3j 


16 


li 


■30 


6.2s 


I.2S 


3 


1 


Ij 


.22 


5.25 


I. OS 


2l 


ft 


I 


•IS 


3 -95 


.79 


2l 


A 


i 


.I2i 


2.95 


.59 


If 



Cast Steel 



Plough Steel 



Diameter, ins. 


Approxi- 
mate cir- 
cumfer- 
ence, ins. 


Approxi- 
mate 
weight 
per ft. lbs. 


Approxi- 
mate 
strength, 
tons of 
2000 lbs. 


Proper 
working 
load, tons 
tons of 
2000 Ibs^ 


Diameter of 
drum or 

sheave in ft. 
advised 


a 


4l 


355 


63 


12.6 


II 


If 


41 


3 


53 


10.6 


lO 


li 


4 


2^45 


46 


9^2 


9 


li 


3l 


2 


37 


7^4 


8 


I 


3 


1.58 


31 


6.2 


7 


1 


2j 


I .20 


24 


4.8 


6 


3 


2i 


.89 


18.6 


3.7 


S 


a 


2i- 


.75 


IS^4 


3^1 


4f 


f 


2 


.62 


13 


2.6 


4l 


A 


ij 


.50 


10 


2 


4 


i 


Ij 


• 39 


7^7 


1-5 


3i 


A 


u 


• 30 


5^5 


I . I 


3 


f 


li 


.22 


4.6 


.92 


2i 


ft 


I 


•IS 


3^5 


.70 


2i 


A 


1 


• I2i 


2.5 


.50 


li 



Diameter, ins. 


Approxi- 
mate cir- 
cumfer- 
ence, ins. 


Approxi- 
mate 
weight 
per ft., lbs. 


Approxi- 
mate 
strength, 
tons of 
2000 lbs. 


Proper 

working 

load, tons 

of 2000 

lbs. 


Diameter 
of drum or, 
sheave in ft. 
advised 


i^ 


4i 


3-55 


82 


16.4 


II 


li 


4i 


3 


72 


14.4 


10 


1} 


4 


2.45 


60 


12 


9 


li 


3i 


2 


47 


9^4 


8 


' 


3 


I.S8 


38 


7-6 


7 


i 


2i 


I .20 


31 


6.2 


6 


i 


2i 


.89 


23 


4^6 


5 


H 


2j 


• 75 


18 


3^6 


4f 


1 


2 


.62 


16 


3.2 


4i 


ft 


li 


• 50 


12 


2.4 


4 


i 


I^ 


■39 


10 


2 


3J 


A 


li 


.30 


7 


1.4 


3 


1 


ll 


.22 


5.9 


1.2 


2i 


ft 


I 


• IS 


4.4 


.88 


2i 


A 


J 


• I2S 


3.4 


.68 


li 



Improved Plough Steel 



Diameter, ins. 


Approxi- 
mate cir- 
cumfer- 
ence, ins. 


Approxi- 
mate 
weight 
per ft., lbs. 


Approxi- 
mate 
strength, 
tons of 
2000 lbs. 


Proper 

working 

load, tons 

of 2000 

lbs. 


Diameter 

of drum or 

sheave in ft. 

advised 


li 


4i 


3^55 


90 


18 


II 


i| 


4i 


3 


79 


16 


10 


il 


4 


2.45 


67 


13 


9 


ij 


3i 


2 


52 


10 


8 


I 


3 


1.58 


42 


8.4 


7 


i 


2f 


I . 20 


33 


6.6 


6 


a 


2i 


.89 


25 


5 


5 


ii 


2i 


• 75 


20 


4 


4f 


-1 


2 


.62 


I7i 


3.5 


4i 


ft 


li 


• SO 


13 


2.6 


4 


3 


li 


.39 


II 


2.2 


3i 


re 


Ii 


.30 


7i 


1.5 


3 


1 


Ii 


.22 


6i 


1^3 


2i 



It is quite usual for purchasers to specify in their inquiries that the 
diameters of the pulleys and drums must bear a certain relation to 
the diameter of the rope, but this stipulation is not sufficient in itself 
without some consideration being also given to the arrangement of 
the rope and pulleys. 

If the generally accepted ratio of seven circumferences, or twenty- 
two diameters, of the rope for the diameter of the barrel be assumed as 
suitable for the drum and pulleys as in Fig. 25, then the diameters 
for the other figures, to give equal dtirabiUty, should be as shown in 
Table 14. 

To make the comparisons quite fair between the different arrange- 
ments it must now be pointed out that, owing to the increased number 
of falls of rope adopted in Figs. 27 and 29, the size of the rope maybe 
reduced as shown in Table 15 while retaining the same factor of 
safety. 

Combining the figures given in Tables 14 and 15 will give drum and 
pulley diameters as shown in Table 16. 

The noticeable feature in the last table is that whether two, four, 
or six falls are adopted, the diameter of the drum and pulleys should 



ROPES 



161 



Table 14. — Ratio of Diameter of Pulleys and Drums to 
Circumference of Rope to Give Equal Durability 



Fig. No. 


Ratio of pulley and drum diameter to 


rope circumference 


24 


4 to I 


25 


7 to I 


26 


8 to I 


27 


9.5 to I 


28 


II to I 


29 


10 to I 


30 


II to I 



Table 15. — Relative Rope Circumference Allowing for 
Smaller Ropes Due to Increased Number of Falls 



Fig. No. 


Number of falls 


Relative rope circumference 


24 


2 


140 


25 


4 


100 


26 


4 


100 


27 


8 


70 


28 


12 


57 


29 


8 


70 


30 


12 


57 



remain about the same if the ropes are to have equal durability (com- 
pare Figs. 27 and 28 with Fig. 25). It is clear that very large pro- 
portions are necessary to insure a reasonable life for ropes on cranes 
with many falls of rope. Reference to Fig. 26 and Fig. 29 in Table 
16 shows the increase that should be made in the diameter of the drum 
and pulleys if a reverse bend occurs in the run of the rope. 

In Fig. 25, as already mentioned, the ropes make two bends at the 
lower pulleys to one at the drum, and therefore, if the lower pulleys 
are made of the same diameter as the drum, they will be responsible 
for two-thirds of the wear and tear of the rope. It is usually difficult 
to increase the diameter of the working barrel or drum of a crane. 



Table 16. — Drum and Pulley Diameters Resulting from a 
Combination of Tables 14 and 15, and Still Assuming That 
100 Represents the Condition in Fig. 25 



Fig. 
No. 


Ratio of pulley and 

drum diameter to 

rope circumference 

according to 

Table 14 


Relative circum- 
ference of rope as 
per Table 1 5 


Resultant pulley 

and drum diameter 

assuming 

Fig. 25 = 100 


24 


4 


140 


80 


25 


7 


100 


100 


26 


8 


100 


114 


27 


9i 


70 


95 


28 


11 


57 


90 


29 


10 


70 


100 


30 


II 


57 


90 



because to do so affects the ratio of the gearing and also requires a 
much larger framework with a correspondingly greatly increased 
cost of manufacture, but if it is agreed, as a result of Mr. Biggart's 
experiments, that increasing the diameter of the pulley, over which a 
loaded rope passes, by an amount equal to twice the circumference 
of the rope, reduces the evil effects of bending the rope round it to 
one-half, then a simple means of improving the durability of crane 
ropes is immediately at the disposal of the designer, namely, to in- 
crease the diameter of the pulleys in the blocks, leaving the drums of 
the original size, as indicated by Fig. 31. This alteration can usually 
be efife.cted without serious alteration of the design, and may even be 
carried out on existing cranes. 

The result of increasing the diameter of the pulleys, as shown by 
Fig. 31, by an amount equal to two circumferences of the rope, will 
be that the effect of the double bend around the lower pulley is 
halved, and the resultant effect of the three bends will be equal to 




^^__ * ' Ground Line 

British System, Open Drive 



Ground Li ne 
British System, Up and Over Drive 



11 



Fig. S2. — Arrangements of rope drives in efficiency tests. 



162 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



American 



British 



100 

S 90 



80 



70 



i 

if li 



100 



70 



100 



90 



80 



70 



100 
90 

& 80 

a 

0) 

n 

60 



50 



100 
g 90 

H 80 
70 



7— I =^ - 



2500 



3500 4500 

Eope Speed, Ft. per Min. 



5500 



2500 



3500 4500 

Rope Speed, Ft. per Min. 



5500 



Open drive, 50-ft. centers, 6 ropes, j load. 
Relation of efficiency and speed. 









































90 

>> 
S80 

.2 

60 




































































































































f 





























































































25 



50 75 100 125 

Distance between Centers, Ft. 



150 



25 



50 75 100 

Distance between Centers, Ft. 



112 



Open drive, 7 ropes, American, 6 ropes British, 4500-ft. speed, f load. 
Relation of efficiency and distance between centers. 



_J,_— — ^ '^ I p " J 



3 4 5 

Number of Ropes 



LUU 

90 
















































^ 


























80 
70 
fiO 






















/ 


Y 


y 
































/ 






















m 













3 4 5 6 

Number of Ropes 



Open drive, American, so-ft. centers, 4Soo-ft. speed, British 75-ft. centers, 3Soo-ft. speed. 
Relation of efficiency and number of ropes. 
100 r 



























OiJen 

— i — 


!)rive^ 




.-rf 


















L— 


_Ul 


an 


1 Over 


Dri 


ve 












^ 


l 


> 


C 


» 
























/ 


7^ 
































/ 


} 

































90 



80 



60 



50 



40 

















































, 


I \ 






































y 


^ 


^ 




Tin 


md 


Over E 


riv< 



































/ 


y 


^ 






























/ 


/ 
































' 


' 



































0123 4 56^8 

Number of Ropes 

loo-ft. centers, 451 

Comparison of efficiencies 

Fig. 2>2>- — Efficiencies of American 

* 
two only and the relative life of the rope will be increased by 50 per 

cent., or the drum diameter might be reduced by an amount equal 
to 1.2 times the circumference of the rope with a corresponding 
reduction in the size of the framework of the crab or winch, while 
still retaining a relative life for the rope equal to Fig. 25. In this case 
the diameter of the lower pulleys would only require to be about one 
circumference of the rope larger than the original size of Fig. 25. 

In making the foregoing comparisons of diameters of drum and 
pulleys with different arrangements of rope it has been assumed that 
the hook is raised to the full height available at each lift. This, 
however, is not the case in actual practice, the majority of loads 
not being raised one-half this height. 

This consideration brings to light another great advantage of 
Fig. 31 as compared with any of the others. Where, as is usually 
the case, the average height of lift in a shop does not reach half the 
maximum available, then that portion of the rope which passes under 



0123456789 
Number of Ropes 
oo-ft. speed, f load, 
of open and up-and-over-drives. 
and British systems of rope driving. 

the lower pulley does not reach the upper drum, and accordingly is 
only subject to the wearing action of the two bends at the lower 
pulley. If, therefore, the effect of the bends at the lower pulley is 
reduced to one-half, by the proposed increase in diameter of the 
pulley, then the actual life of the rope will be doubled, instead of 
only being increased by 50 per cent, as was first assumed. 

Where there are more than two falls of rope, as in Figs. 27 and 28, 
the effect of increasing the diameter of the pulleys by an amount 
equal to two circumferences of the rope is also very marked, re- 
ducing the effect of the seven bends in Fig. 27 to four and a half, 
with corresponding increase in the lift of the ropes. This shows up 
the fault of those designers who adopt large drums (in order to ob- 
tain the great length of rope entailed by high hfts) and are yet con- 
tent to make the pulleys of small sizes, when they could enormously 
increase the durability of the rope by the adoption of larger pulleys 
at little extra cost. 



ROPES 



163 



lOU 














~^ 






























' 


90 


■"■ 




























^ 


■^ 


































80 
70 















































20 



40 



60 



80 100 120 140 160 180 200 
Brake Horse Power 
American open drive 4500 ft. speed 75 ft. centers s ropes. 
Relation of efficiency and load . 



100 

, 90 

80 






























3boo 


ttiSpe 

y 


,d 






-*J 


^ 




-* 
















, 


500 


^t. 


3pf 


ed 




;? 


y 





































































80 100 120 140 

Brake Horse Power 

Comparison of efficiencies of exact and differential drives. 

100 
90 



80 



I 80 



90 



360 



180 
Rope Tension. Pounds 
American open drive, 100 ft. centers, i rope, full load 
Relation of efficiency and rope tension. 





_ 


^ 










AnJeriian |Op^n Drive ^ 


















nglishlOpin— 
















■ 




} 






°"'S 






Ameri 


can[Up 


an 


dO 


ver Drivel 




^ 


<^ 




^ 


^ 


r 
I— 


— ' 


-Ei 


-1; >. nn and 


Over I 


;d^ 








/^ 


'A 


r' 




























A 


Y 


































'/ 





































1 




2 




3 


4 




5 




3 




7 




3 


9 



Fig. 



34-- 



Number of Rose 
100 ft. centers, 4500 ft. speed, | load. 
Comparison of efficiencies of four general plans 
of rope driving under uniform conditions. 
Various data from rope-drive efficiency tests. 



When the rope makes a reverse bend at the barrel, as in Figs. 26 
29 and 30, the barrel ought to be increased in diameter to counter- 
act the effect of the reverse bend. Thus, if in each of these cases 
the diameter of the drums were made larger by an amount equal to 
two circumferences of the rope, the durability of the rope would be 
equal to Figs, 25, 27 and 29 respectively. 

The "lay" of the strands and the lubrication of the rope when in 
use have each a considerable effect upon durability, Mr. Biggart's 
experiments showed Lang's lay ropes to have more than double 
the life of those of ordinary lay, and ropes that are oiled last more 
than twice as long as when this precaution is neglected, as already 
mentioned. ,The superiority shown by Lang's lay naturally gives 
rise to the question as to why it is not exclusively used. The ex- 
planation given by rope makers is that such ropes must be very 
carefully handled to avoid "kinks," and also they are found to be 
more liable to "spin." 

Efficiency of Rope Driving 

Very complete tests of the efficiency of rope driving were made by 
E. H. Ahara at the works of the Dodge Mfg. Co. {Journal A. S. M. E., 
Aug. 1913). Both the British and American systems were tested 
and in each case the open and the up and over arrangements were 
included, the meaning of these terms being sufficiently explained by 
Fig. 32, which illustrates the constructions tested. The losses of 
the motor, jack shaft and intermediate drive were eliminated by 
taking preliminary readings from the prony brake applied to the jack 
shaft under all the various loads and speeds. One-inch manilla rope 
was used in all the tests. All bearings were of the ring-oiled babbitted 
type. 

The results of the tests are shown in Figs. 33 and 34. Most of the 
charts are self-explanatory, but, regarding the one relating to exact 
and differential drives, it should be explained that in the former 
the grooves of any one sheave were as nearly as possible of the same 
diameter, while in the latter the diameter of each groove was approx- 
imately yj in. less than the preceeding groove, the eighth groove 
being \ in. smaller in diameter than the first one. The limitation 
of the tests of the British system to 112 ft. center distance was due 
to the dragging of the slack ropes on the ground when that distance 
was exceeded. 

The loading of hoisting slings of tnanilla rope as recommended by 
the National Founders Association is shown in Table 17 which gives 
the load for each single rope of the best long-fiber grade. 



Table 17. — Sape Loads of Manilla Rope Hoisting Slings, Lbs. 

When handling molten metal the rope should be 25 per cent, stronger 
than these figures 



Dia., 


Circ, 






A 


/C^ 




ins. 


ins. 






A-60-A 


/^90A^ 


^«iSo*\ 




85 






100 




H 


I 


120 


60 


Vi 


l}^ 


250 


210 


175 


125 


H 


2 


360 


300 


250 


180 


H 


2H 


520 


440 


360 


260 


H 


2M 


620 


520 


420 


300 


I 


3 


750 


625 


S2S 


375 


i\i 


-3W 


1000 


850 


700 


500 


iH 


3'/4 


1200 


1025 


850 


600 


i}^ 


4H 


1600 


1350 


1 100 


800 


m 


SW 


2 100 


1800 


1500 


lOSO 


2 


6 


2800 


2400 


2000 


1400 


2V1 


7^2 


4000 


3400 


2800 


2000 


3 


9 


6000 


5100 


4200 


3000 



The loading of hoisting slings of wire rope as recommended by the 
National Founders Association is shown in Table 18 which gives the 
load for each single rope of plow steel grade having six strands of 
nineteen or thirty-seven wires. For crucible steel rope the loads 
should be reduced one-fifth. 

Table i8. — Safe Loads of Wire Rope Hoisting Slings, Lbs. 

When handling molten metal the rope should be 25 per cent, stronger 
than these figures 



Dia.. 






A 


/\ 




ins. 






A-60-A 


/^^^r\ 


^<3S^ 




1,275 


1,050 




H 


1,500 


750 


\i 


2,400 


2,050 


1,700 


1,200 


»/« 


4,000 


3,400 


2,800 


2,000 


^4 


6,000 


5, too 


4,200 


3,000 


% 


8,000 


6,800 


5.600 


4,000 


I 


10,000 


8,500 


7,000 


5,000 


iH 


13.000 


11,000 


9,000 


6,500 


iV^ 


16,000 


13,500 


11,000 


8,000 


m 


19,000 


16,000 


13,000 


9,50a 


iM 


22,000 


19,000 


16,000 


1 1,000 



CHAINS 



The leading types of chains used for power transmission are shown 
in Figs. I to 9, while Table i by H. E. Haywaed, engineer of 
experiments and tests, Link Belt Co. {Amer. Mach., Aug. 28, Sept. 4, 
19 13 ) gives the uses to which they are put and the limiting speeds 
under which they should run. 

Crane Chains 

The strength of open and stud link crane and cable chains form ed 
the subject of an elaborate investigation and analysis by Profs. G.A. 
GooDENOUGH and L. E. Moore {University of Illinois 
Bulletin No. 18). The authors conclude that the unit 
stresses on which the formulas of Unwin, Weisbach and 
Bach are based are much in excess of the values regarded 
as permissible in machine construction using reasonable 
factors of safety. The formulas proposed by the authors 
for the strength of chain links are: 
P = .4 d'^s (open) . 
P = .S <f2^ (stud). 
in which P = load, lbs., 

rf = diameter of bar, ins., 

5 = permissible unit stress, lbs. per sq. in. 

The following conclusions are of interest as bearing 
upon certain general opinions held by engineers in regard 
to chains. "The introduction of a stud in the link 
equalizes the stresses throughout the link, reduces the 
maximum tensile stresses about 20 per cent, and reduces 
the excessive compressive stress at the end of the link 
about 50 per cent. 

"The stud-link chain of equal dimensions will, within 
the elastic limit, bear from 20 to 25 per cent, more 
load than the open-link chain. The ultimate strength 
of the stud-link chain is, however, probably less than 
that of the open-link chain. 

"In the formulas for the safe loading of chains given 
by the leading authorities on machine design, the maxi- 
mum stress to which the link is subjected seems to be 
underestimated and the constants are such as to give 
maximum stresses of from 30,000 to 40,000 lbs. per 
square inch for full load." 

The loading of hoisting chains, as practiced by the Illinois Steel 
Co., is given in Table 2. The loads given are uniformly one-tenth 
the breaking loads. This company requires all chains to be annealed 
at least every six months. 

Table 2. — The Loading or Hoisting Chains 



made, which is the cheaper quality, will be found very accurate in 
pitch and shape of links. The hand-made varies a little in pitch of 
links and the links will be found to vary considerably in shape, 
especially in the weld, which is at the end of the link. 

When it is intended to use a hand-made chain, the sprocket casting 
to be used is sent to the chainmaker who makes the chain over the 
wheel, fitting each link into place; thus making what is known as 
hand-made wheel chain. However, regardless of which chain is 
to be used, it is preferable to adopt the sizes given in manufacturers' 




Fig. 7- 
Roller. 

Figs, i to g. — Leading types of chains. 



Fig. 9- 
Morse Silent. 



Size, ins. 


Safe load, lbs. 


Size, ins. 


Safe load, lbs. 


i 


30s 


li 


I0,S2S 


1 


690 


If 


12,350 


i 


1,230 


ii 


14.32s 


1 


1,920 


If 


i6,4SO 


\ 


2,76s 


2 


18,715 


I 


4.925 


2i 


22,440 


n 


5.92s 


2j 


27,70s 


i\ 


7.310 


2| 


33.530 


If 


8,830 


3 


39,89s 



See also the end of this section. 

The lay-out of sprockets for crane chains is thus explained by A. W. 
Jenks, Chief Engr. Vulcan Iron Works {Amer. Mach., March 24, 
1910). 

Cable chain is either hand-made or machine-made. The machine- 



catalogs for machine-made chain, as the proportions of the links have 
been adopted after long experience. 

Should the chain be on hand, it is wise to measure over some 18 
or 20 links, and obtain an average pitch per link. Extreme acciuacy 
must be used in all operations or the sprocket will not turn out right. 
The following example. Fig. 10, covers the design of a lo-tooth or 
lo-pocket sprocket for i-in. chain. It is first necessary to obtain 
the dimensions A and B. The chain catalogue gives the length of 
link W for i-in. chain, as 45 ins. and the width of link w as 3^ ins. 
From the length we find that ^ = 3 f ins. and .B = i f ins. The points 
XX are the~pin centers upon which the links revolve when passing 
over the sheave. 

The next step is to find the pitch diameter D, which passes through 
these points XX is follows: 

Let N = number of teeth or of pockets in whole wheel. 
= 10 
A = distance, center to center of link length. 

= 3l ins. 
B = distance, center to center between two links. 
= I f ins. 



164 



CHAINS 



165 



Angle y = 



Tan angle z = 



1 80° 

N 
i8o° _ 

10 
sin y 



18" 



-J-+COS y 
.30902_ 



1-75 
3-75 



-=.21797 



+ .gSio6 



The tangent .21797 corresponds to an angle of 12° 17' 46' 
sine of which =.2 1297. 

A _ 3-75 
sin z .21297 



the 



Pitch diameter =- 



: 17.6081 ins. 



As the pockets are not to be machined, allow ample clearance; 
make the pocket at least i in. longer than the link, which will leave 
the width of tooth at the center line of the wheel 5 in. The groove 
in the center of the wheel which accommodates the vertical links, 
should be amply wide and deep enough to permit the links to fall into 
a natural position. 

It must be noted that the tooth as drawn, is not the shape of the 
tooth at the side of the central groove, but at the center line. The 
patternmaker will find it easier to work from this imaginary place, 
but explain with a note, so that he will not misunderstand. The 
shape of the tooth faces can be found by considering that each link 
lifts on its center X, until it comes in line with the next link, when 
the two lift on the second center, until in line with the third and so 
on. However, make the tooth somewhat thinner than the contour 
thus found, or it will be diificult to get the chain into the wheel. 







Table i. — ^Types and Uses of Chains 








Highest speed for 




Type 


Classes 


Where used 


general use, 
ft. per min. 


Description and Qualifications 




Open link 


Cranes, dredges 


ISO for 


Used for heavy loads moving at low speeds, rough work; power applied by 






hoists and 


I in. 


drums, one end of chain secured to drum, or by pocket wheels, both ends of 


Crane 




slings. 


and larger 


chain free. 


or ^ 
open link 


Spreader or 


anchors. 


100 




stud 


moorings. 


on capstans 




■ 


hand chains 


hand hoists 


3S0 


For application of hand power to hoists, etc., used endless, runs on pocket wheels 
and rag wheels. 


Detachable 


Ewart with 


Power trans- 








hook joint 


mission, eleva- 
tors and con- 
veyors 


600 


Used for power transmission at moderate speeds. Buckets and flights are 
attached to chain by means of "attachment links" when chain is used for 
elevating and conveying. 


"Link Belt" 


Closed joint 


Same as Ewart 
for dirty places 


600 


Used chiefly for power transmission in dirty and gritty locations. Made, in 
best type, with hardened-steel pins and bushings. 


' 


Machine 


Power 




Machine-made steel roller chains are much more accurate than malleable-iron 




made 


transmission 




chains and will run at higher speeds and loads than "Link Belt." 


Roller ^ 








Rollers give better sprocket action than solid joint. Wheels are cut and good 
machine-made roller chain may be compared with cut gearing. 




Cast malleable 


Elevators and 


600 


Malleable-iron rollers with telescoped mal.-iron end bars (tubular), the halves 




and steel 


conveyors 




of end bars telescoping one into the other. Steel pin passes through tubular 
end bar. Substantial and durable. 
Bearing surface of joint is given maximum possible area through use of seg- 
mental hardened-steel bushings extending throughout entire width of chain 




Extended 






and bearing on a cylindrical pin which is free to rotate. Shape of link and 




bearing 


Power 




wheel tooth is such that elongation is compensated for and sprocket action 
is very gentle, allowing chain to run quietly at high speeds. 
Superior to cut gearing at equal speeds and loads. Runs on cut sprocket wheels 
of special form. 


High speed 


Rocker joint 


transmission 


1200 


Link form substantially similar to the above with the same quiet running 


silent 


Plain 
bearing 


\ 




and compensating action but with each joint provided with specially de- 
signed roller bearing. 
Link form similar to above, joint bearing formed by round hole in link with 
round pin, affording half the bearing surface given by extended bearing con- 
struction in chains of equal width and equal diameter pin. 



It is advisable to use at least five decimal places in all quantities 
the result is very greatly affected by their absence. 

Having the pitch diameter, the wheel can be laid out. It is 
desirable to make a full-sized layout, if only as a check upon the 
computations. 

Divide the circle into 10 parts for the 10 teeth. But two or three 
links need be drawn to obtain all necessary dimensions for the shape 
of the pocket. The horizontal chain link is a chord of 33 ins. length 
on the pitch circle; the vertical link is a chord of if ins. length. 
The axis of the vertical link passes through the centers XX of the 
two adjacent horizontal links. The diameter E across the flats can 
now be measiured and this is really the most important dimension 
of the wheel. 

It would be wise, as a check, to space off alternate chords of if ins. 
and 3 J ins. around the entire wheel. Of course, there should be 
10 of each. The distance E can also be found by computing the 
cosine of angle z with radius = 8.80405 ins. or half the calculated pitch 
diameter and deducing from the result i the diameter of the link 
material. 



Wheels are sometimes made with pockets for the vertical links, 
but it is preferable not to use them, as, when the wheel becomes a 
little worn, the vertical link has a tendency to pry the horizontal link 
from its bearing. 

It is better that the chain be too tight than too loose, as some 
stretch will occur in the first few days of operation. 

Care must be used in the calculations; approximations will not do, 
as the errors multiply. 

The attachment of chains to hoisting drums by the common method 
shown in Fig. 11 is pointed out by G. E. Flanagan {Amer. Mach., 
Oct. 23, 1902) to be defective. The fault lies in drilling the hole 
for the spur of the chain anchor in the groove, in place of through 
solid metal beyond the chain groove as shown in Fig. 12. The first 
method subjects the anchor spur to a bending stress several times 
greater than is done by the second, and may cause the failure of the 
connection. Crane drums should be so proportioned that from one- 
half to a full coil of chain will remain upon the drum with the hook in 
the lowest position in which it is possible for it to sustain a load, and 
in this case only a fraction of the full stress will come upon the anchor; 



166 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 







H = d 






A = pitch of chain + d 
B= pitch of chain — d 

J? = pitch diameter 

£ = diameter across fiats 

F=dX.7S 
G=d 

Fig. io. — Proportions of sprockets for cable chains. 



but although this requirement may be observed when the machine 
is first installed, conditions may be changed afterward, as by digging 
deeper pits in the foundry, and thus the entire load may come directly 
upon the chain anchor, which ought to be fully capable of meeting 
such an emergency. It will be noted that Fig. 12 requires a link of 
extra length on the end of the chain in order to reach the spur in the 
position shown. Figs. 11 and 12 are both open to the objection that 
the drum may be rotated far enough to bring the pull of the chain as 
shown in dotted lines of Fig. 12. 
In this case the load will not come 
upon the spur at all, but principally 
upon the nearest tap bolt, and it will 
be increased by a leverage depending 
upon the distance from the chain to 
this bolt. A better arrangement than 
either is shown in Fig. 13. 

Fig. 14 shows a section of a 
grooved drum, with proportions. The 
thickness of the metal below the 
bottom of the groove is determined 
by treating the drum as a hollow 

cylindrical beam with the load concentrated in the middle, 
links should not bottom in the groove. 

Hand chains are used endless, hanging from hoists, etc., with the 
lower loop free. The rims of the wheels over which the chain passes 
are called rag wheels. These rims are simple in design, usually 
having a V-groove with about 60 deg. included angle, with ridges 
cast radially along the inner sides of the V to provide gripping points 
for the chain links. 



The Ewart Chain 

The Ewart chain with hook joint. Fig. 4, is used for power trans- 
mission at moderate speeds. According to Mr. Hayward (Amer. 
Mack., Aug. 28, 1913) the thickness of the tooth of the sprocket wheel 
must be such as to give the hook portion of the link some freedom 
between teeth. If the tooth space were made to conform to the shape 
of the hook, the slightest stretching of the chain under load or through 
wear would cause the chain to ride up on the flanks of the teeth, and 
eventually the chain would be broken by riding over the crowns. 
Conversely, the wider the tooth space the greater the amount of 




Fig. 14. — Section of chain drum. 

stretch or wear that can take place before riding. On wheels of a 
large number of teeth the space may be considerably wider than on 
small wheels, as the wear on the tooth flanks is more distributed, 
each tooth coming into action once in each revolution of the wheel. 
The wide tooth space is also desirable in large wheels because it 
permits of the greatest possible elongation of the chain before the 
increased pitch of the chain causes it to ride. 

The working load which may be applied to a given link belt depends 
in each case upon speed of chain, cleanliness of location, and character 
of the load. The best method of rating a chain is by applying a 
factor, varying with the speed, to the average breaking strength of 
the given chain. Table 3 gives factors that have been determined 
by exhaustive experiment and by use: 

Table 3. — Speeds and Working Loads for Ewart Chains 



Chain speed in ft.' per min. 


To obtain working load divide average 
ultimate strength by 


\ 
200 f 
200 1 
300 J 
300 1 
400 f 
400 \ 
500 f 
500 1 

600 r 
600 1 
700 / 


6 
8 
10 
12 
16 
20 




Fig. II. 



Figs, ii to 13.- 



FiG. 12. 

—Anchors for crane chains. 



Fig. 13. 



The 



If the chain is to be subjected to shock or is to work in a gritty 
place, especially in elevators and conveyors where coarse or gritty 
materials are being handled, the factors must be increased beyond 
those given in the table. In lookingover the catalog of manufacturers 
of link belting the wide range of sizes and types often makes the 
selection of chain difiicult; the following suggestions may simplify 
the problem. 

In selecting chains for power transmission, first determine the 



CHAINS 



167 



I 



diameter of the small sprocket wheel, keeping it as small as possible. 
Select a chain of medium pitch with the proper breaking strength 
and iind the number of teeth in the sprocket wheel of the diameter 
selected. If possible use wheels with more than eight teeth; smaller 
wheels cause rapid wear on both wheels and chain through the large 
angle of articulation when the links enter and leave the wheel. 

Note that a chain of medium pitch is desirable for ordinary trans- 
mission purposes. Short-pitch chains will not permit of suflScient 
tooth space in the wheel to allow for much elongation of the chain 
through wear, also the joints are greater in number in the same length 
of chain, and the same amount of wear per joint will cause greater 
elongation than in medium pitch. Short-pitch chains are designed 
primarily for applications where backlash of the chain on the wheel 
is objectionable. The long-pitch chains are only desirable for trans- 
mission purposes where the chain speed is very low. 

At equal chain speeds on equal-diameter sprockets the long-pitch 
chains hammer on the sprocket wheels much harder than the medium 
pitch, make much more noise, and, strength being equal with medium 
pitch, do not make as durable a drive. Long-pitch chains are ap- 
plicable principally to elevator and conveyor work. Where the chain 
speeds, compared with ordinary power transmissions, are low, 
seldom exceeding 200 ft. per min., the diameters of the sprocket 
wheels are governed by consideration of the material being handled 
rather than the energy transmitted. 

Cast-iron sprocket wheels in the rough are likely to be irregular 
in tooth spacing. Unequal shrinkage, rapping of the pattern in 
molding and inaccuracies in the pattern often cause the wheels to be 
incorrect in pitch and diameter. In an ordinary gray-iron casting 
the best way to secvure a good wheel is to cast a little large in diameter 
and grind the periphery down to properly fit a piece of standard chain. 

Hard-rim sprocket wheels are furnished by some manufacturers. 
These are cast from special iron in such a manner as to make the rims 
and teeth extremely hard and tough while the hubs are soft for boring 
and keyseating. These wheels when properly made, are a decided 
economy in spite of their slightly higher price, as they last several 
times longer than the gray-iron wheels. Care should be given to 
their selection, however, as they are subject to some troubles not 
present in the ground soft-iron wheels. 

The face of the wheels, both on the teeth and on the root diameter, 
should be parallel to the bore of the wheel and the edges of the wheels 
should be free from fins or other projections. The iron is too hard 
to grind over the entire face of the wheel, and projections will cut 
the chain to pieces very quickly. 

The design of a sprocket wheel for link belting is not a difficult 
matter, but the patterns are expensive and the production of a good 
wheel in the foundry requires considerable skill. For the use of 
those who wish to make their own wheels, the following is offered: 

Having determined the number of teeth desired and the pitch of 
the chain, the next step is to find the diameter of the pitch circle, 
Fig. 15. Careful measurements should then be made of the chain 
to determine the dimensions shown in Fig. 16. The root diameter 
of the wheel, as shown in Fig. 17, is the pitch diameter less 2XA. 
The flanks of the sprockets or teeth may be made straight from just 
above the root line to a little above the pitch line and should be in- 
clined at such an angle as to give the hook of the link ample clearance 
in leaving and entering the wheel. 

It is not necessary to make the tooth at its base conform to the shape 
of the hook. The straight-tooth flanks will cast more regularly 
and will form a more uniform bearing for the chain than if the hook 
and tooth forms were similar. The thickness of the tooth at the 
pitch circle determines the amount of wear and stretch that may take 
place in the chain before it begins to ride the wheel; each tooth comes 
into action once in a revolution of the wheel, and the wear on a tooth 
is proportional to the number of times it comes into action. There- 
fore, the wear on a wheel with a small number of teeth is greater 
than on one with a large number, making heavy teeth necessary in 



small wheels. Furthermore, the number of links in mesh with a 
wheel of a small number of teeth is less than in a wheel with a large 
number, making it possible to reduce the clearance without taking 
from the useful life of the chain. 

Table 4 expresses this difference in percentages of the available 
tooth space in the chain, as shown in B-D, Fig. 16 




Of 



o 



b'b = Pitch of Chain 



N= Number of Teeth 
Q^^-P Pitch K Jdius/ 



X)" 



i OB 



SiaY '■ 
Pitch Diam. 



Fig. 16. 
Proportions of Links 



/■ 




Flo. 15. 

Pitch Circle Diameter 



k-c. 



Fig. 17. 
Root Diameter 




Fig. 18. — Section of rim. 
Figs. 15 to 18. — Laying out sprockets for the Ewart chain. 

Table 4. — Number and Thickness of Tooth 



Number of teeth in wheel 


Per cent, thickness of teeth at pitch line 


8 to 12 


75-80 of tooth space in chain 


13 to 20 


70 of tooth space in chain 


21 to 3S 


65 of tooth space in chain 


36 to 60 


55-60 of tooth space in chain 



The straight flank of the tooth may be continued to nearly the 
total height of the tooth and then curved over to form a flat crown, 
or a rounded crown may be used as shown in the dotted lines in 
Fig. 14. 

Fig. 18 shows a section of a typical sprocket-wheel rim. The 
dimensions may be expressed approximately in terms of the dimen- 
sions of the link, thus: 

B = W — ^W up to I in., which is sufScient for wide chains 

2 
H = 2.sP 
D= .7W 
E = i.5W 

- F=^ 
3 
G = .6Pr 



168 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



These proportions are necessarily approximate. The wide range 
of designs and sizes makes it impossible to formulate a satisfactory 
method that will apply to all cases. 

The proper use of the Ewart chain has been explained by S. B. Peck, 
Vice President Link Belt Co. {Amer. Mach., May 14, 1908) as follows: 

When considering the relative merits of different methods of run- 
ning chain-drives, the drive should be considered as a whole, and the 
action noted at four points: a, entering point on the driver; b, releas- 
ing point on the driver; c, entering point on the driven; d, releasing 
point on the driven, as shown by abed, in Fig. 19. 

In this discussion the action at a point is. said to be good when all 
the articulation or bending takes place in the joint of the chain, Fig. 



In Fig. 26 we have the driver large, the driven small; hence b and d 
are the teeth in action. Chain runs bar first; action at a good, at b 
bad, at c good, at d bad. 

In Fig. 27 we have the same sprockets as in Fig. 26, but the chain 
runs hook first. Here the action at a is bad, but the fact that the 
hook is not in contact with a tooth face at this point makes the 
consequent wear of little account. The action at b is good. The 
action at c is bad, but this is on the slack side of the chain and this 
bad action causes no wear. The action at d is good. 

It is thus seen that there are two very bad points (J and d) where 
the chain runs bar first and only one serious trouble (a) when running 
hook first. Therefore, always run hook first in such a case. 




Fig. 21. Fig. 25. 

Figs. 19 to 29. — The correct use of Ewart chain. 



Fig. 29. 



20. The action is said to be bad when, in bending, the link rubs on 
the sprocket, producing wear on the sprocket and outside or external 
wear on the hook, Fig. 21. 

Another fact is also to be remembered: There is never more than 
one tooth in action at any one time. No matter how carefxiUy the 
chain and sprocket may be made, as soon as the load comes on, there 
is a change caused by stretch and wear. 

We can predetermine which tooth shall be in action by making 
the pitch of the wheel either larger or smaller than the pitch of the 
chain. Thus, on the driver. Fig. 22, the wheel pitch being smaller 
than the chain pitch, the entering tooth does all the work. In Fig. 23 
the conditions are reversed: the wheel pitch is the larger and the 
releasing tooth does the work. On the driven the same thing holds, 
except that here conditions are reversed. 

When the wheel pitch is smaller than the chain pitch the releasing 
tooth does the work. Fig. 24, and when the wheel pitch is larger 
than the chain pitch the entering tooth does all the work. Fig. 25. 

For the best work the pitch of the driver should be larger than the 
pitch of the chain. Fig. 23, and the pitch of the driven should be 
smaller than that of the chain, as in Fig. 24. The releasing teeth b 
and d are, therefore, the working teeth, and the chain can seat at a 
and c quietly and take the load gradually as the wheel revolves. 

Having considered the question of wheels, we will now regard 
the drive as a whole to determine whether the chain shall be run 
bar first or hook first. 



Consider a drive when the sprockets are such as are usually 
furnished: These are ground to fit the neiv chain; when the latter 
stretches, both driver and driven are small as compsired to it, and 
teeth a and d are now in action. 

In Fig. 28 we have such a pair of wheels with the chain running 
bar first. The action at a is good; at b it is bad, but as there is no 
tension on the chain at this point, this is not objectionable. At c the 
action is good; at d it is bad. In this case, therefore, it would seem 
that the wear would be confined to the driven wheel and this is so in 
actual practice. 

Only wear is on driven, caused by the bad action at d, this forms 
hook on d and breaks chain. 

Observe the same wheels with the chain running hook first, Fig. 29. 
The action at a is bad; at b it is good; at c it is bad, but not objection- 
able, because, as before, there is no tension at this point; action at d 
is good. Thus all the wear would seem to be on the driver as a result 
of the action at a. This is found to be the case; hence both theory 
and practice show that with the chain running bar fiTSt driven wheel 
wears, while with chain running kook first driver wheel wears. 

Now, it is found that because the wear at d, running bar first, is 
caused by the link slipping up the tooth, it tends to undercut and 
form a hook and thus break the chain. On the other hand, the wear 
at a, when running hook first, is caused by the link slipping down the 
tooth, and the wheel will wear out completely without endangering 
the chain. It has also been proved that the driver, running hook 



CHAINS 



169 



first, lasts several times as long as the driven wheel when running 
bar first. As the driven wheel is in nearly every case much larger 
than the driver and the consequent wear on each tooth is less, it 
would seem that if the chain were run so as to wear the driven, the 
wear on the two wheels would be equalized. This would be poor 
practice for the reason that the driver, being smaller, is more cheaply 
replaced, and the repair account will, therefore, be less running 
hook first. 

In elevators, the head wheel acts as a driver, and the foot wheel 
simply as an idler, because it is doing no work. Therefore, run the 
chain bar first so as to favor the driver. On conveyers one wheel is 
always an idler, comparatively speaking, and the same reasoning 
holds as for elevators: the chain should run bar first in all cases. 

These remarks apply equally well to all closed-end pin chains; 
the dosed end corresponds to the hook and the pin end to the bar 
of the Ewart chain. 

In general, therefore, on drives run hook first. On elevators and 
conveyers run bar first. 

The Roller Chain 

The following data relate to the practice of the Diamond Chain and 
Manufacturing Company. When selecting a roller chain the follow- 
ing considerations apply: 

The pitch of the chain should not be greater than the speed of the 
smaller sprocket will allow. To find the maximum sprocket speeds 
refer to the column of max. r.p.m. of Table 6 from which it will be 
seen that a short pitch chain allows a higher sprocket speed than a 
long pitch, a light-weight chain allows a higher speed than a heavy 
weight of the same pitch, and a wide chain allows a higher speed than 
a narrow one of the same pitch. 

It was formerly supposed that the chain speed must not exceed a 
certain limit under pain of rapid wear. A long series of observations 
and experiments by the Diamond Chain and Manufacturing Com- 
pany has proven that chain speed has little to do with the destructive 
action, but that high sprocket speed combined with long pitch is 
very noisy and destructive, because of the impact between link and 
sprocket tooth. Hence the importance of selecting a chain of the 
shortest possible pitch. The following approximate formulas 
apply: 

(900 \ 
— : — I 
r.p.m. J 

900 

in which P = pitch of chain, ins. 

The weight of the chain is also important, although it is not pos- 
sible to exercise such a wide range of choice in weights as in pitches, 
especially as a short pitch chain is also comparatively light. But, 
when the sprocket speed is high and there is more than one weight of 
chain of the same pitch and width to choose from, it is well to select 
the lighter chain provided its rivet area and ultimate strength are 
ample. 

Width of Chain. — In general a wide chain of short pitch is better 
than a narrow one of longer pitch; but where the sprockets are apt 
to run considerably out of alignment, as in motor trucks or armature 
shafts of electric motors, or where a crossed chain is desired, a narrow 
chain must be selected because of its great flexibility laterally. The 
frictional losses in a chain drive are less for rivets of small diameter 
than for those of large diameter: Hence, if the shearing strength of a 
small rivet is great enough for the service required, it is better to get 
the required rivet area by increasing the length of the rivet than by 
increasing its diameter. 

The projected rivet area is the product of the rivet diameter and the 
length of the bushing, or block, in which the rivet turns. 

The chain pull should not in general be greater than 1000 pounds 
per square inch of projected rivet area, but for slow speed it may 
sometimes run as high as 3000 lbs. per sq. in. with fair results. 
It should not exceed one-tenth of the ultimate strength of the chain, 



H 



Maximum r.p.m. of sprocket- 



(a) 
(&) 



since there are but few cases in which the load is so uniform that a fac- 
tor of safety of at least 10 is not necessary. Indeed, where the power 
is suddenly applied the required factor of safety may run as high as 40. 
Formulas for the calculation of chain velocity, chain pull, horse- 
power, etc., are given below, in which 

i\^ = No. of teeth, T = chain pull, lbs. , 

i' = pitch, ins., 5 = r.p.m. of sprocket, 

i7 = horsepower, F = velocity of chain, ft. per min., 

D = pitch diam. of sprocket, ins. 

SNP 
V = —— or .26iy,DS (approx.) 



12 

12F 
' NP 



396,000 Xfl^ 
TNP 



33,000 Xfl" 39,600 XF 
1 = ri or 



SNP 



H- 



VT 



SNPT 



33,000 396,000 



12V sg6,oooXH 
^~ SP ^^ SNP 
- 12F 396,000 XH 

SN °^ SNT 



D = .si8XNP (approx.) or 



1 26,000 X^ 

fs 7 



(approx.) 



(c) 
id) 
(e) 
(/) 
(?) 
W 
(i) 



Tables 5 and 6 will be useful in the selection of the proper chain 
to transmit a given horse-power at a given speed and chain tension. 
If the following four maxims are properly observed in the design of 
a chain drive, much of the trouble with respect to noise and undue 
wear can be avoided. 

1. Keep the pitch as short as the load will allow. 

2. Avoid the use of less than fifteen teeth, unless the sprocket speed 
is relatively much lower than that given in Table 6. 

3. Select a wide chain in preference to a narrow one, excepting in 
cases where the chain is crossed or where the sprockets must run 
considerably out of alignment. » 

4. Select a light chain in preference to a heavy one if strength and 
rivet area are adequate. 

The alignment of the sprockets should be as nearly perfect as pos- 
sible; otherwise both chain and sprocket teeth will wear more on one 
side than on the other, and the drive will be noisy and short-lived. 

The center distance should be adjustable wherever possible in order 
to take up slack due to elongation from wear. A little slack, how- 
ever, is an advantage, as it allows the chain links to take the best 
position on the sprocket teeth, and reduces wear on the bearings. It 
is a curious fact that when the center distance is such that the span 
of the chain on the tight side is an exact multiple of the pitch the 
efficiency is higher and the chain runs more smoothly. Oftentimes 
when the slack side of the chain fails to run in a smooth curve, a 
slight alteration of the centers will correct the trouble. For a satis- 
factory drive the center distance should not be less than one and one- 
half times the diameter of the larger sprocket, nor more than sixty 
times the pitch; but much depends upon speed and other conditions. 

An adjustment of the center distance equal to the pitch of the chain 
is all that will ever be necessary. If not more than half of this 
amount can be provided, an offset link may sometimes have to be 
inserted in order to make the chain the proper length. 

Idler sprockets should be used only where the conditions make it 
imperative. It is as important that idlers should be kept within 
the proper limits of speed and number of teeth as either the driving 
or driven sprockets. Although the idler carries practically no load, 
the effect of impact between tooth and roller is the same, and the 
teeth will wear with surprising rapidity if the speed is too high, the 
number of teeth too low, or if not properly mounted in correct 
alignment. 

When an idler is used for the purpose of taking up slack, it should 
be placed against the slack side of the chain, preferably, but not 
necessarily, between the two strands of the chain. It should be a 



170 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



sprocket rather than a roller. If used on the tight side of the chain 
to lessen vibration, it should be placed on the lower side in such a 
position as to allow the chain to run in a straight line between the 
two main sprockets. A flat steel plate mounted below the tight side 
of the chain to guide it in a straight path will be found a very satis- 
factory substitute for an idler sprocket as a means of checking 
vibration. 

Vertical drives for chains have been condemned unduly. Experi- 
ence shows them to be as satisfactory as horizontal drives excepting 
in cases where the centers cannot be adjusted to take up the slack 
in the chain. The same remark applies to oblique drives, and it 
makes little difference in any drive whether the tight side is above 
or below. 

Crossed chain drives have been used with success in a number of 
cases, notably in aeroplanes. The chains should be narrow, how- 
ever, the center distance should be suflficiently long, and some means 
should be provided at the place where the chains cross to keep them 
from rubbing together. They are sometimes run through crossed 
tubes. But at best a crossed chain does not make an ideal drive. 

The encasing of chains in dust and oil proof housings is very de- 
sirable, as it affords means of continuously lubricating the chain 
and actually prolonging the life of both chain and sprocket from 
200 to 300 per cent. A chain case with an oil bath need not be very 
expensive, and the advantages which result with respect to increased 
efficiency and reduction of noise, as well as longer life, wiU generally 
make the investment a profitable one. 

As to material for sprockets, they may be made from machinery 
steel, cast steel, alloy steel, semi-steel or malleable iron, gray iron, 
brass or bronze. In short, any metal that makes a good gear will 
make a good sprocket. Cast iron is being used more and more for 
sprockets on account of its cheapness, ease of machining, and good 
wearing qualities. 

Cast teeth have the same advantages for sprockets as for gears. 
They can be recommended only for low speeds, and where cheapness 
of production is a matter of greater importance than smooth, quiet 
running and long chain life. 

A new chain should not be applied to an old or much worn sprocket, 
as the chain will be quickly ruined and an unsatisfactory drive is sure 
to result. A sprocket that has worn to a hooked form of tooth exerts 
at both entering and leaving contact a wedging action that cannot be 
resisted by any chain. 

Fixed distances between sprockets must be maintained and the 
alignment of sprockets should be perfect. 

After extended tests, the Diamond Chain and Manufacturing 
Company has introduced a new form of sprocket tooth which elimi- 



Addendum=J^ p(.6-Tan-22. ) 5^^ 




Fig. 30. — Sprocket tooth form for roller chain. 

nates much noise and wear. In the old form the inevitable wear of 
the chain and the resulting elongation of the pitch, necessitates clear- 
ance in the sprocket teeth. The new form applies the same principle 
as that of silent chain sprocket teeth, the chain engaging the teeth 
at a progressively increasing diameter as the chain pitch increases, 
and always without clearance. 

This form of sprocket tooth is shown in Fig. 30. The pressure 
angle (or angle between the direction in which the chain pulls and 
a normal to the tooth outline) is kept constant at about 20 deg. 



Sprockets with a greater number of teeth can be used. Cutters for 
these teeth are known as Diamond Universal Sprocket Cutters and 
are defined by specifying the pitch and the roller diameter. Other 
dimensions, such as thickness, outside diameter, hole and keyway 
are standard. They are made for the following ranges: 

7-8 teeth 

9- I I teeth 

12-17 teeth 

18-34 teeth 

35 and over teeth 

The outside diameters for these sprockets are less than those usu- 
ally tabulated. Instead of adding the roll diameter to the pitch 
diameter to obtain the outside diameter, the following formula is 
used: 

Outside dia. = pitch dia. Xi' ( (0.6 — tan "irp) 

in which P = pitch of chain, ins., 
iV = number of teeth 

To Calculate Length of Chain 

in which P = pitch of chain, 

C = center distance in pitches, 
iV = number of teeth on large sprocket, 
» = number of teeth on small sprocket, 
-L = chain length in pitches. 

As a chain cannot contain a fractional part of a pitch the next 
whole number above the calculated number of pitches must be used. 
If it is an odd number, an offset link must be used in all cases except 
for the block center and twin roller chain. The chain length in 
inches is found by multiplying the number of pitches by the itch. 

To Calculate the Center Distance for a Tight Chain 

If the center distance can be suited to the length of the chain, an 
even number of pitches should be chosen to avoid the use of an offset 
link, and the proper center distance may be calculated from this 
formula: 

P 



C = j I 2L-N-n+\/i2L-N-n)^-.&24{N-ny 

in which C = center distance, ins.,. 

L = length of chain in pitches. 



(0 



Both the above formulas are approximate, but the error will 
amount to only a few thousandths of an inch for most cases, and in 
all cases the error is less than the variation in length of the best chains. 

Calculation of Sprocket Wheel Diameters for Roller Chains and 
Built-up Block Chains 



Pitch diameter = - 



180° 



(m) 



Sin- 



N 



Bottom diameter = pitch diameter— Z) (w) 

Outside diameter = outside diameter -f-P ((0.6— Tan -rr-j (o) 

in which N = number of teeth in sprocket, 
P = pitch of chain, ins., 
D = diameter of roller, ins. 

Sprockets cut with the Diamond Universal Cutter should have 
bottom diameter . 003 in. to . 005 in. less than the above in order to 
provide for variation in size of rollers and for dirt. 



CHAINS 



171 



Calculation of Sprocket Wheel Diameters for Roller 
Chains and Bushing Chains. For Block- 
center and Twin-roller , 



Tan C = 



i8o° 

'IT 



■'-{■ cos 



i8o° 

N 



Pitch diameter = - 



(3) 



ir) 



sin C 

Bottom diameter = pitch diameter— 6 (5) 

Outside diameter = pitch diameter+4(o.6 — tan KC) (0 

in which iV = number of teeth. 

6 = diameter of round part of chain block 

(.325 in. for I in. P, and .532 in. for ij^ in. P) 
B = center to center of holes in chain block 

(.400 in. for I in. P, and .564 in. for iH in. P) 
A = center to center of holes in side bars 

(.600 in. for I in. P, and .936 in. for iH in. P) 

For practical purposes the following formulas will give the outside 
diameter within .001 in. of the correct dimension: 
For i-in. pitch block chains 

i25iV 



O.D.=P.D.+ 
For I H-in. pitch block chains 

O.D.=P.D.-\- 



45N-72 

9 -^-15 
16N 



{u) 



{v) 



Calculations may be abbreviated by the use of Tables 5 and 6. 
Although the outside diameters are given to three places of decimals 
extreme accuracy is not needed in this dimension. The pitch diame- 
ter is not needed in connection with the machining of a sprocket and 
is given only as a check on the other dimensions. The bottom diame- 
ter is the most important. They should never exceed the amounts 
given in the tables but may be several thousandths under. To 
allow for dirt they should be from .003 to .005 in. less than the 
tabulated values. 

Table 5 of sprocket diameters for L-in. pitch is carried as high 
as III teeth, and the pitch diameters are given to four places of 
decimals. From this table the pitch diameter and outside diame- 
ter can be computed for any sprocket of other than i-in pitch. 
For example, let it be required to find the diameter of a 70-toothed 
sprocket for No. 762 stud chain, which has a pitch of .326 in. and 
a stud diameter of .128 in. A 70-toothed sprocket, i in. P, lias a 
pitch diameter of 22.2892. Multiplying by .326 gives 7.266 ins. as 
the pitch diameter for a .326 in. pitch chain. Subtracting the stud 
diameter from this gives the bottom diameter 7.138 ins. The 
outside diameter given in the table is 22.867. Multiplying this by 
.326 gives 7.45s ins. as the outside diameter required. 

As an exam-pie in the calculation of a chain drive, suppose it is re- 
quired to transmit 12 h.p. from a motor at 885 r.p.m. to a line shaft 
at 360 r.p.m.; center distance 40 ins.; maximum allowable diame- 
ter of sprocket on line shaft, 18 ins. 

From column five of Table 6 the longest pitch that can be used for 
a sprocket speed of 885 r.p.m. is i in.; and from column four it 
is seen that there is no chain with a pitch less than i in. that has a 
normal capacity as high as 12 h.p. We will therefore try to use -a 
I-in. pitch chain. 

In the table of sprocket diameters the greatest number of teeth 
that can be used on the large sprocket is found to be 54. The veloc- 
ity ratio -x^ = . 2458. The nearest fraction to this that can be used 

is ^%3. Now a 13-toothed sprocket is undesirable, unless we can 
do no better. We wiU therefore see what can be done with a %-in. 
pitch chain. The maximum number of teeth that can be used on the 
large sprocket is in this case 73. The fraction ^^^fs is equal to .2466, 
and is very close to the ratio required. The number of teeth for a 



Table 5. — Sprocket Diameters for Roller 
See Text for Use with Other Pitches. 
I-in. pitch 



Chains 



No. 
teeth 


Pitch 

diatn. 


Out- 
side 
diam. 


s 

OOP 


-2 p 


No. 
teeth 


Pitch 
diam. 


Out- 
side 
diam. 


6 

.- ".2 

^-^^ 
■2 • 
.0 


S 

Oin^ . 
,0 




Ins. 


Ins. 


Ins. 


Ins. 




Ins. 


Ins. 


Ins. 


Ins. 


6 


2.0000 


2.332 


1.438 


I-37S 


59 


18.7892 


19.363 


18.227 


18. 164 


7 


2.3048 


2.677 


1.742 


1.680 


60 


19. 1073 


19.681 


18.545 


18.482 


8 


2.6 13 I 


3-014 


2.051 


1.988 


61 


19.4255 


20.000 


18.863 


18 . 700 


9 


2.9238 


3-347 


2.362 


2.299 


62 


19.7437 


20.318 


19. 181 


19.019 


10 


3.2361 


3-678 


2.674 


2. 611 


63 


20.0618 


20.637 


19 . 499 


19-337 


II 


3 -5495 


4.006 


2.987 


2.924 


64 


20.3800 


20.955 


19.818 


19.65s 


12 


3-8637 


4-332 


3-301 


3.239 


6S 


20.6982 


21.274 


20. 136 


19-973 


13 


4- 178s 


4-657 


3-616 


3-554 


66 


2 1. 0164 


21.593 


20.454 


20-291 


14 


4.4940 


4.982 


3.932 


3-869 


67 


21.3346 


21.911 


20.772 


20.610 


15 


4-8097 


5-30S 


4.247 


4-185 


68 


21.6528 


22.230 


2 1 . 09 I 


20.928 


16 


5-1259 


5-627 


4.563 


4-501 


69 


2 1.97 10 


22.548 


21.409 


2 1.246 


17 


S-4423 


5-950 


4.880 


4-817 


70 


22.2892 


22.867 


21.727 


21.564 


:8 


5-7588 


6. 271 


5- 196 


5 • 134 


71 


22.6074 


23. 185 


22.045 


21.882 


19 


6.0756 


6.593 


5-513 


5-451 


72 


22.9256 


23 - S04 


22.363 


22.20 1 


20 


6-3925 


6.914 


5-830 


S-768 


73 


23.2438 


23.822 


22.681 


22.519 


21 


6-7095 


7-235 


6- 147 


6.08s 


74 


23.5620 


24.141 


23.000 


22.937 


22 


7.0266 


7-555 


6.464 


6.402 


75 


23.8802 


24.459 


23.318 


23 -255 


23 


7-3439 


7-875 


6.781 


6.719 


76 


24. 1984 


24-778 


23.636 


23-573 


24 


7-6613 


8. 196 


7-099 


7.036 


77 


24.5166 


25.096 


23.954 


23-892 


25 


7-9787 


8.516 


7-416 


7-3S4 


78 


24.8349 


25-415 


24.272 


24.2 10 


26 


8.2962 


8.836 


7.-734 


7-671 


79 


25. 1531 


25-733 


24-591 


24.528 


27 


8.6138 


9- 156 


8,051 


7-989 


80 


2S-4713 


26-052 


24.909 


24.846 


28 


8.93 IS 


9-475 


8.369 


8-307 


8l 


25-7895 


26.370 


25.227 


25- 16s 


29 


9-2491 


9.79s 


8.687 


8.624 


82 


26. 1078 


26.689 


25-545 


25-483 


30 


9-5668 


10 . 1 14 


9.004 


8.942 


83 


26.4260 


27.007 


25-864 


25.801 


31 


9.884s 


10 . 434 


9.322 


9. 260 


84 


26.7443 


27.326 


26. 182 


26. 119 


32 


10.2023 


10.753 


9.640 


9-S77 


85 


27.0625 27.644 


26.500 


26.437 


33 


10.5201 


H.072 


9.958 


9-895 


86 


27.3807 


27.962 


26.818 


26.756 


34 


10.8380 


11.392 


10.276 


10.213 


87 


27.6989 


28.281 


■27 . 136 


27.074 


3S 


11. 1558 


It. 711 


10.593 


10.531 


88 


28.017 I 


28.599 


27.455 


27-392 


36 


11-4737 


12.030 


10.911 


10 . 849 


89 


28.3354 


28.918 


27.773 


27.700 


37 


11.7917 


12.349 


H.229 


1 1 . 167 


90 


28.6536 


29-236 


28.091 


28.029 


38 


12 . 1096 


12.668 


11.547 


11-485 


91 


28.9718 


29.554 


28.409 


28.347 


39 


12.4275 


12.987 


11.865 


11.803 


92 


29.2900 


29.873 


28.728 


28.66s 


40 


12.74SS 


13.306 


12. 183 


12. 12 1 


93 


29.6082 


30. 191 


29 . 046 


29.083 


41 


13.0635 


13.625 


12. SO I 


12.439 


94 


29.9264 


30.510 


29-364 


29.301 


42 


13-3815 


13 - 944 


12 . 8 19 


12.757 


95 


30.2446 


30.828 


29.682 


29.620 


43 


13-6995 


14-263 


13 - 137 


13 -07s 


96 


30.5628 


31-146 


30 . 000 


29-938 


44 


14.0175 


14-582 


13-455 


13-393 


97 


30.8811 


31-465 


30.319 


30-256 


45 


14-3356 


14-901 


13-773 


13-7 II 


98 


3 1 • 1994 


31-783 


30.637 


30.574 


46 


14-6536 


15 . 2 19 


14 . 09 I 


14.029 


99 


31.5177 


32. 102 


30.955 


30.893 


47 


14-9717 


15.538 


14.409 


14-347 


100 


31.8360 


32.420 


31.274 


31.211 


48 


15-2898 


15.857 


14-727 


14 - 665 


10 I 


32. 1543 


32.739 


31-592 


31529 


49 


15-6079 


16. 176 


15-045 


14-983 


102 


32.4726 


33.057 


31.910 


31.848 


50 


15.9260 


16.495 


IS -363 


IS -30 I 


103 


32.7909 


33-376 


32.228 


32. 166 


SI 


16.2441 


16.813 


15.681 


15.619 


104 


33 . 109 I 


33.694 


32.547 


32.484 


52 


16.5619 


17. 132 


15-999 


15-937 


los 


33-4274 


34.012 


32.865 


32.802 


S3 


16.8803 


17-451 


16.318 


16.255 


106 


33-7457 


34-33 1 


33.183 


33-12 1 


S4 


17- 1984 


17-769 


16.636 


16.573 


107 


34.0640 


34-649 


33-501 


33-439 


SS 


17.5166 


18.088 


16.954 


16 . 892 


108 


34-3823 


34.968 


33.820 


33.757 


56 


17.8347 


18.406 


17.272 


17.210 


109 


34.7006 


35.286 


34- 138 


34-076 


S7 


18. 1529 


18.725 


17.590 


17.^528 


no 


35.0189 


35.605 


34.456 


34-394 


S8 


18.4710 


19 . 044 


17-909 


17.846 


1 1 1 


35.3371 


35-923 


34-775 


34-7 12 



^-in. pitch chain would then be 18 for the motor, and 73 for the 
line shaft. 

The High Speed Silent Chain 

The preliminary design of Morse silent chain drives may be made in 
accordance with Table 7, which has been supplied by the Morse 
Chain Co. 

{.Continued on page 173, first column.) 



172 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 6. — Data for the Design of Diamond Chain Drives 




1 The normal pull given in the table corresponds to a pressure of looo lbs. per sq. in. of pro- 
jected rivet area. The maximum pull is about three times the normal pull, but does not exceed 
one-tenth of the ultimate strength of the chain. 

The projected rivet area for any chain may be found by dividing the number representing the normal 
pull by 1000. For example, the normal pull for chain No. 149 XH in. is 83 lbs., and the pro- 
jected rivet area is .083 sq. in. 

The normal horse-power given in the table corresponds to the normal pull at a chain speed of 800 ft. 
per min. 

The maximum horse-power given in the table is the maximum only when the chain pull does not 
exceed the limit given in the sixth olumn. 

The dimensions L and H are given to enable the designer to determine the amount of space 
required for a given chain drive. Columns C, T, E and R pertain to the design of the cross-section 
of the sprocket teeth. F (in figure) is the diameter of the rim or hub. It should not exceed pitch 
diameter minus I. iXfl, since it is necessary to allow clearance for the side bars of the chain. The 
last six columns have reference to lettered dimensions of the illustration. 









Diameter 


'Normal 


Max. 


Normal 


Weight 














Chain number 




Pitch 


of roller or 


and 


max. 


and 


max. 


per ft.. 


L 


H 


C 


E 


T 


R 








block, ins. 


h. 


P- 


r.p.m. 


pull. 


lbs. 


lbs. 














In. 


In. 






Normal 


Max. 




Normal 


Max. 


In. 


In. 


In. 


In. 


In. 


In. 


In. 


9-4SXM6 


I 


Block 


.325 


.8 


2.4 


22 14 


32 


96 


• 435 


.23 


.332 


Me 


.056 


'Me 


2^4 


9-45 X Vi 


I 


' ' 


.325 


I.O 


3. 1 


2316 


41 


124 


.27 


.493 


.332 


546 


.056 


'=/64 


2^4 


45 XW 


I 


' ' 


• 32s 


• 5 


1.6 


2 102 


22 


65 


. 16 


• 353 


.332 


5'ie 


.056 


%i 


2^4 


6 1-7 IX )4 


I 


f Twin 
1 roller 


• 325 


I. I 


3-0 


1734 


44 


120 


• 25 


• 50s 


.377 


I5'e4 


.090 


J64 


IJ62 
































6 1-7 IX Me 


I 


" 


.325 


1.3 


3.0 


1972 


55 


120 


.29 


• 5675 


• 377 


^U 


.090 


'W4 


"/62 


6 1-7 IX H 


I 


" 


■ 325 


1.6 


3.0 


2 170 


65 


120 


.33 


.630 


.377 


^%i 


.090 


'564 


"/62 


65-75 X)^^ 


Yi 


Roller 


.306 


I. I 


3.0 


22 12 


44 


120 


.28 


• 505 


.382 


562 


■ 04s 


U4 


'%4 


65-75 X Me 


Vi 


" 


• 306 


1.3 


3.0 


2608 


55 


120 


■ 30 


• 5675 


.382 


562 


■ 045 


1^4 


^%i 


- 6S-75XW 


'A 


" 


.306 


1.6 


3.0 


2904 


65 


120 


.32 


• 630 


.382 


562 


■ 04s 


1564 


"Ai 


80 XH 


'A 


/ Built 
[ block 


.306 


2. I 


6.2 


2400 


83 


250 


.42s 


.723 


•447 


562 


■ 04s 


'564 


'J64 


10 IX Me 


I 


Block 


■ 32s 


1.4 


4.0 


2352 


56 


160 


• 34 


595 


• 380 


Me 


■ 056 


562 


nu 


loiXH 


I 


** 


■ 32s 


1.7 


4.5 


2404 


68 


180 


.38 


.658 


.380 


546 


056 


'5.62 


21.64 


loiXJi 


I 


" 


.325 


2.3 


S.o 


2556 


92 


200 


.46 


.783 


.380 


546 


.056 


'562 


nu 


lOiX^^ 


I 


/ Double 
1 block 


■ 32s 


4.6 


7.5 


27 10 


183 


300 


.82 


1.423 


.380 


Me 


.056 


1562 


2^4 


102 X Vi 


I 


Block 


• 32s 


1. I 


3.4 


2135 


45 


135 


.33 


.493 


.380 


Me 


.056 


'564 


2^4 


102 X Me 


I 


*' 


• 32s 


1.4 


4,0 


2225 


56 


160 


.38 


.595 


.380 


Me 


.056 


562 


2^4 


102 X% 


I 


" 


.325 


1-7 


4.5 


2318 


68 


180 


.42 


.658 


.380 


Me. 


.056 


'J.62 


2^4 


102 X}^ 


I 


•■ 


■ 325 


2.3 


5.0 


2453 


92 


200 


.50 


.783 


.380 


546 


.056 


'562 


2^64 


103 XM 


I 


*' 


.325 


I. I 


3.4 


2135 


45 


135 


■ 33 


.493 


.380 


54 e 


.056 


ae 


21-64 


103 X Me 


I 


" 


■ 325 


1.4 


4.2 


2225 


56 


169 


.38 


.595 


.380 


54e 


.056 


562 


2^4 


103X^4 


I 


" 


■ 32s 


1.7 


5. I 


2318 


68 


205 


■42 


.658 


.380 


546 


.056 


1J.62 


25.64 


103 X!4 


I 


" 


.325 


2.3 


6.3 


2453 


92 


250 


.50 


.783 


.380 


546 


.056 


1562 


2564 


103 X Vz 


I 


f Double 
1 block 


• 325 


4.6 


9.4 


2598 


183 


375 


.90 


1.423 


.380 


Me 


.056 


'562 


21/64 


lOSXm 


iH 


Block 


■ 532 


3.3 


9.9 


12 13 


133 


398 


.89 


.898 


■534 


15^4 


.090 


2564 


1^62 


105 XJ^ 


iM 


" 


.532 


4. I 


12.4 


126 1 


166 


497 


1.03 


1.023 


.534 


^Hi 


.090 


3564 


IJ62 


1 47- 149 XM 


56 


Roller 


.400 


2 . I 


6.2 


1857 


83 


250 


.475 


.697 


.551 


Me 


.056 


'5(^4 


H62 


147- 149 X% 


56 


" 


.400 


2.7 


8. I 


1993 


108 


325 


.619 


.822 


.551 


Me 


.056 


1^62 


11.62 


147-149 XJ.^ 


m 


** 


.625 


6.3 


19. 


636 


253 


760 


1.58 


1.325 


.900 


56 


. 113 


2564 


■He 


I47-I4PX?^ 


iH 


*' 


.625 


7.3 


21.9 


688 


292 


877 


1.69 


1.450 


.900 


56 


. 113 


3564 


'He 


147-149 XM 


iH 


" 


.62s 


8.3 


24.8 


730 


331 


994 


1.80 


1.575 


.900 


56 


. 113 


'Me 


"He 


153 X Me 


H 


'* 


• 469 


2.6 


7-9 


1400 


106 


317 


.71 


.870 


.612 


.225 


.068 


562 


'562 


153 X% 


H 


" 


.469 


3.0 


9.0 


1481 


120 


359 


.76 


.933 


.612 


.225 


.068 


'W2 


1562 


153 X^ 


H 


" 


.469 


3-8 


II. 


1608 


147 


442 


.86 


1.063 


.612 


.225 


.068 


1562 


1562 


153X56 


H 


" 


■ 469 


4.4 


12.5 


1701 


175 


500 


.96 


I. 181 


.612 


.225 


.063 


3564 


1562 


153 X'/^ 


54 


/ Double 
\ roller 




7.6 


18.8 


1600 


295 


750 




1.828 


.612 


.225 


.068 


'562 


1562 


154 X'/^ 


I 


Roller 


.62s 


6.3 


19.0 


1863 


253 


760 


1.68 


1.32s 


.820 


'564 


.090 


2564 


1%2 


154X56 


I 


" 


.625 


7.3 


22.0 


929 


292 


877 


I. 81 


1.450 


.820 


'564 


.090 


Me 


'%2 


154 XM 


I 


" 


.62s 


8.3 


24.8 


983 


33 1 


994 


1.94 


1.575 


.820 


'564 


.090 


4564 


17,62 


154X56 


I 


/ Double 
\ roller 


.625 
.625 


14.6 


42. 5 


975 


585 


1700 


3.289 


2.621 


.820 


'564 


.090 


546 ■ 


1562 


IS5X?6 


I 


Roller 


5625 


4.4 


13.2 


888 


17s. 6 


526.8 


1.07 


1.062 


.734 


'564 


.090 


"/62 


"762 


155 XM 


I 


" 


.5625 


5.3 


15.8 


980 


2 10.8 


632.4 


I. 17 


1. 187 


.734 


1564 


.090 


2564 


"/62 


155X56 


I 


" 


.5625 


6. I 


18.4 


1052 


246 


738 


1.27 


I-3I2 


•734 . 


'5'e4 


.090 


Me 


'562 


155X1.4 


I 


/ Double 
I roller 


• 562s 


10.5 


31.6 


II06 


422 


1200 


1.84 


2. 121 


• 734 


1564 


.090 


^546 


'562 


157X54 


1K2 


Roller 


• 750 


9.9 


29-7 


526 


396 


1 189 


2.41 


1.739 


1.097 


mi 


.135 


'He 


i54e 


IS7X I 


l'/2 


" 


■ 750 


12.3 


36.9 


570 


492 


1476 


2.74 


1.994 


1.097 


2564 


. 135 


'Me 


'54e 


i6oX56 


i'/4 


*' 


• 750 


8.7 


26.2 


615 


350 


:049 


2.54 


1.6 14 


I. 181 


56 


• 113 


Me 


iHe 


160 XU 


iM 


" 


.750 


9.9 


29.7 


640 


396 


1 189 


2 .69 


1.739 


I. 181 


56 


• 113 


'He 


iHe 


160 X I 


iVi 




• 750 


12.3 


36.9 


717 


492 


1476 


2.99 


1.994 


I. 181 


56 


• 113 


'Me 


iHe 


160 X 54 


iM 


/ Double 

1 roller 
Roller 


• -750 


19.8 


59.5 


680 


793 


2400 


4.851 


2.972 


I. 18 I 


56 


• 113 


'He 


IHe 


162 X ¥i 


l>/2 


.875 


13-0 


39.0 


448 


520 


1560 


3.89 


1-951 


1.407 


m-i 


• 135 


'He 


'Me 


162 X I 


m 


** 


.875 


15.7 


47.2 


500 


629 


1888 


4- 15 


2.201 


1.407 


2564 


• 135 


'Me 


'Me 


164X1 


154 


** 


I. 000 


18.0 


54.0 


388 


720 


2 160 


4,96 


2.249 


1.657 


"A2 


562 


2562 


6H64 


I64X I 


i?4 


/ Double 
\ roller 


I. 000 


36.0 


■ 108.0 


412 


1440 


4000 


8.75 


3.929 


1.657 


1J62 


562 


2562 


«H4 


168 xiH 


2 


Roller 


I. I2S 


24.4 


73. I 


334 


975 


2925 


6.32 


2.651 


1.907 


1562 


. 180 


1562 


1562 


168 xiH 


2 


f Double 
1 roller 


I. 125 


55.8 


167.3 


349 


223 I 


6000 


11.56 


4.666 


1.907 


'562 


. 180 


1562 


1562 


700x1.6 


Vi 


fStud 
1 chain 


. 105 


. I 


.4 


5854 


5.8 


17.3 


.098 


.385 


.200 


5^4 


.022 


%4 


564 


702X?-ie 


.326 


" 


. 128 


. 2 


.6 


4228 


8.6 


25-7 


.156 


.459 


.270 


562 


.029 


'H64 


iHe4 


705 X M 


.385 


" 


. 148 


.3 


1.0 


4014 


13.3 


40. 


.246 


.580 


■ 340 


H 


■035 


'564 


m.i 


705 X Me 


.389 


' ' 


.148 


.4 


I. 


3957 


13.3 


40.0 


.273 


.755 


■ 340 


M 


■ 03s 


25^4 


1564 


7o8X?4e 


'/2 


' ' 


. 128 


. 2 


.6 


3138 


7.4 


22.3 


.078 


.451 


. 192 


562 


■045 


'V44 


' 5 



CHAINS 



173 



Table 7. — Data for the Design of Morse Silent Chain Drives 












Notes 


Pitch, ins 


1 


tV 


1 


1 


f 


9 
10 


IT% 


ij 


2 


3 




1. Number o£ teeth =r. 
Exact outside diameter = D. 

When T has less than 20 teeth, D = 

pitch diameter. 
When T has more than 20 teeth, £> = 

pitch diameter + (2 X addendum). 

2. Use sprockets having an odd number of 

teeth whenever possible. 

3. When specially authorized, a larger 

number of teeth than shown may be 
cut in large sprocket. 

4. Thickness of sprocket rim, including 

teeth, should be at least i . 2 times the 
chain pitch. 

5. The number of grooves in the sprocket, 

their width and distance apart, varies 
according to pitch and widthof chain. 
In every case leave the designing and 
turning ot these grooves to the Morse 
Chain Company. 

6. The width of the sprocket should be \ 

to J in. greater on small drives, and i to 
5 in. greater on large drives than nom- 
inal width of the chain. 

7. An even number of links in the chain and 

an odd number of teeth in the wheels are 
desirable. 

8. Horizontal drives preferred; tight chain 

on top desirable for short drives without 
center adjustment. 

9. Adjustable wheel centers desirable for 

horizontal drives and necessary for ver- 
tical drives. 

10. Avoid vertical drives. 

11. Allow a side clearance for chain (parallel 

to axis of sprockets and measured from 
nominal width of, chain) equal to the 
pitch. 

12. Maximum linear velocity for commercial 

service 1200 to 1600 ft. per minute. 

These data are for use in preliminary de- 
sign. Engineering features should always be 
submitted to the Morse Chain Company 
for approval before ordering. 






Minimum number of teeth: 

Small sprocket driver 

Small sprocket driven 


13 

17 


13 

17 


13 

17 


13 
17 


13 
21 


15 
25 


15 
29 


17 
29 


17 
31 


17 
35 


19 
37 


Desirable number of teeth in 
driver sprockets. 


IS-17 


IS-17 


IS-17 


17-21 


17-21 


17-23 


17-23 


17-27 


17-31 


19-31 


21-31 


Maximum number of teeth in 
sprockets. (See Note 3). 


75 


85 


99 


109 


115 


125 


129 


129 


129 


131 


131 


Desirable number of teeth in 
driven sprockets. 


35-SS 


3S-SS 


55-75 


55-75 


55-85 


55-95 


55-105 


55-115 


55-115 


55-1 I 5 


55-1 IS 


To find pitch diameter of 
wheel multiply number of 
teeth by (ins.). 


.1195 


.127 


.159 


.199 


.239 


• 2865 


.382 


• 477 


• 636 


• 955 


1.2732 


Addendum. For outside diam- 
eter of sprockets 20 to 130 T. 
(See Note i), ins. 








■ 05 


.06 


.075 


.09 


. 12 


• 15 


.20 


•30 


.80 


Maximum r.p.m 


3000 


2600 


2400 


1800 


1200 


HOC 


800 


600 


400 


250 


100 


Tension per inch width chain, 
lbs: 

Small sprocket driver 

Small sprocket driven 


40 
30 


43 
35 


80 
65 


100 
80 


120 

95 


• 

150 

120 


200 
160 


270 
210 


450 
350 


750 
600 


900 
700 


Radial clearance beyond tooth 
required for chain, ins. 


• 37 


.40 


.50 


.62 


.75 


• 90 


1 . 2 


1-5 


2.0 


3.0 


4.00 


Approximate weight of chain 
per inch wide, i ft. long, lbs. 


■ 7 


.75 


1 .00 


I .20 


1.50 


1.80 


2.50 


3^00 


4^00 


6^oo 


8.00 










.0063 


.009 


.013 


.023 


.035 


.058 


.145 




















.16 


.25 


• 35 


• 45 


•7 


1 .0 


2,0 


4.0 












Approximate Weights for Solid and Armed Sprockets 
r = number of teeth. 
F = face in inches. 

C = constant in pounds per inch in face per tooth as per table. 
Weight of armed sprocket = TXFX C. 

Add 25 per cent, for split and 50 per cent, for spring and split sprockets. 
Weight of solid pinion = T^X (F + i) XC. 









Note (l) does not apply. Pitch and out-side diameter are the same or equal. 



It should be borne in mind that the silent chain is practically a 
flexible rack, and gives a positive drive. Its use, therefore, is unde- 
sirable where the necessity of some slip exists, such as would be 
found in driving punching presses where the accumulated speed of 
the balance wheel does the work of each stroke. In drives having 
an infrequent shock load, due to accident or lack of uniformity of 
material to be worked, a safety or shearing pin sprocket is fitted as 
a safeguard. 

Iti regularly intermittent service, such as air-compresser driving, 
a spring sprocket wheel is always desirable and sometimes necessary. 
In drives subject to sudden overloading and heavy shock, a shearing 
or safety pin is often fitted. These chains are regularly used for 
transmitting loads up to 1500 h.p. 

The following observations explain more fully some of the informa- 
tion given in the table: 

The limitation of the desirable number of teeth in the large 
sprocket, given in the fourth line of the table, is intended to give a 
reasonable provision for the increased pitch of the chain due to use. 
As is well known, the chain gradually engages the sprocket at increased 
diameters as its pitch increases, and, with too large sprockets, the re- 
duced ratio of pitch to diameter reduces this provision below the de- 
sirable limit. The call in note 7 for an even aumber of links in the chain 
is intended to eliminate the special link which an odd number of links 
require. This can usually be brought about by a slight adjust- 



ment of the center distance. The call in the same note for an odd 
number of teeth in the sprockets is intended to provide a hunting- 
tooth effect, by which all parts of the sprocket-tooth faces are suc- 
cessively engaged by the links. While this is a desirable it is not 
an essential feature, and when exact speed ratios which call for an 
even number of teeth are required, such teeth may be used without 
hesitation. 

The preliminary design oj Link Belt silent chain drives may be made 
in accordance with Tables 8 and g which have been supplied by the 
Link Belt Co. 

In applying silent chains of any type, the following suggestions 
should be considered: 

Drives should not be vertical if such arrangement can be avoided; 
if vertical or nearly so make the center distance between shafts short; 
a long vertical chain will tend to drop away from the teeth of the 
lower wheel, causing bad chain action. 

Provide means for adjusting the distance between shafts. This 
will facilitate the installation of the chain and will, in some cases, 
prolong the life of the drive by making it possible to take up wear. 
On extremely short center drives the adjustability is not as essential 
if care is exercised to make the center distance such as will keep the 
chain without slack. 

Do not run chains tight; initial tension is not necessary, and it 
increases the bearing pressures in both chain and shaft bearings. 



174 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 8. — Capacity of Link Belt Silent Chains 



Speeds in ft per. 
min. 


SOO 


600 


700 


800 


900 


1000 


IIOO 


1200 


1300 1400 


1500 


Nom. 
pitch 


6| 
0:2 
2& 


Face 






















of 
wheel 


H.p 


H.p 


H.p 


H.p 


H.p 


H.p 


H.p 


H.p 


H.p H.p 


H.p 




i 


I 


.58 


.66 


• 72 


.78 


.82 


.88 


.91 


■ 95 










\ 


il 


.87 


.98 


1.07 


1. 16 


. 122 


1.30 


1.38 


1.42 










I 


i^ 


1. 16 


1. 31 


1.43 


i.SS 


1.63 


1.73 


1.82 


1.89 








i" 


li 


il 


1.45 


1 .64 


1.79 


1. 91 


2 .04 


2.18 


2.28 


2.36 










i^ 


2 


1.74 


1.97 


2. IS 


2.30 


2.45 


2.60 


2.73 


2.83 










2 


2i 


2.32 


2.62 


2.86 


3.08 


3-27 


3-46 


3.64 


3.78 










3 


Z\ 


3.48 


3-91 


4.28 


4.61 


4.89 


5.22 


5.46 


5.67 










\ 


Ij 


.84 


■ 95 


1.04 


I . II 


1. 19 


1.27 


1.33 


1.38 


1.42 








3 


If 


1.26 


1 .40 


1.56 


1.70 


1.79 


1.91 


1.99 


2.07 


2.13 








I 


If 


1.68 


1.89 


2.08 


2.25 


2.34 


2.54 


2.65 


2.76 


2.84 






i" 


li 


2i 


2. 52 


2.91 


312 


3.44 


3.57 


3.88 


3.98 


4.14 


4.25 








2 


2f 


3.37 


3.82 


4.17 


4.48 


4-77 


5.10 


S-30 


5-S2 


5.68 








3 


3l 


S.os 


5. 73 


6.25 


6. 75 


7.15 


7.60 


7-95 


8.29 


8.50 








4 


4l 


6.73 


7.64 


8.30 


9.00 


9.53 


10. I 


10.6 


II. I 


II. 3 








I 


If 


2.22 


2.SJ 


2.74 


2.96 


3-15 


3-33 


3 SO 


3.64 


3. 75 








U 


if 


2.77 


3.15 


3-41 


3.71 


3.93 


4.18 


4-37 


4-54 


4-70 








li 


2tV 


3-33 


3.76 


4.12 


4-43 


4.72 


5.00 


S-25 


S.45 


5.62 






f" 


2 


2ii 


4-43 


S.02 


5-47 


5.91 


6.30 


6.67 


7 .00 


7.28 


7.50 








3 


3i4 


6.65 


7.52 


8.22 


8.88 


9.45 


10. 


10.05 


10. 09 


II. 2 








4 


4^ 


8.86 


10. 


10.9 


11. 8 


12.6 


13.3 


14.0 


I4-S 


IS-0 








6 


6J 


13.3 


IS.O 


16.4 


17-7 


18.9 


20.0 


21.0 


21.8 


22. S 








I 


ll 


2.8s 


3.22 


3. SI 


3.78 


4-05 


4-37 


4.48 


465 


4.82 








li 


2i 


3.56 


3.98 


4-39 


4.70 


5.06 


5. 30 


5 60 


5.78 


6.02 








li 


2j 


4.27 


4 -85 


S.27 


S.67 


6. 10 


6.40 


6.72 


6.98 


7.23 






i" 


2 


2I 


5-68 


6.42 


7.03 


7.S6 


8.10 


8.55 


8. 95 


9-31 


9.63 








3 


3I 


8.55 


9 63 


10.5 


II. 4 


12. 1 


18.8 


13-4 


14.0 


I4-S 








4 


a\ 


10.4 


12.8 


14.0 


IS. I 


16.3 


17-3 


17.9 


18.6 


19-3 








6 


t\ 


17. 1 


19-3 


21 . I 


22.8 


24-3 


25.7 


26.8 


27.9 


28.9 








2 


2\ 


7 


7.91 


8.65 


9-33 


10 


10.5 


10.9 


II. 4 


II. 8 








2i 


3i 


9 


10. 1 


II. I 


12.0 


12.9 


13s 


14. 1 


14.7 


IS. 2 








3 


3i 


II 


12.4 


13.6 


14.6 


iS-7 


16.5 


17.2 


18 


18.6 






I" 


4 


4i 


IS 


16.9 


18.6 


20 


21.5 


22.5 


23.5 


24.6 


25-4 








S 


si 


19 


21.5 


23. 5 


25.2 


27.2 


28.7 


29-7 


311 


32.1 








6 


6i* 


23 


26 


28. S 


30.5 


32.9 


34-S 


36 


37.6 


38.9 








8 


8H 


31 


34-9 


38.4 


41.2 


44-3 


46.3 


48.5 


SO. 7 


52.4 
16.4 








2 


3 


9.7 


II. 


II. 9 


13 


13.8 


14.6 


IS. 3 


15-9 


16.7 






3 


4 


15-3 


17.3 


18.7 


20.3 


21.7 


22.9 


24.2 


25 


25-7 


26. 5 






4 


5 


20.8 


23.5 


25-S 


27.6 


29.6 


31-2 


32.6 


34-1 


35.1 


36.2 




li" 


S 


6 


26.3 


29.8 


32.3 


3S.I 


37-5 


39-7 


41 .6 


43.2 


44- S 


4S.8 






6 


7 


31.8 


36.2 


39.1 


42-7 


4S.3 


48.2 


50.3 


52.2 


53.8 


SS-S 






8 


9 


42.8 


48.5 


52.7 


57.2 


61.2 


64 


67.8 


70.3 


72. S 


74-6 






10 


II 


54-1 


61.3 


66.5 


72.2 


77.1 


81.2 


85.6 


88.7 


91.4 


94.1 






3 


4 


20. I 


22.7 


24-7 


26.9 


28.7 


30.3 


31.8 


33 


34 


3S 


35.7 




4 


5 


27.5 


31. 1 


33.7 


36.6 


39-1 


41.2 


43.4 


4S 


46.4 


48 


48.7 




5 


6 


34.8 


39.3 


42-7 


46.3 


49-5 


52. 3 


55 


57 


58.7 


60.7 


61.6 


zh" 


6 


7 


42.2 


47.6 


51.8 


S6.3 


60 


63-4 


66.5 


69 


71. 1 


73. 5 


74.7 




8 


9 


56. 7 


64.2 


69.7 


75.7 


81 


85.2 


89-7 


93 


95-8 


99 


lOI 




10 


II 


71.4 


80.7 


87.7 


95.2 


102 


107 


113 


117 


121 


124 


127 




12 


13 


86 


97.3 


106 


115 


123 


129 


136 


141 


145 


ISO 


IS2 




6 


7A 


56.1 


63. 5 


69 


75 


80 


84-3 


88.8 


92 


94.8 


97.5 


99.6 




8 


9A 


75.7 


85.6 


93 


lOI 


108 


114 


120 


124 


128 


131 


134 


2" 


10 


lift 


95.2 


107 


117 


126 


136 


143 


151 


IS6 


161 


165 


169 


12 


13ft 


114 


129 


141 


153 


164 


172 


182 


188 


194 


199 


204 




14 


iSft 


134 


152 


I6s 


179 


191 


201 


212 


220 


227 


233 


240 




16 


I7ft 


154 


174 


189 


205 


220 


231 


243 


252 


260 


267 


273 




6 


7i 


73 


82.7 


90 


98 


104 


110 


116 


120 


124 


127 


130 




8 


9\ 


100 


113 


123 


133 


143 


ISO 


158 


164 


169 


174 


178 


2\" 


10 


1I5 


126 


143 


155 


168 


180 


190 


200 


207 


213 


220 


224 


12 


I3l 


153 


173 


188 


204 


218 


230 


242 


251 


259 


266 


272 




14 


I5i 


179 


204 


220 


240 


255 


270 


284 


294 


303 


313 


318 




16 


17J 


206 


235 


253 


274 


294 


310 


326 


338 


348 


359 


36s 



Table 9. — Data for the Design of Link Belt Silent Chain Drives 

This table is based on standard practice and is to be used for preliminary 
design only. Consult the makers before final design is adopted. 





*^ 




c 


Driving wheel 


Driven 
wheel 




















c 


2 "^ 
0, ■" 


§1 




5 


0) 


d 


a- 




ID 


in "ia 





•« 

^ 




J3 


0) 

"o 




^^ 


n 


6 
2 












6 
2 


6 


•2| 
^.6 


a ° 
■0 


n CO 
c 


6 
2; 


M 

2 h 


s 


. a 


E ° 


« 


6 

S 
'S 


6 

3 


.2 -o 

J3 Hi 



3 


a 




0. ^ 
< ■3 

e 











s 


a 




f" 


. 12 


t" 


i" 


17 


100 


2260 


1200' 


19 


ISO 




1" 


.16 


h" 


1" 


17 


100 


183s 


1300' 


19 


ISO 


S 2-3 
-w e 


s" 


.20 


1" 


I" 


17 


100 


1467 


1300' 


19 


150 


i" 


.24 


3// 


I" 


17 


100 


1223 


1300' 


19 


ISO 


i" 


■ 32 


i" 


5 


17 


100 


918 


1300' 


21 


150 




li" 


.40 


li" 


i" 


17 


100 


791 


1400' 


21 


ISO 


.2 S) "0 


Ih" 


.48 


li" 


1" 


17 


100 


705 


1500' 


21 


ISO 


2" 


.637 


2" 


If" 


19 


100 


474 


1 5 00' 


23 


ISO 


«l 


2i" 


.796 


2h" 


i¥' 


19 


100 


378 


1500' 


23 


150 



The loading of hoisting sling chains as recommended by the National 
Founder's Association is shown in Table 10, which gives the loads 
for each single chain of the best grade of wrought iron, hand made, 
tested, short link grade. 

Table 10. — Safe Loads of Hoisting Chain Slings, Lbs. 

When handling molten metal the chains should be 25 per cent 

stronger than these figures 



Diam. of 
iron, ins. 






y^eo-A 


yAm\^ 


,x-*i1o>\ 




425 






SOD 




Yi 


600 


300 


H 


1,200 


1,025 


850 


600 


H 


2,400 


2,050 


1,700 


1,200 


H 


4,000 


3,400 


2,800 


2,000 


% 


5,500 


4,700 


3,900 


2,750 


% 


7,500 


6,400 


5,200 


3,700 


I 


9,500 


8,000 


6,600 


4,700 


iH 


12,000 


10,200 


8,400 


6,000 


m 


15,000 


12,750 


10,500 


7,500 


m 


22,000 


19,000 


16,000 


11,000 



BRAKES 



The retarding moments of hand brakes may be obtained from the 
formula by E. R. Douglass {Amer. Mach., Dec. ig, 1901): 

M=P{K-i)- (a) 

in which M = retarding moment, lb. -ins. 

P = force pulling free end of brake, lbs., 

rf = diameter of brake drum, ins., 
K = a, factor such that 
com. log K = .oojsSfc, 

/= coefficient of friction, 

c — angle of drum embraced by band, degrees, 

all as shown in Fig. i. 

The value of M may be found from Fig. 2, which is plotted for 

P=i and d=io. To use the chart find M for the arc of contact 

and the assumed value of / and multiply the result by the value of 

d 
P and by the ratio of — The plotted values of/ are: 

Cork inserts on metals, /= about 
Leather on metals, / = about 

Leather on wood, /= about 

Metals on wood, / = about 

Metals on metals, /= about 

See also end of section. 

The pulling force at the free end of the strap being known, that at 
the fixed end may be found from Fig. 4. 

The width of the brake band may be found from the following 
formulas, adapted from those by R. A. Greene (Amer. Mack., Oct. 8, 
1908) which give the practice of the Browning Engineering Co., 

4MX 



•33 

•4 

■3 

.2 

.2 



F = - 



d^S 



(b) 



in which P = width of drum face, ins., 

If = retarding moment as found from (a) or Fig. 2, 

X = a, factor from Table i, 

<i = diameter of drum, ins., 

S = a. limiting factor which should not exceed 65. 

Assume a brake drum of diameter (^ = 30 ins., a puUing force 

P = i5oo lbs., an arc of contact of 260 deg. and a metal strap on a 

metal drum. From Fig. 2 we find, for a pulling force of i lb. and a 

drum diameter of 10 ins., a value of M of 7.3, giving for the actual case 



M-- 



= 7.3XiSooxg 



=32,850 Ib.-ins. 
From Table i we find the value of X to be 1.68 and hence from (6) 
^_ 4X3285oXi.68 
900X65 
= 3.8 ins. or, say, 4 ins. 
Next we check against the speed by the formula: 

i? = 5XiX.262Xr.p.m. (c) 

in which R= a, factor which should not exceed 54,000 according to 
Yale and Towne practice, or 60,000 according to Brown Hoist prac- 
tice. If r.p.m. = 100 we find: 

-R = 65X3oX. 262X100 
= 51,090 

which, being below the limiting value, we conclude that the brake 
will answer although near the limit. Had the value of R gone above 
the limit it would have been necessary to assume a wider drum and 
by substitution in (b) found a smaller value of 5 which, substituted 
in (c), would have brought the value of R below the limit. 



The differential hand brake, Fig. 3, is more used in Europe than in 
the United States, where some attempts to use it have met with fail- 
vae because its principle was not correctly grasped. In Fig. 3, a 
represents the weight to be lifted by the rope drum b. At c is the 
brake drum, the ends of the brake strap beng at d and e. The direc- 
tion of motion when lowering the load being that shown by the arrow 



\ ^^A^ ^ / / / 




















\^ \ \ / 1 1 




















\ ^v \ / / / 


















1 


\ ^i V' / / 




















^\ \ / / 


















' 


\^ \</ ^ 






































j 




^ "^ --^ 


















1 




^y^^^ 


















1 




^-■\^,/^ 


















r 




r<'^^^^ 4i 


















1 






















1 


























1 






Fastened TiG. i. 




















J 






































40 


























1 






































1 




















































































































/ 








































































/ 












35 






























































/ 








































































' 








































































/ 










































































/ 








































































1 
















30 


























































/ 






































Plotted for 
P = l 

d: = io. 


















































































/ 




























































/ 
















/ 










































/ 


















/ 


25 






















































/ 
















/ 






















































/ 


















/ 






















































's. 


/ 
















/ 






















































,/// 
















/ 
























































^ 


/ 
















/ 
























































/ 
















/ 










20 






























































/ 












^ 














































/ 














/ 


























































/ 














/ 




























































1 






























s 










































/ 
































a 15 








































/ 










"> 


/ 


/ 




















^ 






































/ 










(^V 






























































/ 










V 
























y 






































/ 










/ 
























y 




10 


































/ 










y 






















y 






































/ 










/> 




















y 










































/ 








/I 




















y 


' 










































V 








y 












^ 


















































/ 
























^ 














































y 








y 












(^ 




















































/ 






y^ 












--^ 




















































/ 
















L-' 


■^ 






















































^ 




L^ 








■^ 






























































it. 




-^ 
























































- 


^ 


rf 


g 


















_ 


























_ 


_ 






_ 


_ 




_ 











90 



270 



360 



180° 
Angle o 

Fig. 2. — The retarding moment of band brakes. 

it follows that the end d of the strap carries the load, this end being 
commonly attached to the frame and the application of the brake 
being made by tightening e, which is attached for that purpose to the 
end/ of a bell crank which is operated by the brake lever g. With 
the differential brake, however, the end d is attached to an additional 
arm h of the bell crank, the action being that the tension in d tends to 



175 



176 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



tighten e and thus apply the brake, and it is on the ratio between 
the arms / and h that the design hinges. 

The two ends of the brake strap are under the same conditions 
as the slack and the tight sides of a belt. It is obvious that if the 
strain on d be, for example, twice that on e, and if the length of h 
be slightly more than half that of /, the brake will apply itself when 
the drum is released from the° engine which hoisted the load, and if 
the load is to be lowered at all the brake must be released by hand. 
On the other hand, if the length of h be slightly less than half of / 
the brake will not apply itself, the action of the tension on d serving 
merely to reduce the pressure which must be applied to g in order to 
hold the load. 



Table 


I. — ^Factors for 


THE Width of Band Brakes 


Degrees 


X 










/=.2 


/ = .3 


/=-4 


i8o 


2,14 


1.64 


1.40 


I9S 


2.03 


1.56 


I -35 


2IO 


1-93 


i-So 


1.30 


240 


1.76 


1.40 


1.23 


250 


1.72 


1-37 


I. 21 


260 


1.68 


1-35 


1. 19 


270 


1.64 


1.32 


1. 18 


280 


1.60 


1.30 


1. 17 


290 


1-57 


1.28 


i-iS 


300 


1-54 


1.26 


1. 14 



Table 2. — Coefficients foi! 


Differential Band 


Brakes 


Fraction gf cir- 
cumference em- 
braced by brake 
strap 


Value of coefScient of friction 


.18 


•28 .2,2, 


• 47 


Ratio between arms 


•S 
.6 

• 7 
.8 


1.76 
1.97 
2. 21 
2.47 


2.41 
2.87 
3-43 
4.09 


2.82 

3-47 
4.27 

S-2S 


4.38 

5.88 

7.90 

10.62 



Table 2 is the work of H. A. Vezin and represents the practice of 
the F. M. Davis Iron Works Co. {Amer. Mack., Nov. 23, 1905). It 
gives the ratios which the arms / and h must bear to each other in order 
to give the limiting condition between those described; that is, in order 
that the brake may be self applying, the arm h must be slightly 
longer, and in order that it may need help applied to lever g, it must 
be slightly shorter than the figures given by the table. 

For coefficients of friction other than those used by Mr. Vezin, 
Mr. Brown's chart for the ratio of the tensions, Fig. 4, may be used, 
as it gives the same results as Mr. Vezin's table. 

Certain band-brake calcidations are more readily made with the aid 
of Fig. 4, by Jas. A. Brown {Amer. Mach., Apr. 19, 1906), than with 
Fig. 2. The relation of the tensions in the two ends of the strap is 
given by the formula: 

T, 



K = l 



?• ^ = 2.729 /«• 



in which ri= lesser tension, 

r2 = greater tension, 

/ = coefficient of friction 

n = fractional part of drum embraced by strap. 

Tj 
The value of AT = ^may be ob tamed directly from Fig. 4. Find the 

product of /X« on the right-hand vertical, trace horizontally to the 
curve and then down and read the value of K. For example, with a 
coefficient of friction of .33 and one-half the circumference in con- 
tact, the product is .165, giving a value for K of 2.82. This 
chart is also useful in belt and other calculations relating to wrap- 
ping friction. 



The retarding moment of block brakes. Fig. 5, after wear has brought 
about uniform contact, may be found from the formula (E R. Doug- 
lass, Amer. Mach., Dec. 26, 1901): 

M=^hJQdf 
in which M = retarding moment for one block, Ib.-ins., 

Q = force pressing block on drum, lbs., 
d = diameter of brake drum, ins. , 
/= coefficient of friction 
I 



/=- 



<t) 



b being the angle subtended by one block, degrees. For more than 
one block multiply by the number of blocks. Values of / may be 
taken directly from Fig. 5. Values of/ have been given in the dis- 
cussion of band brakes. 




Fig. 3. — The principle of the differential band brake. 

The retarding moments of axial brakes, Fig. 6, after wear has taken 
place, may be found from the formula (E. R. Douglass, Amer. Mach., 
Dec. 26, 1901): 

M = X/^' 
4 

in which If = retarding moment for one block, Ib.-ins., 

X = force pressing block on disk, lbs., 

(i = external diameter, ins., 

(i' = internal diameter, ins., 

/=coefl&cient of friction. 

For more than one block, as in the Weston multiple disk brake, 
multiply by the number of friction surfaces in contact. Values of / 
have been given in the discussion of band brakes. 

The surface of brake drums should also be sufficient to provide 
for the dissipation of the heat generated without undue rise of temper- 
ature, and this obviously depends on the frequency of the service. 
No comprehensive study of this subject has been made so far as the 
author is aware. According to E. R. Douglass {Amer. Mach., Dec. 
26, 1901) for such brakes as are used on electric cranes, where the work 
is severe and constant, good results are obtained with a provision 
of I sq. in. of wood or leather frictional surface for every 200 
or 250 ft.-lbs. of energy to be absorbed. When the brake is less often 
called into service these figures may be much exceeded. The brakes 
of railway cars, which operate under the most favorable conditions 
for keeping cool, are of metal and are used less frequently, are required 
in extreme conditions to absorb as much as 20,000 ft.-lbs. per sq. in. 
of brake shoe. This service tears off and ignites the metal and the 
shoes must be frequently replaced. According to P. M. Heldt 
{Horseless Age, Aug. 28, 1912), hub brakes of automobiles (pleas- 



BRAKES 



177 



ure cars) should have i sq. in. of surface for each 15 lbs. weight of 
the car, while on commercial vehicles the ratio should be i sq. in. 
for each 30 lbs. weight of car. Assuming 20 and 10 miles per 
hour respectively, as the average speeds to be dealt with in stopping 
the cars, these figures give 240 and 120 ft.-lbs. of energy per sq. 
in. of surface. 

The band brake is not so much used on large hoisting engines as 













— 1 








~ 




~^ 




~ 




































































































































--■ 






































' 








































J-n 


' 




















_ 




— 


_ 


-. 




-Tr 




.- t 






/ 


^ 






^ 




— 




- 


-^ 


— 


— 


^ 






— 




-X 






r 


/' 










s 






















M 




i::::^ 










/ 




^ 






















;;? 




in 
















EflEec 


ti 


re 


T 


ull 






y 




t 


\ 




7 












I 


=t;- 


T 










::::^::::::: :. . 




T 


% 


>. 


/ 






























y. 




^ 














/ 




















;' 










V. 


"-> 






/ 
































*Si 


. 




c* 
















> 




'P 


^A^ 




























" 






I 






































y 








































y 






































/ 














































' 


































































A 






































/ 






































A 












































































A 








































1 






































/ 


1 




































/ 




1 


































/ 






1 






















■ 










/ 








1 






























/ 










1 






























/ 










1 
1 




























/' 












1 


























/ 














1 
























/ 


f 














1 
























/ 
















1 
























7 








































~ 7 


^^ 














1 

J_ 

























.35 



30 



.25 



4 5 

Values of K 



Fig. 4. — The ratio of the tensions in band brakes. 



1.5 



1.0 

































^ 


-^ 


























^ 


--- 


^ 




























^ 


^ 





























































^ 


-^ 


'^ 














































































































- 










































































- 


















1 





































ing movement. The block brake withdraws posit vely and leaves a 
large portion of the drum exposed, thus favoring the dissipation of 
the heat. 

A Superior Hoisting Brake 

A superior and very large brake is shown, in principle, in Fig. 7. 
It was applied by the Nordberg Mfg. Co. to the main hoisting engine 

of the Tamarack Mining Co. 
{Amer. Mach., Sept. 21, 1899), a 
brake being applied to each end 
of the hoisting drum. 

The brake consists of a pair of 
jaws AAi, adapted to grip the 
brake wheel, the surfaces in con- 
tact being basswood and cast- 
iron. The jaws are supported by 
a pair of carriers or anchors, BBi, 
which, as they have to prevent 
the jaws from partaking in the 
rotation of the drum when the 
brake is applied, have to be 
securely anchored into the foun- 
dation. The jaw A carries a pair 
of levers CCi, the short ends of 
which connect by means of rods 
DDi to end of jawvl, while the 
long arms connect to a lever E 
by means of two rods FFu E 
carries a weight G sufficiently 
heavy to furnish the braking 
power needed. By a steam device 
H and lever / the weight can be 
applied or released. Lever E is 
held in the jaw A, but as the pins 
on which rod F connects to lever 
E axe equidistant each side of the 
fulcrum, there is no reaction due 
to the forces in rods FFi on jaw ^ . 
All parts shown on Fig. 10, except 
steam device and rock shaft for 
lever I, are in duplicate at the two ends of the drum. In order to 
secure a parallel motion of the jaws to prevent the lower portion 
gripping first, the parallel rod / is used. Its action is plain if it is 
stated that its length is equal to that of the carrier B, making the 
action of the brake like a parallel motion vise. K, Ki, K2, K3, are 



.20 



15 



10 



10 




30'^ 



60 



90° 
Values of b 



120 



150° 



180 



Fig. 5. — The coefiicient of block brakes. 

formerly, partly because of the fact that it completely surrounds the 
drum and interferes with the free dissipation of the heat and partly 
because it does not positively withdraw from the drum when 
loosened, but consumes power and generates heat during the hoist- 
12 



Fig. 6. — Axial brake notation. 

set screws to limit the motion of the brake jaws The steam device 
H is single acting, the steam releasing while the weight sets the 
brake. The connection to hand lever is by a floating lever, 
whereby the piston is caused to follow the motion of the hand lever. 
It is plain that this type of brake is always ready to go on, even if 
the steam should fail, and it is thus as reliable as a hand brake. 



178 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



It is, in fact, more reliable, as it will not allow the engine to be 
started without steam being first turned on. Accidents have hap- 
pened from the opposite arrangement. 



H\ 



5 




Fig. 7. — Nordberg arrangement of large block brakes. 

Automatic Brakes 

One of many constructions of automatic load brakes is shown in 
Fig. 8, by A. D. Williams (Amer. Mack., Aug. 20, 1903). In action, 
the tendency of the load to run down locks the brake, 
which revolves freely in the hoisting direction. When 
lowering, the motor counteracts this tendency of the load 
to lock the brake and the load cannot move until its 
action on the brake is overcome by that due to the 
motor. The brake absorbs the acceleration due to 
gravity on the load, so that it drops at a speed deter- 
mined by the motor. 

Fig. 8 shows a brake of the Weston type. 5 is a 
. ratchet ring free to revolve when hoisting, but held by 
dogs or pawls from turning in the lowering direction. 
^ is a retaining ring for holding the pawl ring on the rim 
of the clutch jaw K. Inside of B, between K and the 
thrust collar H, are eight washers, foior of wh"ch, marked 

F, are of brass and are free to turn in either direction. 
The remaining four washers are steel, two, marked D, 
being keyed to the pawl ring B by the feather C, and 
two, marked E, being keyed to the collar // by the feather 

G. The clutch jaw K is keyed to the shaft P and held in 
place by the nut N. The thrust collar H is secured to 
a flange on the pinion M. A clutch jaw to mate with K 
is formed on one end of the pinion M, and its bore is 
threaded to fit the screw cut on the shaft. The enlarged 
portion of the shaft beyond the pinion is a bearing 
whose far end takes the thrust due to the screw. There 
is a bearing beyond the nut N also. 

The action of this brake is due to the downward pull 
of the load on the pinion. The shaft being stationary, 
presses the thrust collar tight against the washers, so 
that the whole brake is locked from turning in the 
lowering direction. Any motion in this direction 
causes the dogs to engage with the ratchet ring, and no further 
downward motion is possible unless the motor is started to lower. 



Upon starting the motor to lower it turns shaft P and relieves the 
pressure on the washers, and as soon as the motor overcomes this 
pressure sufficiently to permit the load to revolve the washers E, it 
will fall. The friction between the washers overcomes any tendency 
of the load to accelerate. The chamber in which the washers are 
enclosed must be kept flooded with oil, but, nevertheless, considerable 
heat is developed. In hoisting the disks are clamped by the screw 
and the whole brake revolves. 

For brass and steel washers the pressure between the surfaces 
should not exceed 100 lbs. per sq. in. 

When designing a brake of this character (G. F. Dodge, Ajhct. 
Mack., Oct. 29, 1903) the maximum load is known from the capacity 
of the crane. Assuming some reasonable values for the outer and 
inner radii of the disks, within the limits of clearance we have at our 
command, we find the average radius of the disks. Dividing the 
moment of the load by this radius, we obtain the pull in pounds that 
the disks must resist at this radius and if we divide this by N times 
the coefficient of friction (which for lubricated surfaces may be taken 
at .05), we obtain the axial pressure necessary to hold the load, iV 
representing the number of rubbing surfaces and being assumed 
such that the pressure per square inch of disk is within reasonable 
limits. The axial pressure having been determined, it is then left 
to find the angle of the helical cam such that it will produce a little 
more than this pressure under the influence of the load upon the pin- 
on, the excess being but a trifle more than sufficient to offset the fric- 
tion between the helical cams. Too small a pitch simply adds to 
the work done in lowering and a consequent generation of heat. 

The Prony Brake 

A construction of Prony brake (due to Professor Sweet) which has 
come into large use, is shown in Figs. 9 and 10. The brake drum is 
provided with internal flanges about 2 ins. high, forming an annular 




Fig. 8. — The Weston multiple-washer load brake. 

trough for the water which absorbs the heat. Supply and waste 
pipes are provided as shown in Fig. 9, the latter having its end 



BRAKES 



179 



flattened to act as a scoop. When in action the centrifugal force 
causes the water to revolve with the drum. 

The location of the tension screw and of the gap in the brake band 
should be at the bottom as in Fig. lo, and not at the top as in Fig. 9. 
(E. J. Armstrong, Chf. Engr., Ball Engine Co., Amer. Mach., Aug. 19, 
1900.) Thus located, it is subjected to the initial wrapping tension 
only, and is free from the irregular gripping action. This makes it 
feasible to introduce a spring as shown, which, in turn, accomplishes 
the purpose of the more complex compensating devices of which 
many have been made. Mr. Armstrong, who has had much experi- 
ence with Prony brakes, finds no difiiculty in maintaining loads of 
200 h.p. for indefinite periods and with a greatly improved degree of 
steadiness. 

The area of the brake surface of Prony brakes may be deduced from 
the experiences of Mr. Armstrong and of the Union Gas Engine Co., 
which latter company also has a com- 
plete outfit of brakes for testing its en- 
gines {Amer. Mach., July 27, 1905). In 
both cases the brakes are of the Sweet 
pattern, Figs. 9 and 10, and the brake 
blocks are of maple. 

One of the Union Gas Engine Co's. 
brakes having a brake drum 30 ins. 
diameter by 20 ins. face has been found 
capable of absorbing continuously 140 
h.p. at 350 r.p.m., whUe, under con- 
tinuous work, it took fire when loaded 
with 150 h.p. 

Mr. Armstrong has a brake with a 
drum 48 ins. diameter by 16 ins. face, 
which absorbs 200 h.p. "without very 
much trouble from the blocks catching 
fire. At 225 h.p. it takes fire every minute or two and 260 h.p. is 
the absolute limit with one man handling a garden hose and devoting 
himself to putting out the fires." The blocks are kept well greased 
with fallow. 

E. H. Waring has pointed out {Amer. Mach., Nov. 30, 1905) that the 
ultimate capacity of the Prony brake is measured by the capacity 
of the water to absorb the heat, which is measured by the drum 
surface alone. When operating below the ultimate capacity, the 
brake absorbs power in proportion to its speed but, as an 
increase of speed does not increase the surface in contact with 
the water, such increase, after the capacity is reached, while 
adding to the work put into the brake, does not add to its capacity 
to absorb it. 

With Mr. Armstrong's brake 200 h.p. are obviously about equiva- 
lent (perhaps a little more than equivalent) to 150 h.p. with the Union 
Gas Engine Co's. brake. The Armstrong brake has a total drum 

48X16X3-1416 x,-u A- -A A u 

= 16.755 sq. ft. which, divided by 200, 



Bass, beech, poplar, and maple are found more satisfactory for 
brake blocks than harder woods. 

The Rope Brake 

The rope brake has found much favor of late years for small and 
medium capacities. It is very flexible as regards capacity and is 
practically free from chattering. 

For the area of the drum surface in relation to capacity, the author 
has no special data. The limiting figures already given for the 
Prony brake (.09 sq. ft. per h.p.) will serve as a guide, remembering 
that, as ropes naturally take fire more readily than wood blocks, 
some increase in the surface provided is advisable. 

Additional data for the design of rope brakes are given by Prof. 
J.,C. Small WOOD {Power, May 28, 191 2) as follows: 




Fig. 9. 

Figs. 9 and ic- 



B.h.p.=- 



surface of 



144 

gives .0838 sq. ft. per h.p. Similarly the Union Gas Engine Co's. 
brake has a drum surface of — — = i?.o8 sq. ft. which, 

144 OH 

divided by 150, gives .0872 sq. ft. per h.p. From these data we may 
fairly conclude that .09 sq. ft. per h.p. marks the limit where firing 
is imminent, and this value is further fortified by Mr. Waring's 
value of .1. 

Mr. Armstrong considers that the amount of block surface exerts 
an influence. His brake has 25 3Xi6-in. blocks — covering almost 
exactly one-half the circumference. This figure for the Union Gas 
Engine Co's. brake is unknown. The concordance of the figures, 
however, is a clear index of their reliability. 

For brakes without water cooling no data are available, but, 
obviously, the constant should be greatly increased. It is, in fact, 
practically impossible to absorb much power continuously without 
water cooling. 



Fig. 10. 
■Customary and improved constructions of the Prony brake. 

The simplest form of rope brake consists of a rope wrapped around 
a fly-wheel or pulley on the shaft the power of which is to be measured, 
as in Fig. 11. The ends of the rope are attached to some stationary 
apparatus through spring balances for measuring the pull which is 
created by previously tightening the rope. The difference between 
the tensions on the two ends is the net force overcome. To vary 
this force, it is necessary only to change the initial tightness of the 
rope. 

The horse-power is obtained from the formula 

2 X radiusX 3- 1416 X forceX r.p.m. 
33000 
or 

B.h.p. = .00019 X radius X r.p.m. X force 
Strictly speaking, the radius of the brake should be taken as the 
radius of the wheel plus the radius of the rope, but, in most cases, 
the radius of the wheel only is sufficiently accurate. 

It is seen that since only the difference between the rope tensions 
is needed it is not necessary to measure them separately. Separate 
measurenent, however, allows a form of brake which is easier to 
make, although it is not so convenient to use. 

The form of brake shown in Fig. 11 may be applied to a vertical 
shaft, and the rope tensions are measured by the spring balances 
SS, the turnbuckle being provided to vary the tensions. This brake 
is suitable for small torques and high rotative speeds such as yielded 
by an electric motor or a steam turbine. For large torques, at least 
one of the spring balances must be replaced by a measuring device 
having a larger capacity. 

The force on the slacker side of the rope is generally small and 
therefore a spring balance is sufficient to measure it. The use of 
two spring balances, even with small torques, is objectionable as 
they are apt to allow bodily motion of the rope. This may result in 
chattering. 

In Fig. 1 2 one of the balances and the turnbuckleare dispensed with 
by using dead weights on the rope extremity having the greater 



180 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



tension. The horse-power is varied by adding or removing these 
weights. This brake has the advantage that an increase of length 
of the rope does not affect the brake load since such an increase would 
be accompanied by a lowering of the weights only, the tensions 
remaining the same. 

The weights should be provided with a stop P, to prevent an acci- 
dentally excessive friction from raising or throwing them. The 
spring balance of Fig. 12 may be replaced by another but smaller 
set of dead weights. Then, when weight is added to one side, enough 
should also be added to the other to produce equilibrium. It is an 
awkward matter, however, to do this nicely, as unavailable sub- 
divisions of weights are at times required. 

The construction of Fig. 12 maybe modified by using a single 
heavy weight on the left side and a set of small weights on the right. 
The heavy weight rests on a platform scales so that the rope tension 
on this side is the diiierence between the platform-scale reading and 
the weight on the scales. The brake load is varied by changing the 
weights on the right side. This device has been found very satis- 



be carefully determined with the ropes detached, and this weight 
subtracted from the readings taken during operation. 

Generally, if a brake load is to be carried by an engine for any 
length of time, some provision must be made to withdraw the heat 
generated by the friction to prevent the rope charring or catching 
fire. This is usually accomplished by feeding water into the trough 
formed by internal flanges on the rim of the fly-wheel or pulley to 
which the brake is attached. Usually, another pipe with a scoop- 
like entrance is arranged to withdraw the heated water. Very often, 
however, this is unnecessary, since, if boiling is allowed, the water 
supply may be adjusted so as to equal the evaporation; that is, 
evaporation disposes of the water without allowing the rope to get 
too hot. If the fly-wheel is not too small for the power absorbed, 
this means of disposal will be found eSective. 

Air radiation sometimes provides ample cooling when the brake 
pulley is large in comparison to the power absorbed. This is particu- 
larly the case when the brake load is to be applied for a short time 
only. 




Fig. 14. 



Figs, ii to 15. — Rope brakes. 



factory in that fine regulation may be secured with very uniform 
resistance. In this case the net force overcome at the rim of the fly- 
wheel is the heavy weight minus the sum of the scale reading and 
the small weights. 

• In some cases there is not room to hang weights from the fly-wheel, 
in which case a brake of the type shown in Fig. 13 is applicable. 
The heavier tension is measured by the platform scales, being trans- 
mitted from the rope end by a lever L, having equal arms. The 
fulcrum is preferably a triangular piece« of steel in order to avoid 
friction, but may be a pin, as shown. The handwheel is used to 
make fine adjustments of the load by tightening the rope; the turn- 
buckle may be used for coarse adjustments. 

Fig. 14 shows a form of brake in which the difference of tensions 
is measured directly from a single scale reading. The ends of the rope 
are attached to the cross pieces CC of a wooden frame which rests on 
a platform scales. It is important that stops 55 be provided to 
prevent the Ufting of the framework, if it is possible for the engine 
to reverse. It should be noted that the weight of the frame should 



Some safeguard is needed generally to prevent the rope from slip-, 
ping off the pulley. The rope, like a belt, tends to run to the center 
of a crowned pulley, and where only a single or half turn is used on 
such a pulley no other provision need be made to keep it on. For 
large powers, where a number of ropes are necessary, it is well to 
provide some other means to accomplish this purpose. 

An externally flanged pulley will do this simply; if one is not avail- 
able, blocks Uke that shown in Fig. 15 may be fitted to the rope. 
This shows a block suitable to a single turn, or less, of double rope. 
Enough of these blocks should be fastened to the rope to hold it 
securely to the wheel. 

Except for light powers, it is better to use double rope, as this pro- 
vides more surface to resist wear without altering the desired relation 
of the tensions. 

It is preferable to attach the device for adjusting the rope tension 
to that end upon which the tension is smaller, namely, the end which 
points in the direction of rotation. The preference is made because, 
the force being less, it requires less effort to change it. 



BRAKES 



181 



If two spring balances are used the springs should be of different 
stiffness; otherwise their vibrations are likely to synchronize and a 
chattering will result. 

The design is usually adapted to the size of pulley or fiy-wheel at 
hand. It is first necessary to ascertain if the available pulley is 
large enough. 

To determine the size and number of ropes, there must first be found 
the net force which the brake must handle; that is, the force which 
will be indicated upon the scale, or the difference of the forces if two 
scales are used. To find this: 

Multiply the horse-power by 5250 and divide the product by the 
radius of the wheel in feet and the number of revolutions per minute. 
The final quotient will be the net tension. 

It is well here to emphasize the meanings of the terms net force 
and rope tensions. The net force is the effective force overcome by 
the engine. The rope tensions are the forces existing in the ends of 
the rope and their difference equals the net force. The tension in the 
tight end of the rope is therefore greater than the net force, and the 
rope must be strong enough to carry this tension. 

Table 3. — Ratio of Tensions in Rope Brakes. Calculated foe 
A Coefficient of Friction of . 4 



Number of turns 
rope 


Ratio of greater ten- 
sion to net force 


Ratio of greater to 
lesser tension 


1 

i 


1.40 
1. 18 


3-Si 
6.59 


I 


1.09 


12.3 


li 


I OS 


23.2 


i^ 


1.02 


43-4 



The relative values of the rope tensions depend upon the number 
of turns around the wheel and the condition of the rubbing surfaces. 
Table 3 gives quantities that will reduce the calculations for design 
in the general case. 

The data apply to well worn manila rope on smooth pulleys. This 
table gives the ratios of the brake forces for various numbers of rope 
turns. From it is seen, for instance, that with a half turn, the greater 
rope tension is 1.4 times the net force and 3.51 times the lesser tension. 
It follows that to find the greatest tension resisted by the rope: 

Multiply the net force by the figure in the second column of Table 3 , 
corresponding to the number of turns in the first column. 

Using this result, a suitable rope to carry the load may be selected 



from Table 4. This gives the working strength of good manila rope 
of three strands, a factor of safety of about five being used. The 
rope is listed according to its largest diameter. 

Table 4. — Strength of Ropes 



Diameter, ins. 


Working strength, lbs. 


1 


300 


1 


4SO 


i 


700 


i 


1000 


I 


1300 


li 


1700 


i| 


2100 



In selecting the rope provision should be made for the weakening 
effect of wear. 

Coefficients of Friction 

The most recent determination of the coefficients of friction for the sub- 
stances commonly used in brakes are those found by the tests of the 
National Brake and Clutch Company as follows: 



Materials 



Coefl&cient 



on dry metal. 



on oily metal. 



Metal and cork 
Leather and cork 
Fiber and cork 
Metal and cork 
Leather and cork 
Fiber and cork 

Fiber on dry metal 

Fiber on oily metal 

Leather on dry metal 

Leather on oily metal 

Charred leather on oily metal . 

Metal on dry metal 

Metal on oily metal 



•35 



•32 

.27 
. 10 
•23. 
■15 
.08 

■IS 
.07 



The coefficient will vary with the condition of the contacting sur- 
faces. Smooth and unyielding surfaces offer less resistance than 
rough and yielding ones. In metal to metal contacts different metals 
are usually employed for the opposing surfaces, as bronze and steel 
in plate clutches and cast iron and steel in brakes of the shoe type. 



FRICTION CLUTCHES 



Every small and medium-sized planer is an illustration of the 
perfection of action of a shifting belt acting as a friction clutch when 
properly proportioned and under loads that are not too great. The 
shifting belt, when applied to lathe and other machine-tool counter- 
shafts, is not satisfactory because its speed is too low, leading to the 
loss of that smartness and promptness of action characteristic of 
planer belts. Were counter-shaft belts driven at the speeds of planer 
belts, their action would be just as satisfactory. 

The most satisfactory analysis of friction clutches known to the 
author is that by John Edgar (Amer. Mach., June 29, 1905). Mr. 
Edgar takes as his design constant the product of the coeflScient of 
friction and the unit pressure between the surfaces and thereby 
eliminates preliminary assumptions of that most uncertain factor, 
the value of the coeiScient of friction. His constant is, of course, 
equal to the unit tractive force of the surfaces, this way of looking 
at it giving a more tangible idea of its meaning. The analysis was 
originally offered for expanding ring clutches in which an internal 
split metal ring is expanded against the interior surface of a surround- 



The example is for h.p. = 40, Edgar's constant =50; r.p.m. = 100 
giving, for diameter=ioins., breadth = 3. 23 ins. or, for diameter =20 
ins., breadth = .8 in. 

The values of Edgar's constant given on the chart have been ob- 
tained as follows : 

For expanding ring clutches, metal on metal, Mr. Edgar compared 
actual clutches with the formula and found the value of C to range 
between 50 and 100. Table i of dimensions of actual clutches of 
the same type is supplied by C. L. Utcher {Amer. Mach., June 24, 
1909), column II giving Mr Edgar's constant, having been added 
by the author. In all of Mr. Utcher's cases the expanding rings are 
of cast-iron while the rings into which they expand are of cast-iron 
or low carbon steel (about 35 points carbon). Mr. Utcher says 
that "in case 9 clutch fails on very heavy cuts which are quite within 
the capacity of the machine otherwise" and for this reason the 
average has also been given with this clutch omitted. The average 
value thus obtained agrees quite closely with Mr Edgar's lower 
value, while the other cases, excluding 9, indicate that his higher value 



Table i. — Dimensions of Expanding Ring Clutches 



Case 


H. p. of 
belt 


Maxi- 
mum h. p. 
of cut 


R. p. m. 

of 
of clutch 


Diameter 

of clutch, 

ins. 


Width of 

clutch, 

ins. 


Surface 

of 
clutch, 
sq. ins. 


Surface 

speed of 

clutch, ft. per 

min. 


Tangential 

force of 

resistance 

R 


Radial force 

to produce R 

F=jR 


Pressure on 
surfaces of 
clutch, lbs. 
persq. in. 


Edgar's con- 
stant from 
column 2 


Columns 


I 


2 


3 


4 


S 


6 


7 


8 


9 


10 


II 


I 


8 


4 


600 


3 


I 


95 


430 


310 


3.100 


325 


32. S 


2 


8 


4 


300 


5 


I 


16 


390 


335 


3,350 


210 


21 


3 


16 


8 


400 


6 


li 


24 


630 


420 


4,200 


180 


18 


4 


24 


12 


400 


7 


1} 


28 


730 


540 


5,400 


200 


20 


S 


8 


4 


30 


8i 


Ij 


40 


67 


1,970 


19,700 


495 


49.5 


6 


8 


4 


30 


9 


li 


35 


71 


1,860 


18,600 


525 


52.5 


7 


16 


8 


25 


10 


li 


47 


65 


4,000 


40,000 


850 


85 


8 


24 


12 


20 


12 


I5 


57 


63 


6,300 


63,000 


1,115 


III 


9 




1\ 


6 


16 


li 


63 


25 


9,900 


99,000 


1,570 


157 






Average of all 












1 








61 


Average omit- 
ting 9. 




















48.7 




1 

















ing metal drum, but it is applicable to nearly all types, materials and 
duties, provided the constant is obtained from successful clutches of 
the type and subject to the duty in question. Mr. Edgar's formula is: 

^(i^SX r.p.m. 



h.p. = C- 



in which 



40120 



C = Edgar's constant = coefficient of friction X radial pres- 
sure, lbs. per sq. in. = tractive force of friction surfaces, 
lbs. per sq. in., 

rf = diameter of friction surfaces, ins., 

h = width of friction surfaces, ins. 

This formula may be solved for several types of clutches by Fig. i 
by Prof. J. B. Peddle {Amer. Mach., Aug. 8, 1912) the use of 
which is as follows: 

Join the given horse power with the suitable value of Edgar's con- 
stant and note the intersection with axis I; join this intersection with 
the given r.p.m. and note the intersection with axis II. Any values 
of diameter and breadth lying in a line passing through the intersec- 
tion on axis II will transmit the given horse power at the given speed. 

182 



is admissible, especially as Mr. Utcher says, "from careful observa- 
tion of the condition of the clutches after several years' running, I can 
assert that, without doubt, any of the clutches is capable of trans- 
mitting the full power, as given in column 2, for an indefinitely long 
period of time." 

In Mr. Utcher's clutches the surface of the rings was interrupted 
by grooves about a quarter of an inch in width cut transversely in 
order to permit the lubricant to escape. Experiment shows, he says, 
that this practice increases the driving power of the clutch about 20 
per cent, for the same applied pressure. C. W. Hunt {Trans. A. S. 
M. E., Vol. 30) cut such grooves \ in. wide by -^ in. deep in increasing 
numbers and found a progressive improvement in the prompt engage- 
ment of the clutch until the grooves were spaced a little more than 
I in. apart. 

The diameter of clutches of this type should be large in proportion 
to the width, and the expansion ring should be stiff enough to prevent 
its expansion by centrifugal force. The pressure shotdd be applied 
by some form of toggle or bell crank mechanism by which the pressure 
increases rapidly at the end of the movement. Thus equipped, 
Mr. Utcher says that in his clutches "in no case is the span of move- 



FRICTION CLUTCHES 



183 



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184 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



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FRICTION CLUTCHES 



185 



I 



ment more than 15 ins., nor the pressure needed more than can be 
comfortably applied by one hand." 

Within the limits of uncertainty of the value of the coefficient of 
friction, the value of the tangential force required to push apart 
the ends of the expanding ring may be obtained by the relation that 
exists between radial and tangential forces, as in steam boilers, for 
example. Thus we have 

• . ,. Cdb 

separating force, lbs. >= t' 

f being the coefficient of friction and the remaining notation as before. 
For the imperfectly lubricated metal siurfaces used in clutches of 
this type, / may be taken at .1. 

For clutches with jaws of wood working with drums of cast-iron and 
intended for occasional engagement (line shaft and similar service) 
Henry Souther (Trans. A. S. M. E., Vol. 30) gives dimensions of 
four clutches by the Dodge Mfg. Co. of powers ranging between 
25 and 98 h.p. at 100 r.p.m. The wood blocks were of maple and, 
with the dimensions substituted in Mr. Edgar's formula, the clutches 
yield values of C of 16.4, 15.9, 14. i, and 16.9 respectively, or an aver- 
age value of 15.8, for which we may use 16. 

When the wood blocks do not embrace the entire circumfer^ence, 
suitable correction must be made for the value of h when substituting 
in the formula or when using the chart. Thus, if the blocks embrace 
one-half the circumference, the value to be used for h is one-half the 
actual width of the blocks, and so for other fractions of the circum- 
ference embraced by the blocks. 

For the cone clutch it should be remembered when applying the 
formula, that the pressure factor in C is the normal pressure per sq. 
in., and that, for d, the mean friction diameter is to be used. The 
same values of C are applicable for the same materials and services. 

For iron on iron surfaces, values of C have been given and those 
for other surfaces follow and have been incorporated in Professor 
Peddle's chart. 

For cone clutches having wood on iron surfaces and used under the 
conditions of frequent service, two hoisting clutches by the Lidger- 
wood Mfg. Co. were examined by the author and gave values for 
C of 9.1 and 10.4 respectively, of which the mean is 9.75 or, say, 10. 
For leather faced cone clinches with the opposite engaging surface 
of metal and used under the conditions of frequent service, a hoisting 
clutch by the C. W. Hunt Co. was examined by the author and, 
under successful operating conditions, gave, for C, a value of 14. 
On one occasion this size of clutch was overloaded and the leather 
facing failed. The value of C for this condition works out at 20 ^, 
indicating that, for the materials used, 14 is a safe value. Mineral 
tanned leather is used by the C. W. Hunt Co. for clutch facings, 
and is found to be more serviceable than oak tanned leather. For 
the latter material a smaller value would seem appropriate and, in 
the absence of other data, we may, for it, take 12 as a safe value. 

For leather faced cone clutches with the opposite engaging surface 
of metal, imder the conditions of automobile service, six automobile 
clutches of medium and moderate powers (4 cylinders, of which the 
largest was sXSxi ins.) were examined by the author, calculated not 
jated horse-powers being used. The resulting values of C were 2.01, 
3.32, 2.83, 2.4, 2.85, and 2.26, the average being 2.61 or say 2.5. 

For the axial pressure required to engage cone clutches the author 
has shown {Amer. Mack., Aug. 8, 1912) that 

Cjrdb sin (f) ^ 



axial pres. = 



/ 



4> being the angle, degrees, between the axis and the conical surface, 
and this formula is true for any material and any service, suitable 

1 The author is convinced that the formula in which a factor, the co- 
efficient of friction times the cosine of the angle, is introduced to provide 
(oi overcoming the endwise sliding friction of the cones on each other, is 
erroneous. It is a matter of common experience that a shaft, when in mo- 
tion in its bearings, may be traversed endwise by a force so small as to be 
negligible, and it would seem that the same action takes place in a clutch. 



values of C and / being used. The following values of / are fair 
average values. Iron on iron dry .2; iron on iron imperfectly lubri- 
cated .1; wood on iron, .2; leather on iron, .3; cork inserts on iron, 33. 

Professor Peddle's second chart, Fig. 2, may be used in place of 
this formula. 

The values of the axial pressure obtained from the formula or 
chart being those which wiU just drive, some surplus should be added. 

The angle of the cone is of importance as regards freedom of dis- 
engagement. For metal on metal surfaces, the dividing angle 
between sticking and non-sticking is about 6 or 7 deg. measured 
between the axis and the conical surface, according to A. J. Shaw 
(Amer. Mach., June 11, 1887). For free disengagement, the angle 
should be not less than 10 deg. For wood on metal surfaces the 
angle should not be less than 20 deg. (C. W. Hunt, Trans., A.S.M. 
E., Vol. 30). A common angle for the leather and metal surfaces of 
automobile clutches is 12 J deg., but such clutches are always 
held in engagement by a spring and disengaged by foot pressure. 
According to C. W. Hunt (Trans. A. S. M. E., Vol. 30), leather faced 
cone clutches with angles of 18 to 20 deg. are fitted with an operating 
device for disengagement. 





Fig. 3. — Incorrect construction 
of cone clutch. 



Fig. 4. — Correct construction 
of cone clutch. 



A defective construction of cone clutch is illustrated in Fig. 3 which 
shows the effect of wear. The correct construction is shown in Fig. 
4. The extension of the male and female ends are at an equal angle 
from the acting svuface, the result being an avoidance of the shoulder 
and an increase of surface as the parts wear. 

Cone clutches should have holes provided for the escape of air 
from between the cones. 

The Rockwood Mfg. Co. employ tarred fiber and cast iron. For 
comparatively frequent starting and stopping, they employ a normal 
pressure of 100 lbs. per square inch of surface in order to limit the 
generation of heat, while for occasional stopping and starting they 
employ 200 lbs., such values being practicable because the pressure 
is distributed over the entire surface. The minimum safe value of 
the angle of the faces with the shaft in order to insure free disengage- 
ment is 8 deg. If increased end thrust pressures are not objectionable, 
slightly larger angles, which lead to reduced wear, may be used. The 
value of the coefficient of friction for these materials is .125 giving 
values of 12.5 and 25 for Edgar's constant C, for 100 and 200 lbs. 
normal pressure respectively. The Rockwood Mfg. Co.'s formulas 
for the starting capacity of these clutches having an angle of 8 deg. 
are as follows: 

For 100 lbs. per sq. in. normal pressure: 

h.p. = .000312 D'^WN 

For 200 lbs. per sq. in. normal pressures: 

h.p. = .000624 DWN 

in which, D = diameter, ins. 
W = face width, ins. 
N = r.p.m. 



186 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



The end thrust required to start loads is: 

For loo lbs. per sq. in. normal pressure 

P = 43.722DW 

For 200 lbs. per sq. in. normal pressure 

P = 87.444DTF 

in which P = end thrust, lbs. 

For Weston {multiple disk) automobile clutches running in oil, four 
clutches were examined by the author, three having six and one four 
cylinders, without finding any difference of practice characteristic 
of the number of cylinders used. The highest powered car examined 
had six 4f Xs? in. cylinders. The clutches had from 31 to 51 
disks, of outside diameters ranging between 8 and X2 ins., with face 
widths of I to I in. 

In this type of clutch the mean diameter of the friction surfaces 
is to be treated as d in the formula, while for h the actual radial width 
multiplied by the number of rubbing contacts, that is, the number 
of disks — not twice the number of disks — is to be used. The width 
given by the chart is this product, which is to be divided by the 
number of rubbing contacts to obtain the width of the disks. Should 
large clutches run beyond the scale of the chart, the horse-power 
may be divided by 2 and the resulting number of rubbing contacts 
be multiplied by 2. As these clutches run in a bath of oil a low 
value of C is to be expected. 

The resulting values of C were .888, 1.09, .565, and .565, the aver- 
age being .777 or, say, .75. The highest value was found for the 
highest powered car and one of the lowest values for the lowest 
powered car, although the second lowest value was found for the 
next to the highest powered car. 

Multiple disk clutches should have narrow rubbing surfaces — 
preferably not over -^ the diameter, though this ratio is, in some 
clutches, as low as ^V- With wide surfaces the unsatisfactory 
action of step bearings is introduced. Moreover, with wide sur- 
faces, the bite of the central feather and the uncertainty of disen- 
gagement are increased. For both reasons the results are more 
satisfactory if the required siurface is obtained by increasing the 
number of disks instead of the width of the surfaces. 

For midtiple disk dry plate clutches as used in automobile service, 
F. M. Heldt examined six examples, the resulting mean valve of C 
being 3. The materials used in this construction, which is displacing 
the lubricated type of clutch, are asbestos fabric against steel. From 
8 to 20 friction surfaces are employed. 

The axial thrust of automobile clutches is limited to a low valve 
by the fact that it must be released by the pressure of the foot and 
without undue effort. This, in turn, places a low valve on C. So 
far as the friction surfaces are concerned, there is no reason to suppose 
that the valves of C might not equal those used with the same mate- 
rials in hoisting engine service. 

The Lane friction clutch, which is in wide use in the United States 
on mine hoists employs the principle of wrapping friction. The 
strap, however, goes but once (effectively a little less than once) 
around the drum and is lined with wood blocks. The relation of 
the tensions on the two ends of the strap may be obtained from Mr. 
Brown's chart, Fig. 4, of the section on brakes, which see, using a 
coefficient of friction of .2 for wood on iron. 

The contracting band clutch used on some automobiles is subject 
to the same analysis as the Lane clutch, from which it differs only 
in materials and dimensions, provided the coeflScient of friction be 
known. 

The Ball Friction Clutch or Ratchet 

The design principle of this device is thus explained by E. H. Fish 
{Amer. Mach., Sept. 21, 1911). Due probably to a misunderstand- 
ing of its action this clutch has not risen in favor as much as it de- 
serves. Fig. 5 represents in outline the essentials of such a ratchet. 
The balls or rollers a a a a are placed between a ratchet wheel b and 



a smooth cylindrical shell c. In the position shown in Fig. 5 the 
right-hand ball drops down into the space below it by its own weight. 
This one ball is the one that will start the ratchet working. Power 
being applied to b with motion in the direction of the arrow, this ball 
acts as a wedge between c and b and starts the shell rotating. If the 
resistance is too great the ball will roll between b and c untU it is 
squeezed sufficiently tight so that it drives or something breaks. 




Fig. 5. Fig. 6. 

Figs, s and 6. — Ball friction clutch or ratchet. 

If the angle a, Fig. 6, between the tangents to the curves of ratchet 
and shell at points of ball or roller contact is less than, twice the angle 
of repose, the clutch will either drive or break. That this last is true 
can be seen from Fig. 6, where a baU is held by friction only between 
two surfaces. The forces acting on the ball are: First, its weight, 
which may be neglected as it is very small and is inactive during part 
of the revolution; second, PP the normal or squeezing forces and fP 
— fP is the force of friction incident to the normal forces. It will be 
seen that these are the forces which act while the clutch is stationary. 
In motion they change, fP on the ratchet side remaining a propelling 
force while fP on the side of the shell is counteracted by the tendency 
of the ball to drive the shell. That is, the driving force on the shell 
is at all times sufficiently less than fP so that the ball wUl roU no 
further into the opening. Angle /3 in Fig. 6, is evidently equal to the 
angle whose tangent is / (the coefficient of friction) and is also evi- 
dently equal to J^ a; therefore, a must be made slightly less than 
twice the angle of friction. 

On the other hand a ratchet which does not let go on the back 
stroke is often of little use as a ratchet, which in turn indicates that 
if a is much less than twice the angle of friction it will not let go. If 
a is greater than this the weight of the ball or roller is the controlling 
influence that may cause it to work or fail. This weight, as we hav.e 
seen, is too small to be depended upon. The angle of friction varies 
with the lubrication and may easily be two or three times as great 
with one oil as with another, to say nothing of still greater variation 
in case of no oil at all. Of course, if there is a considerable resistance 
to backward motion of the shell, the ratchet can be forced to let go. 
That is the usual condition but a condition that militates against 
its use in many places; for example, in hand forges where the resist- 
ance to turning backward is so little that the clutch is very apt to 
lock unless an artificial friction is put on the impeller shaft. For 
such situations the clutch is not suited, but there is an abundance 
of places where there is sufficient natural resistance to backward 
motion so that it can be used. 

The theory of its design is very simple. The driving force is any- 



FRICTION CLUTCHES 



187 



thing up to fP at the circumference of the shell where/ may be taken 
as .03 to .05 according to conditions, material and lubrication. Using 
the lower value we find that P is ssJ'^ times the pulling force. For 
example, if the pulling force is to be 100 lbs. at the inside of the shell, 
P will be 3333 lbs. This is the crushing force on the ball. This 
estimate should be increased by a liberal factor of safety in selecting 
balls, remembering that only one ball at a time can be relied upon to 
do the work. 

Claw clutches are more cheaply made if they have an odd number 
of teeth (Professor Sweet, Amer. Mack., Mar. 17, 1910). To mill a 
clutch with an even number of teeth it is necessary to set the mill- 
ing machine twice: first to cut one side of the teeth and then the 
other; but to cut an odd number of teeth it is necessary to have 
only a plain mill, thick enough so that twice through will cut the 
wide part of the gap, and thin enough so that it will pass through 
the small part. This will be best understood by referring to Figs. 
7, 8, 9 and 10. 

Fig. 7 shows a finished three-tooth clutch; Fig. 8 a single cut with 



a proper thickness milling cutter set with one side exactly central; 
Fig. 9 the second cut, which finishes one tooth; and Fig. 10, the third 
cut, which finishes the other two teeth, completing the job. 

A clutch with any odd number of teeth can be finished in the same 
way, and if one side of the cutter is exactly central the clutch will 
be a mechanical fit. 




Fig. 7. Fig. 8. Fig. 9. Fig. 10. 

Figs. 7 to 10. — Milling a claw clutch with an odd number of teeth. 



CAMS 



The following methods for laying out cams, except when other- 
wise specified, are those of C. F. Smith {Amer. Mack, 1905) 

The usual motion given by a cam, when the cam roller is mounted 
on a radius arm, is that given by a crank and connecting rod to a 
cross head, the length of the radius arm being the equivalent of 
that of the connecting rod When the roller is mounted on a slide 
the motion becomes that of a crank and slotted cross head (Scotch 
yoke) . 

For high speed cams this motion requires modification as will be 
explained later. 

The use of a spring for effecting the motion in one direction is 
sometimes desirable. In some cases the spring performs the return, 
while in others it performs the operating movement. When a spring 
performs the return movement, the failure of the spring to act leaves 
its driven part in the operating position and a wreck may be the 
result, while, if the spring performs the operating movement, a failure 
to act leaves the driven part withdrawn from the operating position 
and serious consequences are less probable. A conspicuous illus- 
tration of springs performing the operating movement is found in 
the linotype, adoption of the plan being due to considerations of 
safety. 

An objectionable feature of springs, especially for the return move- 
ment, is that the efiort of extending or compressing them is added 
to the effort required to do the work. Except that the milling 
cutter must be kept to size, there is no greater difficulty in making 
a cam effect both movements than one. 

Omitting consideration of unusual cases, cams are of two types 
drum or barrel and face or radial, of which the former have the 
advantage that they are of smaller diameter for a given angle of 
cam groove, and the latter that their action on the roUer is more 
perfect. 



Laying Out Drum Cams 

To lay out a drum cam operating a roller mounted on a slide, pro- 
ceed as in Fig. i, in which the rectangle o, B, C, 12 represents the 
development of that part of the cam's periphery in which the move- 
ment is to take place. The movement of the roller is shown full 
size by the line o B, and this movement is to be performed while 
the cam turns through an arc of which BC is the development. 
Upon A o equal and parallel to 5 o, a semicircle, called the throw- 
circle, is drawn equal in diameter to the movement of the roller and 
this semicircle is divided into any number of equal parts — in this 
case 1 2. The developed arc BC is then divided into the same number 
of equal parts and projecting lines from the points on the semicircle 
give by their intersections with the corresponding verticals, points 
on the center line of the cam groove as indicated by the small circles. 
While the cam turns from 5 to C the movement of the roller is pre- 
cisely the same as would be given by a crank turning through the 
semicircle o, 6, 12. The return movement may or may not be per- 
formed in the same interval of time, but its groove wiU be laid out 
in the same way. Should the movements be performed in the same 
time the movement of the roller is precisely the same as that of a 
cross head, but with a pause at each end of the stroke, and, should 
they be performed in different times, the same will be true except 
that one movement will be made more quickly than the other. 

The angle of the tangent with the center line of the groove should 



not exceed about 30 degrees, as indicated in Fig. i.^ In laying out 
the drawing the distances oB and BC are given as has been stated, 
and it is desirable to avoid the necessity of laying out the curve in 
order to determine this angle, and this determination may be made 
in several ways. Thus with the angle equal to 30 degrees, there is 
a constant ratio of 2.72 between the length of the lines Bo and BC. 
Bo being given by the conditions of the problem, it is only neces- 
sary to multiply it by 2.72 in order to obtain the length which BC 
must have in order to make the final angle 30 degrees. The length 
of BC, considered as a fraction of the entire circumference, is also 
known from the conditions, and from this the entire circumference 
and the diameter of the cam may be quickly determined. Thus, BC 
being 2.72 times the throw and occupying say n degrees of the cir- 
cumference, we have: 

360 
arcumf erence = 2 . 7 2 X throw X 



diameter = 



2. 72 X throw X360 

3.1416XW 
312 X throw 



approximately. 



The length of BC may also be determined with a protractor 
from the fact that, with the angle of the tangent to the cam curve 
equal to 30 deg., the angle Co 12 will be equal to 20 deg. 12 min. 
or, for practical purposes, 20 deg. This ratio of 2.72, and this 
angle of 20 degrees, are strictly correct for drum cams, of which the 
rollers move in straight lines only, but they may be used for cams 
of which the rollers are guided by radius arms without important 
error. For face cams these constants are modified, as will be ex- 
plained later. 

It will be observed also that near the point of tangency the curve 
and tangent coincide very closel}^ and we may take advantage of 
this to obtain a graphical method which is accurate enough. Having 
divided the end circle, lay out the triangle abc, the height being 
projected from the throw circle and the hypothenuse being drawn 
at 30 degrees, when the base gives the length of one of the divisions 
of the base line. The same result may, of course, be obtained from 
the triangle cde. The chief use of the constants is in the prelim- 
inary layout of the chart, as will be explained later. When laying 
out the curves on the individual cam drawings the graphical method 
is preferable. 

If the roller is supported, by a radius arm, as it usually is, the pro- 
cedure is slightly modified as shown in Fig. 2. The base line and 
throw circle are divided as before, but instead of drawing perpen- 
diculars from the base line divisions, arcs of circles are struck through 
them with a radius equal to that of the proposed radius arm, as in- 
dicated by the line ab and centers c, d, e, etc., and the intersections 
of these arcs with the projection lines from the divisions of the throw 
circle form the locating points for the center line of the cam groove 
as indicated by the small circles. The angle of this line at its middle 
should, as in the former case, be not greater than about 30 degrees, 
as shown, and is predetermined as in Fig. i. 

Laying Out Face Cams 

When the groove is upon the side or face of the cam disk the pro- 
cedure is shown in Figs. 3 and 4. Fig. 3 corresponds with Fig. i 

' Some designers increase this figure, going even to 60 degrees for light 
work. The monotype — a conspicuous example of high class cam construc- 
tion — ernploys angles as high as 37 degrees. 



188 



CAMS 



189 




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it 


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in that the roller is mounted upon a slide which moves 
radially with the cam disk. The cam is here required 
to move the roller the distance oB while turning 
through the angle BC. The throw circle o, 6, 12 is 
drawn with a diameter equal to the radial movement 
of the roller and is divided as before into equal parts. 
The cam angle is similarly divided and radius lines 
o, I, 2, 3, etc., are drawn. The division points of 
the circle are projected to its diameter and arcs 
struck from the center of the cam and from the feet 
of the projection lines give by their intersection with 
the radial lines the points of the center line of the 
cam groove; the limiting angle appearing as before. 
In determining this angle for this style of cam the 
lower triangle abc should be used in preference to 
the upper one cde, as the tangency is closer below 
the middle point than above it. As with the drum 
cam this angle is regulated by varying the diameter 
of the cam disk. 

In laying down this style of cam upon the chart 
the figure B, o, 12, C must, of course, be represented 
by a rectangle of which the length is properly made 
equal to the outer arc BC. The constants which 
have been given for drum cams become for this 
construction 3.23 and 17 deg. 15 min., or, for 
practical purposes, 17 deg.; that is, the length of 
the rectangle should be at least 3.23 times its height 
and the angle of its diagonal with its base should not 
exceed 17 deg., showing that face cams must be 
larger than the drum style in order to have an 
equally favorable angle. 

WhEe the constants given above apply to face 
cams, the extreme radius of a face cam groove is not 
much less than the radius of the piece^ there being but 
a wall of metal and the radius of the roller between, 
and the convenience of a uniform outside diameter 
of cams is so great that this may be neglected, the 
curves of the chart being laid out as though the 
cams were all to be of the drum style and then 
make both face and drum cams of the same outside 
diameter. The face cam grooves will thus be slightly 
steeper than if they were of the drum style, but in 
only a few, and probably in no case, will the angle 
extend 30 deg. Were this angle to be materially 
exceeded in one of the face cams, especially in an 
important one doing heavy work, the whole set should 
be redesigned and enlarged 

In Fig. 4 the roller is carried by a radius arm of a 
length ab. An arc cd is so struck as to pass through 
the center of the cam shaft if possible. Sometimes 
this is impossible, but the practice should be departed 
from only in case of necessity. The center a located, 
the arc ef is struck from the center of the cam disk and 
from centers located on it and with a radius equal to 
the length of the arm the arcs Bo and C12 are struck, 
such that the arc ij is the angle of cam movement 
during which the roller movement is to take place. 
The arc ef between the extreme centers gh of the 
arcs Bo and C12 is then divided as in previous cases 
and arcs o, i, 2, 3, etc., are drawn from the division 
points as centers. The throw circle o, 6, 12 is then 
drawn and the cam curve is quickly located. The 
limiting angle is again shown. 

Two-step Cams 

Face cams giving a movement in two steps are laid 



190 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



out as in Fig, 5. The driven crank arm is centered at 
A, its extreme positions being B and C with an inter- 
mediate dwell at D. The first movement is from B to 
D and the second from Z> to C. The cam roller is 
carried by an arm pivoted to the end of the crank arm 
and having a forked end which straddles a square guide 
block which rides on the cam shaft. 

The extreme throw is laid out on the horizontal cen- 
ter line and the lines B and C are drawn from A to give 
the same movement on each side of the vertical center 
line. The arc EF is drawn and the position D of the 
arm at the intermediate dwell is laid down. Chords GH 
are drawn and on them throw circles are drawn each of 
which is divided into parts as before, the division points 
being projected to the arc EF, The extreme positions 
of the arm which carries the roller are drawn in at I and 
/. The extreme positions of the roller are at K and L, 
and an arc KL tangent to / will give approximately the 
path of the roller which may thus be treated as though 
pivoted at the center M of the arc KL, which does not 
conform to the recommendation that it should pass 
through the center of the cam shaft, as it is impossible 
to make it so conform. With radius FK and centers 
at the points of the arc EF projected from the division 
points of the throw circle on chord G, the positions of 
the circles o, i, 2, 3, etc., from the cam-shaft center are 
obtained and they are drawn. Points P, Q are found 
on the arc MN from which to strike arcs o^ and iz^ to 
include the angle of movement of the cam within which 
the first movement of the roller is to take place, and the 
arc PQ is then divided as before and arcs i^, 2^, 2at etc., 
are drawn, the intersections of which with the corre- 
spondingly numbered circles drawn from the cam-shaft 
center give the outUne of the center line of the cam for the first 
movement. The required length of dwell RS, for which the cam 
curve is of course a circle, is then laid out and the profile for the 
second movement is then determined in the same way from the 
throw circle drawn on chord H, but of this details need not be given. 
The limiting angle of 30 deg. appears in both cases. 

The return movement, not shown, is laid out from a third throw 
circle of which the diameter is equal to the sum of those already used. 

The above methods are sufficient to lay out any single-roller drum 
or disk cam of the types illustrated of which the time and extent of 
the movement only are determined beforehand. 

An Example of a Face Cam 

The layout of an actual face cam is shown in Fig. 6, a portion of 
the machine frame being also shown. The cam is to move a slide 
a by means of a fulcrumed lever b, the position of the roller center 
being at c on an arc which is laid out to pass through the center of 
the cam shaft, as akeady advised in connection with Fig. 4. 

The diameter of the cam disk being known or assumed, we draw 
the outer circle of the cam and lay down the thickness of the outer 
retaining wall, and the radius of the roller, and draw the arc no. 
The radial movement of the roller x is laid down as shown giving 
the arc pq as the inner limit of movement of the roller center. 
The throw circle is next drawn in and divided as has been explained 
and arcs from the feet of the perpendiculars from the division points 
are drawn. The arc through the center oi the throw circle gives, 
by its intersection with the arc struck from the' fulcrum h of the 
lever as a center, the location of the mid-position c of the roller. 
In the present case this happens also to be the middle of the return 
curve of the cam, but this is accidental. 

Assuming or knowing that the working movement of the roller 
must be made during an angle oc of movement of the cam, we draw 




an arc de from the center of the cam shaft as a center and passing 
through the center h of the lever fulcrum, and take its radius hi 
in the tram and find centers/, g from which to strike arcs jk and 
Im passing through the cam shaft center and spanning the angle a 
as shown. These arcs, by their intersections with no and pq, 
determine the positions of the cam curve for beginning and ending 



CAMS 



191 



the movement. Dividing the arc /g as before, we find points i, 
2, 3, etc., from which to strike arcs i, 2, 3, etc., which, ''by their in- 
tersections with the arcs from the feet of the perpendiculars through 
the divisions of the throw circle, give the successive positions of the 
roller center. From these centers circles having a radius equal to 
that of the roller define the cam groove. The return movement is 
laid out in the same way and from the same stroke circle, and is thus 
the same as the acting movement, although the two cam curves are 
very different. Those portions of the groove which represent dwells 



The surface of the zinc may he blackened for the purpose (Wm. 
V.Lowe, Amer. Mach.,Feb. 27, 1908) by the use of a solution of four 
ounces of sulphate of copper in one pint of water to which about 
10 drops of nitric acid have been added. Clean any oil from the 
zinc before coating, then pour the solution over it and distribute 
it with a piece of waste. The color is governed by the nitric acid. 
Add acid until the color is right. After blacking rub the surface 
with an oily rag. This makes the color a more intense black. It 
should be dead, without luster. 




Fig. 6. — Laying out a face cam. 



of the roller are, of course, arcs of circles having the center of the 
cam shaft as centers, and are drawn in. 

Making the Templet 

To make the templet proceed as in Fig. 7. 

A piece of thin sheet metal, for which zinc in very suitable, is 
tacked down on the drawing as shown by the shaded outlines. This 
sheet is previously cut to a shape which shall fall within the pitch 
line of the cam groove but without the inner border of the groove. 
Its form is easily determined by laying a piece of tracing paper over 
the drawing and then drawing freehand a line which shall mark the 
desired outline of the zinc. The paper is then trimmed to this line 
and is used as a templet to which to cut the zinc. 



With the zinc tacked over the drawing the horizontal and vertical 
center lines are carefully drawn with a fine, sharp scriber and then 
with a pair of dividers having fine, sharp points, those portions of 
the roller circles which are covered up by the zinc are redrawn on 
the zinc. With an irregular curve the bounding line of these arcs 
is then drawn, though not shown in the illustration, the circular 
portion of the groove which represents dwells of the cam roller being 
drawn with the dividers from the center of the zinc as located by the 
center lines. 

Making the Former 

The outline completed, the metal is carefully dressed down by hajid 
to the outline and the templet is then placed upon the former blank 



192 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



which has previously been faced off. The center lines are drawn on 
the former blank and marked as on the drawing and templet and 
the outUne is then transferred to the blank by a fine, sharp scriber, 
and the former is then dressed down to this line, the bulk of the 
metal being removed in a milling machine. The circular portions 
of the outline are easily followed exactly and the other portions are 
followed as closely as possible, after which they are dressed down 
to the outline as carefully as possible by hand. 



Fig. 8 illustrates the laying out of a drum cam. To make the 
horizontal measurements on this drawing, it is necessary to trans- 
late the positions given in the chart by degrees into inches of cir- 
cumference and, dividing the entire circumference of the chart by 
the number of s-deg. divisions, we find the length of each one of 
them. 

The movement of the roller should take place during 12 s-deg. 
intervals which, translated into ins., are laid down as is the verti- 




FiG. 7. — Transferring the cam profile to the templet. 



The error introduced by the hand work is not important as, the 
circular portions being exact, the movement given by the cam is 
exact, the slight error due to the hand work being in the rate of the 
movement only. 

Laying Out Drum Cams 

The laying out of drum cams cannot be done quite as directly as 
that of face cams, as the layout must be a development and not a 
projection drawing. After laying out the profile, it is transferred 
to a sheet zinc templet, in the same manner as a face cam, which is 
then wrapped around the former on which the outline is scribed as 
i n the case of a face cam. To provide for any minute discrepancy 
between the length 01 the templet and the circumference of the 
former, it is important that the place selected for the joint in the 
templet shall be within a straight part of the groove. Were it 
within an inclined part the result of such a discrepancy would be a 
jog when the templet is wrapped around the former, but by select- 
ing a straight portion this is avoided. 



cal movement giving the rectangle within which the curve is to be 
laid down and to find it we have only to follow the method given in 
Fig. 2. The throw circle is drawn and divided as before. Arcs ah 
and cd are drawn through the corners of the rectangle with the 
length of the radius arm as a center, giving the centers ef. The 
line ef is then divided and the intermediate arcs are drawn, the in- 
tersections of which with the parallels through the divisions of the 
throw circle give points on the pitch Hne of the groove from which 
circles with a radius equal to that of the roller define the groove. 
The dwell before the return movement takes place is then laid down 
and the return curve is drawn in the same way. 

A feature of drum cams which should not be overlooked is the divid- 
ing up of the arc of motion, as shown in Fig. 9, in which the lever 
is laid out as it should be with the arc of motion divided by both 
horizontal and vertical center lines. 

Conical rollers are frequently used for drum cams, the rollers being 
laid out as are the pitch lines of bevel gears. This practice is of 
doubtful value as it leads to an end thrust which cramps the roller 
and leads to wear under heavy loads. Straight rolls are preferable 



CAMS 



193 



and to reduce the theoretically imperfect rolling action, they should 
be short — not much longer than half their diameter. 

The cam should run away from the fulcrum — not toward it. 

The diameter of the roller pin should not exceed one-half that of 
the roller in order to reduce the tendency of the roller to stick on the 
pin and thus wear flat. 



bodies. As falling bodies experience no shock in starting, so cam 
motions laid out to conform to the same law experience no shock 
in starting and the same is true for stopping if the retardation is 
made uniform like the acceleration reversed. 

The simplest method of laying out the gravity cam curve is that given 
in Fig. lo, in which the drawing of the parabola is avoided, by A. 




Fig. 8. — Laying out a drum cam. 




Fig. 9. — Correct division of the arc of motion. 




234 56 78 9 10 11 12 
4 9 16 25 36 25 16 9 4 1 C 
Fig. 10. — Laying out cams to the gravity base curve. 



High Speed Cams 

For high speed cams the throw circle is not a satisfactory base curve. 
It is well known that the crank motion, instead of being an easy, 
is a harsh one. In the center of the movement of an engine cross 
head, where the velocity is highest, the acceleration is zero while 
at the centers where the velocity is zero, the acceleration is at its 
maximum. Moreover, as the center is turned, the acceleration is 
abruptly changed from a negative to a positive maximum. For 
these reasons if there is the slightest lost motion pounding and noise 
result. 

The ideal base curve for high speed cams (note that the dividing 

line between low and high speed cams cannot be drawn) is the gravity 

curve (parabola) which differs from the circle as a base curve for 

cams in that the acceleration given by it is uniform as with falling 

13 



B.LENFEST(4>Mer. ilfac^.,4^n7 13, 1905). Divide the base line AC 
into equal parts as for the throw circle. Draw a line AE at any 
convenient angle and of indefinite length and lay off on it, to a con- 
venient scale, distances from A to F and from E to F proportional 
to the odd numbers i, 3, 5, 7, 9, 11; connect £ to 5 or i^ to the middle 
point G of AB and draw from points i, 3, 5, 7, 9, lines parallel to 
EB or FG, intersecting AB at i, 4, 9, 16, 25, 36. Project these 
points thus found on AB to verticals from points i, 4, 9, 16, 25, 36, 
on AC, and draw the curve (5) through these points thus located 
on the verticals. 

The Cam Chart 

The cam: chart, by which the proper timing and coordination of 
cams is obtained, is such a large subject that its elements only can 
be presented here. The usual procedure in designing a cam-operated 
machine is to begin at its operating point, determining first the move- 
ments required and the general location of the cams and connections 
and then to lay out the chart in accordance with the required move- 
ments. A portion of such a chart is shown in Fig. 11. A base line 
is drawn representing the assumed circumference of the cams, which 
is subject to correction should it be found impossible to get all the 
movement into a circumference without increasing the cam groove 
angles beyond the limiting value. Needless to say, the process 
involves a good deal of trial and error work. 

The base line is divided into S-deg. intervals of which only 22 
are shown in the illustration. A zero line common to aU the move- 
ments is drawn at the left, the cams being treated at this stage as 
though all the rollers were upon the same line and had a common 
zero. It is simpler at this stage to treat all cams as though of the 
drum type. 

The extent of movement of the cam rollers are laid down ver- 
tically and full size. The point in the revolution when each move- 
ment must be begun or completed is laid down and a rise of the 
line from the base line represents this movement. The constants 
that have been given enable the preliminary layouts to be quickly 
made, though, if the movements are at all crowded, the chart curves 
must be laid out from the throw circles and radius arms, as has been 
explained in connection with the laying out of the cams. 

The movements are individually simple, being the simple shift- 
ing of a lever. The laying out of cams thus becomes a matter en- 
tirely separate from and subsequent to the design of the machine 
as a whole. The operating parts' and their movements and the 
location of the cam shaft being determined and the connecting levers 
laid down, the matter, so far as it relates to individual cams, reduces 
itself to the moving of these levers at the right times and by the 
right amounts. The chart deals with these movements only, with- 
out regard to their direction or the connecting mechanism. 

Levers of Unequal Length 

When the cam lever arms are of unequal length (the cam end being 
the shorter) the Lanston Monotype Machine Company, employs 



194 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



the method shown in Fig. 12 {Amer. Mack., Dec. 14, 1905) forla3dng 
out the cam curves. The chart is laid out for the full movement and 
then the line kl representing this full movement is divided, kn being 
the cam movement. Point m being assumed at convenience, lines 
Im and nm are drawn. Points on the chart curve being then pro- 
jected to the line Im and from the intersections down to nm the 
heights of the last intersections above the base line give the distances 
to be used in laying out the pitch curve of the cam. A reverse 
method obviously applies to the reversed arrangement of the arms. 

More Accurate Methods 

Increased accuracy of the former is obtained by the Lanston Mono- 
type Machine Company by the use of an iron drawing board, Fig. 



Constructive Details 

An objectionable arrangement of drum cams is shown in Figs. 15 
and 16 (E. Lawrenz, Amer. Mack., Oct. 12, 1911). Not only is an 
unnecessary side thrust put on the lever and its bearing, but Fig. 16 
shows the cutting of such a cam to be difficult if proper contact with 
the roller is to be obtained — a difficulty which is still greater if the 
cam is to drive the roller in both directions. Fig. 17 shows the 
correct form with proper contact between cam and roller. 

Conjugate Cams 

The inertia of the roller gives rise to serious wear of closed cams 
at high speeds. The direction of rotation of the roller on its pin is 
reversed twice during each revolution of the cam at the points where 







10 


20 


30 


40 


50 


60 


70 


80 


90 


100 




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Fig. II. — A portion "of a cam chart 



40 30 20 10 300 90 




Fig. 12 . — Construction for lever arms of unequal length. 



13, by which the cam outline is laid out directly on the former blank 
and the errors due to a transfer are avoided. The drawing board 
contains a pocket of a depth equal to the thickness of the former 
and a stump of the same diameter as the hole in the former. The 
board has a protractor ruled on its surface which enables angular 
lines to be ruled on the former which is coppered for the purpose. 
The heights of the cam curve as obtained from the chart are laid out 
on the blank which is then dressed down to the scribed outline. 

The radii representing dwells are made to micrometer measure- 
ments from the hole in the former, the inaccuracies of the hand work 
thus affecting the rate of movement only. 

Charts with Separate Base Lines 

Charts with different base lines for the various cams are preferred 
by some designers. An example of this lay out is shown in Fig. 14. 



the roller changes contact from one side of the groove to the other. 
At 140 r. p. m. this action on the monotype was found so destructive 
that with hardened rollers and steel pieces inserted in the cams 
at the places where the greatest wear developed, the usual life of 
some of the cams did not exceed six months. 

The double or conjugate system of cams was invented to meet this 
difficulty by J. Sellers Bancroft {Amer. Mach., Dec. 14, 1905). 
A pair of conjugate cams, a andi. Fig. 18, are keyed to apair of shafts 
which are so geared as to run at the same speed and in the same 
direction as indicated by the arrows. The roller c lies between 
them and is driven in one direction by one cam and in the opposite 
by the other. It wiU be observed that with this arrangement the 
direction of rotation of the roller on its pin is never changed. Its 
speed, of course, varies with the diameter of the cam surface act- 
ing at the moment, and to this extent its inertia comes into play 
to induce sliding, but such changes in speed are small in com- 



CAMS 



195 




083 OiZ 09^ 



Fig. 13.— Iron drawing board for laying out formers. 



/I 



Fig. 14. — Chart with a difierent base line for each cam. 





Fig. IS- ^'°-'^- 

Figs. 15 and 16. — Incorrect cam-lever arrangements. 





Fig. 17. — Correct cam-lever arrangement. 



Fig. 18. — Conjugate cam system of the monotype. 



196 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



parison with reversal and, moreover, they are always gradual, 
whereas the reversal with the usual style of cam is abrupt. 

Such cams of cast-iron are more than twelve times as durable as 
the old style of face cams with steel inserts. 

To insure the cams being true conjugates they are cut on a special 
machine, one cutter acting simultaneously on both cams. 

The reversal of the roller does not take place with cams of which the 
movement in one direction is made by a spring, and hence such con- 
structions are free from wear due to such reversal; on the other hand, 
the spring construction introduces other difficulties at high speeds. 

Spring Adjustments 

The varying resistance of springs due to their extension and the 
resulting varying pressure on the roller may be approximately com- 
pensated by the method shown in Figs 19 and 20, by E. Lawrenz 
(Amer. Mach., Oct. 12, 191 1). 

The cam lever AB with the roller at A swings through the arc AAi, 
the spring S being connected to a third arm OC, so arranged that, as 
the spring is extended and its resistence increased, the effective 
lever arm is reduced in the same proportion that the extension (not 
the total length) in increased. Thus Fig. 20, the free length of 
the spring, being DE = DEi, for position OCi the extension is EiCi 
and the effective lever arm is OFi, while for position OC the extension 
is EC and the lever arm OF. To make the compensation (which is 
exact at the extreme positions and approximate at intermediate 
points) it is only necessary to make 

OFXEC = OFiXEiCi 

To find the required extensions draw a diagram. Fig. 21, in which 
ab=OF and ac=OFi. Through any convenient point d on the 
base line, draw db and dc and extend them. Locate efg such that 
g is equal to the difference between DC and DCi when ef = EiC\ 
and eg = EC. 




Figs. 



Fig. 21. 

19 to 21. — Equalizing the Spring Pressure on Cams. 



This construction is most appropriate with cams laid out on the 
gravity curve system, in which the acceleration of the driven piece 
being uniform, a uniform force is suitable. With cams laid out on 
the throw circle system the acceleration is at a maximum at the be- 
ginning of the movement and the natural action of a spring in giving 
the greatest force at the point where the acceleration is greatest 
is suitable. The acceleration becomes a factor, however, only at 
speeds such that the inertia of the parts is a factor and at speeds 
below this point the construction becomes appropriate for cams laid 
out from a throw circle. 



SPRINGS 



The use of spring tables or charts is greatly facilitated by prelim- 
inary calculation of the data, which may be made graphically as 
explained by B. C. BatcheUer, Chief Engr. New York Pneumatic 
Service Co. {Amer. Mach., Aug. 3, 191 1) as follows: 

In designing machinery in which a spring is required, the designer 
usually knows the length of movement of the spring, the force that 
the spring should exert at some point in its movement, say at the 
beginnjing, and the variation in" force that is allowable throughout 
its movement. These are his fundamental data and before he can 
use the spring formula or tables he must ascertain by computation 
the total compression or extension of the spring from the free length. 
In case the spring is free at one end of its movement, no such compu- 
tation is required, but such is not usually the case. 





^►a 


D 


^ 


^ 

Y 0^.^ 


Pi 


Pi 




_._^_ 


^D^ 




> 


mm§mm 






Fig 


I. 




Fig. 2. 
Figs, i to 3. — The preliminary design of springs. 

Fig. I represents a spring, to be used in compression, exerting a 
minimum force Pi, a maximum force P2, with a movement in length 
D. We wish to know the total amount of compression A. On the 
horizontal line XX, erect two perpendiculars, Pi and P^, of as many 
units in length as the respective forces, and distant apart, D. Through 
the upper ends of the two perpendiculars, Pi and P2, draw an inclined 
line, producing it until it intersects the base line at O. The distance 
A from the point to the perpendicular P2, is the total amount of 
compression of the spring. By similar triangles: 

A P2 



therefore 



D P2-P1 



PiD 



Pi-Pi 



Knowing A and P2 we are now prepared to use the formula or tables 
of springs above referred to. 

Fig. 2 shows a similar diagram for a spring in extension. It is 
obvious that Pi can never equal P2; in other words, we cannot make 
a spring to exert a constant force through a sensible length of move- 
ment, and the less the difference between Pi and P2, the greater must 
be the total amount of compression or extension. Such a diagram 
and computation should be made of every spring, no matter how 
insigniiicant, for they give a clear idea of the limitations of the case 
in hand. 

Example. — Required, a spring to move a piston in one direction 
that is moved in the opposite direction by a definite fluid pressure. 
Fig- 3- 



Let Pi = 150 lbs., 
P2 = i75 lbs.. 



and 



D- 



iTSXii' „3 . 
A — = 81 ins. 



175-150 

We must have a spring of sufficient length, diameter, number of coils 
and size of wire to bear a total compression of 8 J ins., and exert a 
force of 175 lbs. under this maximum compression, without injury 
to the spring. 

Helical (Commonly Miscalled Spiral) Springs 

The carrying capacity and deflection of helical springs of round wire, 
in tension or compression, may be determined from the established 
formulas: 

Sd^ 



P = 8^ 



_ PD^N 
Gd* 



For square wire there is some variation in the coefficients given by 
different authorities. Square wire is disappearing from the best 
practice as it should — the circular section being the more suitable. 
The formulas recommended, if square wire is to be used, are: 

Sd^ 



W-- 



.444- 



D 



F = 5-65- 



Gd* 



in which 

W = carrying capacity, lbs. 
5 = fiber stress, lbs. per sq. in., 
d = diameter of round or side of square wire, ins., 
Z> = mean diameter of coil, ins., 
P = deflection of spring, ins., 
G = torsional modulus of elasticity, 
P = load, lbs., 
N = number of coils. 
These formulas ignore certain secondary stresses, and the propor- 
tions of the springs must be such as to make these stresses negligible 
if the results given by the formulas are to agree with the facts. Thus 
the larger the coil in relation to the diameter of the wire the better. 
In no case should the ratio of these diameters be less than 5. Again 
the smaller the helix angle the closer will the calculated results agree 
with the facts. The formulas for deflection again presuppose that 
the correct torsional modulus of elasticity for the material used is 
employed. The formulas will again give more accurate results 
for tempered steel than for piano- wire springs, in which latter in- 
ternal stresses complicate the conditions. 

Uniformity of practice in the matter of fiber stress is not, of course, 
to be expected. From discussions that have appeared in the columns 
of the American Machinist and elsewhere the following stresses appear 
to be safe and conservative: 

For small springs of hard-drawn piano wire there is good warrant 
for stresses up to 100,000 lbs. per sq. in. For springs of tempered 
steel the following stresses may be used: 



Diameter of steel, ins. 


Stress, lbs. per sq in. 


Up to 1 
i 

1 

1 
2 


75, 000 
70,000 
60,000 
50,000 



197 



198 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



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200 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



It should, however, be said that some large users of springs limit 
the stress to 40,000 lbs. while, on the other hand, the Pennsylvania 
R. R. uses stresses of 60,000 to 70,000 lbs. All these figures are for 
springs subject to moderate shock. For heavy shock they should be 
reduced. Phosphor-bronze may be stressed to 15,000 lbs. and brass 
wire to 5000 lbs., the figures for brass being the least well established 
of all. 

The torsional modulus of elasticity for steel, according to American 
investigators, averages about 12,600,000, while British experimenters 
give the smaller value, 11,000,000. It has been repeatedly proven 
that this constant has the same value for tempered and untempered 
steel. The effect of tempering is to raise the elastic limit and ultimate 
strength, without changing the modulus. The result is that while a 
tempered spring will carry a much heavier load without permanent 
set, the rate of deflection is unchanged. The value of the modulus 
for phosphor-bronze is 6,200,000 and for brass 3,400,000, the figures 
for brass being, again, less well established than the others. Con- 
sidering the miscellaneous compositions that go by the name of 
"brass" definite values for the fiber stress and modulus are not to be 
looked for. 

The accompanying charts. Figs. 4 and 5, by Prof. J. B. Ped- 
dle (Amer. Mach., Aug. 15, 1912) give the same results as the above 
formulas and enable calculations for helical springs to be made with 
great facility. The use of the charts is explained below them. 

The charts may obviously be worked in any convenient direction 
in accordance with the given and required quantities. 

In ordinary cases several trials must be made before a spring of 
the required strength and deflection is found, and it is in the conveni- 
ence of the charts for making these trials that their best feature lies. 

Of course, good sense must be used in all such work. Thus in 
the case of a compression spring it may easily happen that the 
deflection given by the chart will more than close the spring — an 
impossible condition of course. This must be watched for and a 
spring be chosen which will not be an absurdity. The charts are 
equally applicable to both extension and compression springs — no 
initial tension being understood in the case of extension springs. 

The deflection of conical helical springs may be obtained from the 
formula (G. M. Strombeck Amer. Mach., Feb. i, 1912): 



f-=2NP- 



Gd^ 



in which/ = total deflection under load P, ins., 

N = number of coils, 

P = load, lbs., 

Z>i= largest diameter of coil to center of wire, ins., 

Z>2 = smallest diameter of coil to center of wire, ins., 

G = torsional modulus of elasticity, 

d = diameter of wire, ins. 

To design a double or triple helical spring (two or more concentric 

springs) each individual spring to carry an equal part of the total 

load, proceed as follows, (O. A. TnELiN^wzer. Mach., Dec. 27, 1906): 

Let P = total load, lbs. 

iV^ = number of individual springs, 

Z) = pitch diameter of outer coil, ins., 

rf = diameter of wire of outer coil, ins; 
P 
then -T; = load per spring, lbs. 

P 
Design the outer coil for load -r;' draw a line perpendicular to the 

axis of the spring through the center of the cross-section and offset 
.^d to each side of this line on the axis, as in Fig. 6. From these two 
points draw tangents to the circular section of the coil d, when any 

p 
coil di, di, etc., tangent to the two lines will also carry jz lbs. load. 

Theoretically, the two tangents will not be straight lines, but form 
the curve of a cubic parabola. The difference is, however, slight 
and need not be considered for practical purposes. 



Helical Springs in Torsion 

The strength and deflection of helical springs in torsion are usually 
calculated from the equations for the bending of straight beams. 
While these equations are inexact for the conditions, they are much 
simpler than those based on the curved-beam theory and, for springs 
of the usual proportions of wire and coU diameters, they lead to 
errors that are unimportant. 




Fig. 6. — The design of multiple springs. 



The formulas are: 



nd^ 
M = — S 
32 
3667 MDN 

Ct ^=— 

Ed* 



for round wire 



"6~ ' 

2160 MDN \ *«' '^"^'■^ ^'""^ 
"=— EJ* — 

in which If = twisting moment, Ib.-ins. 

d = diameter of round or side of square wire, ins., 

5 = fiber stress, lbs per sq. in., 

a = angle of twist, deg. 

-D = mean diameter of coil, ins., 

iV = number of coils, 

£ = tension (not torsion) modulus of elasticity. 

For springs loaded in this manner square wire is more appropriate 
than round. 

The accompanying charts. Figs. 7 and 8, by Prof. J. B. 
Peddle (Amer. Mach., June 19, 1913) are based on the above formulas. 
Instructions for use will be found below them. Safe fiber stresses 
may be taken at from 80,000 to 100,000 lbs. per sq. in. for tempered 
and from 30,000 to 40,000 for untempered steel. For hard-drawn 
spring brass wire stresses of 15,000 to 20,000 lbs. per sq. in. may be 
used. For steel the usual values of the modulus of elasticity are to 
be used. For spring brass wire, the values of the modulus, according 
to Professor Peddle, range between 13,000,000 and 14,800,000, with 
an average of 14,000,000. 

With a little calculation, the charts are applicable to wire of 
rectangular sections other than square. 

Assume that a wire .2 in. thick (perpendicular to the axis of the 
coil) is to be used, the load being 120 Ib.-ins. and the fiber stress 
60,000 lbs. per sq. in. First find the load which may be carried by a 
square wire .2 in. on a side, namely 80 Ib.-ins. For other widths, 
parallel with the axis of the coil, the load is proportional to the width, 

120 
so that for a load of 1 20 Ib.-ins. the required width is . 2 in. X -q— = .3 in. 

The deflection, on the other hand, will vary inversely as the width, 
Hence, in the example shown on the deflection chart, if we use a wire 
.2X.3 in., instead of a wire .2 in. square, the deflection will be f of a 
revolution instead of one revolution with the number of coils given. 
Or if we must have a deflection of one revolution, it will be necessary 
to increase the number of coils by 50 per cent., which would mean 
37J coils instead of 25. 



SPRINGS 



201 



Elliptic and Semi -elliptic Springs 

The strength and deflection of elliptic and semi-elliptic springs may 
be determined from the formulas: 
■^_nbt^f 

D = ^K full elliptic 

D = -TB'K semi-elliptic 

in which i' = safe load, lbs., 

n = number of leaves (total for semi-elliptic and for one side 

of full elliptic), 
J = breadth of leaves, ins., 
t = thickness of leaves, ins. 
/ = safe fiber stress, lbs. per sq. in., 

L = freelength or projection of one end from center band, ins. 
Z> = deflection, ins., 
£ = modvdus of elasticity. 



I fl— J-2 "I 

^ = (737)-3[_-^ 2r{i-r)-r^ loge rj 

No. of full length leaves 



In these formulas, by Prof. J . B. Peddle, the equivalent plate 
spring is assumed of the trapezoidal form, Fig. 9, instead of, as usual, 
the triangular form, Fig. 10. For structural reasons there must 
be at least one full-length blunt-ended leaf, and the assumption of a 
triangular equivalent, when the number of leaves is few and r, 
therefore, large, leads to errors which may equal 10 or 12 per cent, 
and even more if there is more than one fuU-length blunt leaf. 

It is necessary, in order that the comparison between the ideal and 
actual springs should hold good, to have the points of the shortened 
leaves tapered in width or in thickness, or both, so as to meike the 




rnh 




Fig. 9. 



Fig. 10. 



Figs. 9 and 10. — Equivalent plate springs. 



total No. of leaves 



F-1 



.1-E 



-.7 



.7-r 






' ^ 



.2- 



.1- 



20,000-3^20 000 



10.000. 
8.000- 
6.000- 
4,000-^ 4.000 



2,000^2,000 
■ 1,000 



<i=^2 



1,000- 
800- 
600- 

400- 



200- 

100- 

8.0- 

-''60- 

40- 

20- 
10- 



-.1 



.05- 



kos 



10,000 
8,000 
6.000 



-600 
-400 



-200 . 

CO 

7-100 .0 

-80 ►J 

-60 -w" 
-40 2 



-20 



-10 •- 



-05 



120,000- 
110,000- 
100,000- 

90,000- 

80,O0Ot 

70,000- 

60,000^1 

50.000- 



40.000^-40.000 



,3 30,000 



^ 20,000- 



10.000- 



-120,000 
-110,000 
-100,000 

-90,000 
-80.000 
-70,000 

60,000 
-50.000 



-30.000 



-20.000 



-10.000 



6,OOOJ_5,000 



Connect the given twisting moment with the desired fiber stress. 
The line extended gives the required thickness of wire. The ex- 
ample shows that a square wire .2 in. thick will carry a twisting 
moment of 80 lb. -ins. imder a fiber stress of 60,000 lbs. per sq. in. 

Fig. 7. — Carrying capacity of helical springs in torsion. 



transmission from one leaf to the next one gradual. If this is not 
done and the leaves are blunt-ended, the sides of the ideal plate 
spring would have to be stepped instead of straight. 

The accompanying charts, Figs. 11 and 12, also by Professor 
V%'DX)ix.{Amer.Mach.,Apr. 17, 19 13) give the same results as the form- 
ulas and eliminate the laborious calculations due to the complex form 
of the expression for K. The use of the charts is explained below 
them. 

When comparing the calculated with the actual deflection of leaf 
springs, it must be remembered that the friction between the leaves 
introduces a disturbing factor, the effect of which cannot be 
calculated. 

An examination of the fiber stresses in automobile springs of this 
type was made by David Landau and Asher Golden {Horseless Age, 
Dec. I, 1909). Two cases of alloy steel springs having an elastic 
limit of 184,000 lbs. showed fiber stresses of 43,600 and 56,250 lbs. 
per sq. in., respectively and two cases of high-grade open hearth 
steel springs having an elastic limit of 142,000 lbs. showed fiber 
stresses of 41,200 and 50,500 lbs. per sq. in. Automobile springs are 
protected from extreme overloads by the bumpers which limit the 
deflection. 

The strength and deflection of flat (single leaf) springs may be deter- 
mined from the formulas of Table i, by R. A. Bruce (Amer. Mach., 
July 19, 1900). 

Assuming the length to be determined by circumstances and the 
load and deflection to be given, the simplest method of proceedure 
is to first settle upon the proper depth t in order to secure the requisite 
deflection. The formula for this pmpose is 



/2 

l=aXT- 



(a) 



in which / = length, 

8 = deflection, 
/ = thickness, 

all in inches. 

The value of the multiplier a depends upon the safe stress / and 
the modulus of elasticity multiplied by a number which varies accord- 
ing to the type of spring adopted. The general value of a is given 
for each type of spring in the column under the heading a, but inas- 
much as a good all-round value for/ is 60,000 and for E 30,000,000, 



202 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



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SPRINGS 



203 



a second column has been added, giving the numerical value of a 
for these values of the stress and modulus. They may be confidently 
used for general work, and in cases where springs are not subject to 
alternate bending in opposite directions. The thickness having 
been determined, the breadth in ins. is next found by making use of 
the formula, 

.Wl 



b=cX- 



(6) 



in which W = maximum load, lbs. 
/ = depth, ins., 
/ = length, ins. 

The value of c for the general case is given under the first column 
marked at the head c. For ordinary steel springs a second column 
of values of c has been added, the values assigned to / and E being 
the same as before. The principal dimensions of the spring are there- 
fore easily settled. In order to find the cubic volume of the spring, 
multiply the product oil, b and t by the number given under the 
column marked v. A useful check on the work is to find the energy 
to be absorbed by the spring by multiplying the deflection by half 
the maximum load. If this quantity be divided by the number 
given under the heading R, the result should be equal to the volume 
of the spring in cubic inches. The first column lettered R gives the 
general value of the resilience per cubic inch for a spring of a particu- 
lar type and the second gives the resilience when the stress is 60,000 
lbs. per sq. in. and the modulus of elasticity is 30,000,000. 

The maximum load and the deflection for a given load can be found 
by transposing formulas (a) and (b) as follows 



12 

o=a— 

bi^. 
W=-r 
cl 



Flat Spiral Springs 



ic) 
id) 



The strength and deflection of flat spiral springs (of the watch 
spring type) have been examined by Prof. J. B. Peddle {Amer. 
Mach., Oct. 8, 1914) who accepts the formulas established by J. St. 
Vincent Platts {The Engr., July 18, 1913) as follows: 

„ bt 



N = 



L = 



6f 

{R-rY 
At{R+r) 

TrEjR-ry 
4f{R+r) 



in which T = maximum turning moment, lb. ins., 
fc= breadth of spring, ins., 
t = thickness of spring, ins., 
/ = working fiber stress, lbs. per sq. in., 
N = number of revolutions of arbor., 
L = length of spring, ins., 
E = modulus of elasticity, 
2? = radius of box, ins., 
r = radius of arbor, ins. 

The accompanying charts, Figs. 13 and 14, are based on these 
formulas. Instructions for use will be found below them. The 
manner of attachment of the spring to the box, whether fixed or 
pinned, exerts a complicating influence on the fiber stress. Profes- 
sor Peddle adapts his charts to both conditions by using a fictitious 
fiber stress in place of the working stress. This fictitious fiber stress 



Table i. — Strength and Deflection of Flat Springs 
Notation 

/=safe stress, lbs. per sq. in., 
E = modulus of elasticity, 
R = resilience or energy, in.-lbs. per cu.in., 
W = maximum load on spring, lbs., 
= maximum deflection, ins., 
/ = length of spring, ins., 

l^ 
< = thickness or depth in ins. =aX'r- 

Wl 



6= breadth in ins. =cX 



F = cu. ins. in (useful part of) spring =vXlbt. 



Types of spring used 



General 



\ c \ R 



For f = 60,000 
E = 30,000,000 



I R 




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3E 



J_ 



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ae 



3E 



1.09/ 



4£ 



J_ 
6E 



11 
6E 



.70/^ 
6£ 



6B 



.725/2 



6£ 



6E 



11. 
6E 



.70/' 
6£ 



II 
6E 



72S/2 



6£ 



.33/'' 
6£ 



1.75 
1000 



3 1000 



2.18 
1000 



X 

3 1000 



16 1000 



.54 
1000 



3 X 1000 



40000 



14 



14.52 



6.66 



14 



14.52 



6.66 



2R 

is the working stress multiplied by „ , and is to be used when con- 
sulting the chart. 

It may be noted that considerable latitude is permissible when 
selecting the box and arbor diameters. If a certain ratio is desired 
between them, we have only to move back and forth along the hori- 
zontal line through the intersection on the axis A until the two values 
for the box and arbor diameters intersecting on this line have the 



204 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



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206 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



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Connect turning moment with fiber stress and find intersection with middle axis. Any line through this point will intersect the breadth 
and thickness axes at corresponding values. Example. — Turning moment is io.8 pound-inches and fiber stress 120,000 pounds per square 
inch, then for a spring breadth of 0.6 inches the thickness will be 0.03 inches. 

Fig. 13. — Strength of flat spiral springs. 



SPRINGS 



207 



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Connect number of revolutions to thickness of spring and find intersection with axis A. Connect fiber stress and modulus of elasticity 
and find intersection with axis B. Connect points thus found on A and B and extend line until it intersects the spring length. The sizes of 
the spring box and arbor may be found by taking the intersection with axis A as found above and projecting to the left till an intersection is 
found between suitable values of box diameter (represented by vertical lines) and arbor diameter (represented by curves) interpolating if 
necessary. Example. — If 15 revolutions are required and thickness is 0.03, fiber stress 120,000 lb., modulus of elasticity 36,000,000 then the 
length will be 425 inches. For a box diameter of 8 inches and 2 inches for the arbor an exact intersection is found on the horizontal through 
the intersection on axis A. If, however, this proportion is not satisfactory, other values may be found by interpolation. 



208 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



right proportion. In general, it will be necessary to interpolate 
between the lines as drawn, but this should ofier little dif&culty. 

Materials of and Miscellaneous Information on Springs 

Ordinary carbon steel used for springs, according to C. A. Tupper 
{Amer. Mack., Mar. 24, 1910), has about the following chemical compo- 
sition: Carbon .95 to 1.05 per cent.; manganese .025 to.o4opercent.; 
silicon .12 to .15 per cent.; phosphorus and sulphur not over .03 per 
cent. each. The elastic limit to be expected from such steel varies 
so much with the heat treatment and the methods of tempering used 
that general statements are without value. The highest figures 
observed were given in a paper read before the International Society 
for Testing Materials, September 9, 1909, for steel of approximately 
the above characteristics hardened in water at 1425 deg. Fahr. and 
drawn to 750 deg. Fahr. The diameter of the test piece was .994 in. 
It showed an elastic limit of 240,800 lbs., with modulus of elasticity 
29,220,000 and broke under a deflection at the middle of .744 in. 
It is apparent that the allowable limits of specifications for finished 
springs are rising at a very rapid rate, and what the immediate future 
will bring can only be conjectured. In practice, however, the elastic 
limit actually necessary is very far below the extreme figures just 
cited. For special alloy steels the chemical composition varies widely, 
particularly in relation to the carbon and manganese contents, which 
may range considerably lower. 

Fuller, though older, information regarding the carbon content of 
steel for springs is found in a paper read by Wm. Metcalf before 
the American Society for Testing Materials, 1903, as follows: 

The lower carbons should be put into the lajger bars, because the 
large bars are the most difiicult to harden safely, and the diflaculty 
increases in a geometrical ratio with the increase in carbon. A good 
rule is to put .70 to .90 carbon into bars of more than i in. diameter; 
bars from i to | in., .90 to i.io carbon; bars from 3 to | ins., i.io 
to 1.20 or even 1.30 carbon, and little rods below | in. any high 
carbon up to as much as 1.45. 

Regarding hardening and tempering, Mr. Metcalf says that steel 
of .60 to .90 carbon may be hardened in water; with about .90 carbon, 
a film of oil may be used on the water. From .90 to i.io carbon, 
about 4 or s ins. of oil may be used on the water, and for higher 
steel oil should be used and kept cool by an external tank of 
cold circulating water, or by a coil of pipe inside of the tank with 
cold water running through. 

Tempering must be suited to the carbon; .70 to .80 carbon will 
require very little drawing; .90 to i.io may require the oil to flash, 
and for higher carbons the oil may be biurned oS. Above 1.30 
carbon a heat that barely begins to show color will generally give 
a good spring temper. In tempering, as in hardening, good sense 
and good judgment are the best guides. 

The desirable composition of steel for helical springs according to 
the specifications of the Pennsylvania Railroad is as follows: 

Carbon i.oo per cent. 

Manganese 25 per cent. 

Phosphorus, not above 05 per cent. 

Silicon 35 per cent. 

Sulphur, not above 03 per cent. 

In case the carbon is found to be below .90 per cent., the manganese 
above .50 per cent., the phosphorus above .07 per cent, and the 
silicon below .25 or above .50 per cent., the springs represented by 
the sample or samples will be rejected. Springs made from bars 
three-eighths of an inch or less in diameter need not conform to the 
chemical limits above, but such springs will be rejected if the carbon 
is below .50 per cent., the manganese above i.oo per cent, and the 
phosphorus above .11 per cent. 

The desirable composition of steel for elliptical springs, according 
to the specifications of the Pennsylvania Railroad, is as follows: 



Carbon i.oo per cent. 

Phosphorus, not above 03 per cent. 

Manganese, not above 25 per cent. 

Silicon, not above 15 per cent. 

Sulphur, not above 03 per cent. 

Copper, not above 03 per cent. 

Springs, however, will be accepted which on analysis show the metal 
to contain: 

Carbon, not below .90 per cent, or not 

above i.io per cent. 

Phosphorus, not above 05 per cent. 

Manganese, not above 50 per cent. 

Silicon, not above 25 per cent. 

Sulphur, not above 05 per cent. 

Copper, not above 05 per cent. 

For additional information on steel for springs see Steel. 

Particulars regarding the behavior of springs and the practice of the 
Westinghouse Electric and Mfg. Co. are given by R. A. Peebles, 
Research Engineer of the company (Amer. Mach., May 2, 1912). 

The drawing usually specifies the free height of the spring, the num- 
ber of turns, the size of wire or bar, and either the outside or inside 
diameter of the spring according to where it is to be used. The 
specification provides, however, that the manufacturer, in order to 
obtain the desired combination of load and compression, may vary 
the size of ^vire used, provided the fiber stress figured from the size 
substituted be not more than 10 per cent, greater than the. stress 
figured from the size specified on the drawing. Imperfect workman- 
ship is responsible for the greater number of poor springs, and in 
the majority of cases the faults seem so slight that it is often diffi- 
cult to convince the manufacturer that they are the cause of the 
discrepancies. 

It will be found that of springs which are accurately designed those 
give most trouble which have the smallest number of turns or the 
smallest diameter in proportion to the size of the wire. This is 




Fig. 15. Fig. 16. 

Figs. 15 and 16. — Correct and incorrect constructions of springs. 

because the source of most of the inaccuracy is in the two end turns 
which are set up against the adjacent turns and ground or hammered 
flat to fit the spring seat. 

The end of the wire in the set up end turn should no more than touch 
the next turn as shown in Fig. 15, and not lap against it as in Fig. 16. 
In a large proportion of the springs tested the end turns lap as in 
Fig. 16, often to the extent of J to | of a turn at each end. This 
robs the spring of f to f of an active turn, thus increasing both the 
compression per turn and the corresponding load at the required 
compression. 

It will easily be seen that this is a serious matter in a spring of say, 
four to six turns and is sufficient in itself to throw the spring outside 
the requirements even of a specification which permits a variation 
of 10 per cent, from the specified load. In the case of some large 
springs tested the load at a given compressed length was found to 
vary from 6000 to 14,000 lbs. The load required was 9000 lbs. 
These springs were from jf in. diameter bar and had only 35 turns. 
In some of them the end turns did not touch the adjacent turn at the 
tip. These were the low ones. In others the end turns lapped nearly 
a half turn. 

By far the larger number of springs which fail to pass are too strong. 



SPRINGS 



209 



Springs are seldom below the requirements. This was explained by 
one spring maker on the ground that springs are usually accepted 
by railway companies even when considerably over the specified 
loads. Since the work done for railway companies forms a large 
part of their business, the spring maker usually aims high. This is 
given for what it is worth; as a matter of fact, we have seen that the 
ordinary defects of manufacture are such as would increase the load 
at a given deflection. 

In a discussion on springs {Trans. A. S. M. E. Vol. 23) A. S. Gary 
referred to the many different methods of making springs, the various 
ways of preparing the wire for them, their treatment during manu- 
facture and their treatment after they leave the spring machine. 

Thus, springs are made from hard drawn or hard rolled wire which 
receives no tempering treatment after being coiled. Wire made hard 
by working owes this quality of hardness to the many internal stresses 
it contains. Such springs have a comparatively limited elastic 
limit and are easily fatigued, although the resistance to extension 
or compression, within the elastic limit, is considerably greater than 
with any other kind of wire of the same size. 

A hard drawn or hard rolled steel spring can be much improved 
by a process invented and patented by Mr. Gary's father, by which 
the spring, after being formed and pressed, is heated to a temperature 
between 400 and 700 deg. Fahr. and then rapidly cooled in a blast of 
cold air. A spring of this kind, after this treatment, seems to have 
the internal stresses which were introduced during the coiling and 
pressing processes removed; its elastic limit is materially increased 
and it is less easily fatigued. 

In making compression springs by this process it is found necessary 
to coil them to a considerably greater pitch than is found in the 
finished spring, and then they must be subjected to a sudden overload 
beyond their original elastic limit which reduces their pitch considera- 
bly, and then we obtain a fairly efficient spring, but one inferior to a 
tempered steel spring. 

The best and most durable springs that can be made are formed 
from comparatively high carbon soft steel wire which, after being 
finished, are hand tempered — that is, they are first heated in a char- 
coal fire to a cherry red or slightly higher, and then plunged into a 
liquid bath, which is generally one of oil. They are then carefully 
polished (over more or less of their surface) and held above the 
charcoal fire until the required temper color appears, which color 
differs with the various quaUties of steel used. 

There are many variations of this process, differing in small particu- 
lars, but so delicate are the different manipulations if uniform results 
in any considerable number of these springs are to be obtained, that 
the process has been almost entirely abandoned by spring manufac- 
turers, the best of whom have adopted specially prepared tempered 
wire for their spring stock, and after forming and machining their 
springs they are tempered by the Gary process. 

I might add here that my most uniform results in hand tempering 
springs were obtained by heating and afterwards drawing them by 



passing an electric current through them. The wire composing 
such hand tempered springs is, if they are properly made, free from 
all internal stresses when the spring is at rest, and in properly pro- 
portioned compression springs there is no setting or decrease in pitch 
after the coils are closed tightly one upon the other. 

Another method of making springs is to take steel wire or bars 
and heat them to a lower temperature than a welding heat, then coil 
them hot on the arbor, and before they have an opportunity to cool 
below a dull red throw them into an oil bath. The quality of steel 
used in this process is sufficiently low in carbon to make it unnecessary 
to draw the temper after hardening, but such springs are not to be 
classed as high grade. Most of the heavy car springs are made this 
way. 

In plunging red hot springs into their cooling bath great care must 
be exercised. If they are slowly immersed sideways, one side of the 
spring will often be tempered harder than the other, because of the 
different temperatures of the opposite sides of the spring when they 
are immersed. A similar result is sometimes obtained when long 
springs are slowly immersed endwise, one end of the spring being 
found harder than the other. 

Experience has taught that the most serviceable extension springs 
are those coiled in such a manner as to have their coils, before exten- 
sion, press so closely together as to require the application of a certain 
initial load before the coils begin to open. This initial set is obtained 
by using hard drawn or tempered spring wire and deUvering it to 
the arbor on which the spring is formed in a twisted manner — that 
is, by twisting or revolving the wire around its own axis the same as 
the strands of a rope are twisted firmly together. It has been found 
that the Gary process of tempering does not affect this initial torsional 
strain in extension springs, although the hand tempering process, 
where the spring is heated to redness, destroys it. 

Springs for use in salt water should be made of phosphor-bronze 
in order to avoid corrosion. According to the Brass World 1907, 
if the mixture is rightly made this material cannot be surpassed by 
anything except steel, but if not, it is inferior to yellow brass. 

Experience has taught that if the phosphorus in rolled metal exceeds 
.05 per cent., the bronze is injured. 

The greatest variation in roUed or drawn phosphor-bronze is 
caused by the tin content. A bronze which contains only 3 per cent, 
of tin is inferior to one which contains 8 per cent., although both 
may be phosphor-bronze. On account of the difficulty in rolling 
or drawing phosphor-bronze containing a high tin content, manu- 
facturers wiU substitute a lower percentage if it is possible to 
do it. 

A good spring should contain only copper, tin and a very small 
quantity of phosphorus. 

Those who have had trouble with phosphor-bronze springs should 
ascertain whether their troubles are not caused by the absence of 
the necessary amount of tin, or the presence of zinc. The temper 
is produced by cold-rolKng. 



14 



BOLTS, NUTS AND SCREWS 



The terms lead and pitch, as applied to screw threads, are not always 
clearly defined, the result being confusion in the case of multiple- 
thread screws. A further confusion arises from a loose use of the 
word pitch in the case of single-thread screws which advance the nut 
somewhat near i in. per turn. Thus, while the term 8-pitch means, 
clearly enough, \ in. pitch, the expression i J pitch is not clear because 
it is not known whether the screw is of one and a half turns per inch 
or of 1 5 in. per turn. This form of expression has no proper applica- 
tion to screw threads and should be discontinued. The best form of 
expression, because of its universal application, is | in. pitch, i f in. 
pitch, etc. If the pitch is an aloquot part of an inch, for example \, 
the expression 8 threads per inch is satisfactory, as it cannot lead to 
confusion. 

The confusion between the terms lead and pitch should lead to the 
general, as it already has to considerable, adoption of the usage of 
these terms by the Brown and Sharpe Mfg. Co. By this usage, 
the advance of a screw to one turn is the lead, which is the only term 



Lead or Pitch 



Lead , Pitch 




Fig. 



I. — Single thread. Fig. 2. — Multiple thread. 

Lead and Pitch Screws and Worms. 



they ever use to designate this advance. The turns to an inch are 
obtained by dividing i in. by the lead. Conversely, the quotient of 
I in. divided by the number of turns to an inch is the lead. In other 
words, the product of the lead multilplied by the turns to an inch is 
always equal to i in. 

The term pitch has been limited to designate the distance between 
two consecutive threads or between two consecutive teeth. Divide 
I in. by the pitch and the quotient will be the threads to an inch. 
Divide i in. by number of threads to an inch and the quotient will 
be the pitch. 

The product of the pitch multiplied by the number of threads to 
an inch is always equal to i in. 

The distinction for single,- and multiple-thread worms or screws 
is shown in Figs, i and 2, from the former of which it wiU be seen 
that, for single - but not for multiple- thread screws, pitch and lead 
are dentical (O. J. Beale, Amer. Mach., July 18, 1907). 

Screw Thread Standards 

There is no standard V thread and the continuance of that con- 
struction is a simple nuisance. The taps and dies of different makers 
are not alike and will not interchange, while none of them agree 
with the theoretical or paper " standard." Under these circumstances 
it is impossible to give tables of dimensions of any value and for this 
reason such tables are here omitted. American tap and die makers 
are making a united effort to retire the V thread in favor of the U. S. 
Standard and their efforts should have the support of all. 



Friction of and Resultant Load on Screws and Bolts 

The friction of screw threads formed the subject of experiments by 
Prof. Albert Kingsbttry {Trans. A. S. M. E., Vol., 17). The ex- 
periments were made on a set of square threaded screws and nuts of 
the following dimensions: 

Outside diameter of screw. . . ; 1.426 ins. 

Inside diameter of nut 1.278 ins. 

Mean diameter of thread. i-3S2 ins. 

Pitch of thread J ins. 

Effective depth of nut ij^ ins. 

The conclusions are that for metallic screws in good condition, 
turning at extremely slow speeds, under any pressure up to 14,000 
lbs. per sq. in. of bearing surface and freely lubricated before appli- 
cation of the pressure, the following coefficients may be used: 



Lubricant 


Coefficients of friction 


Min. 


Max. 


Mean 




.09 
. 11 
.03 


.25 
.19 
.15 


.11 




• 143 


Heavy machinery oil and graphite in equal 
volumes. 


.07 



Note that the experiments measured the friction of the thread 
surfaces alone — the friction of the step being eliminated by the 
construction of the apparatus. 

The screws tested were made of various materials — mild steel, 
wrought iron, cast-iron, cast bronze and case hardened mild steel, 
and the nuts of mild steel, wrought iron, cast-iron, and cast brass. 
No material difference was found due to these materials. 

In use, these coefficients should be substituted in the formula by 
which they were calculated as follows: 

^ ^Ttd-fp 

in which Q = tangential force necessary to turn the nut applied at 
the mean radius of the thread, lbs. 
P = total axial load, lbs. 
p = pitch of thread, ins. 
d = mean diameter of thread, ins. 
/ = coefficient of friction. 
For the efficiency of screws as affected by the helix angle of the 
thread, see Efficiency of Worm Gears. The same formulas apply to 
both constructions. 

The resultant stress on bolts due to the initial stress resulting 
from tightening the nut and the addition of a load, such as the 
steam pressure on a cylinder head, depends on the relative elasticity 
of the bolt and the connected parts and is usually indeterminate. 
There are two extreme conditions between which actual cases 
usually lie. The extreme conditions are represented by Figs. 3 
and 4. 

Case I. — Elastic Bolt and Non-elastic Block. — Let the bolt be repre- 
sented by a powerful spring, as in Fig. 3. Let the block be fixed 
and let the nut be screwed up to produce an initial strain of 5000 
lbs., and then let the stirrup with its weight of 5000 lbs. be added. 
Under these conditions, the added weight wiU not increase the 
strain, because if it should the spring bolt would stretch under it, 
the block being non-elastic would notfoUow, the lower washer would 
leave contact with the block, and the supposed increased strain would 
instantly relieve itself — hence there can be no such increase. 

(Continued on page 216, first column) 



210 



BOLTS, NUTS AND SCREWS 



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fO to 


t^ 


Ov N >0 C> 


^ 


Ov 


o 


3 

rr 


























MUM 


" 


(N 


. 




00 CO O '00 NO 


00 




vO 






■^ 
















o 


CO 


t) ,: 






« 


r^ 


f^ 


N O 


■* 


N to O to 


to 


to 


Pt o. 


00 


o o o o 


to 


O 






rt ^ 










c* 


-^ o 


00 


M -<t 0» PO 


o> 


to 


fO o 


PO 


Ov O t^ O 


o> 


o 






















M W M N 


N 


ro 


^ vO 


00 


O -^ r^ w 


o 








w " 


































Si 




























M M M N 


M 


CO 








CS IN 00 t^ 






lO 


00 




CO r- 


o 


to 


00 00 


l> 


T-o 


to 


w 










M « fO 't 'O 


Oi 


M 


Ov 


Ov CO 


o 


O to rf O 


o 


N 


vO ^ 


o. 


r- to O to 


00 


Ov 
CO 


(U 


a 
o 


nJ J5 












W ^ 


o 


00 O PO O 


o 


to 


O PO 


o 


C> PI Ov O 


o 


fs 


2^ 
















W M M 




•N 


ro ^ vO 


r^ o pt so 


Ov 
































M M M 


•^ 


(N 


. 




to Ov >0 N 


00 


c- 




PO lO 






















w 


■O .;! 




1-1 M rf lO 00 


o 


M- 


Tf O N 


PO 


00 00 -^ M 


to 


O 


Tj- vO 


to 


fO O to o 


o 


o 


















r- 


Ch N o O 


to 


o 


SO P^ 


C>1 




PI 






















H M C^ 


N 


CO 


PO to 


r- 


Ov P) to Ov 


CO 


00 






w 


























M M M 


01 


N 


<; o 


sjBnbg 


sis SB ^15 «;B -H 


„» 


H|« 


S^'l^ 


°ts 


"1^ _le» »!<■ 

>o|« "Irt f4» lO|06 


ro -* 




si: 

to 


-i: SB "C sis 

o O t~- t^ 


H|ca 
00 


00 


uoS'BxajT 


relto «1h ioIo ctjn '!« 


H« 


ajsca:* 


< 


Hsa3«i-=B 


HS 


X 


x< 


Se 


?^:S^^U < 


loHe 


■c 








M 




M 


M 


l-H M 


M 


CI IN c^ pj 


PO 


PO 






■^ to to o 


O 


l-^ 


hort 
meter 
hexa- 
n and 
uare 






^HS 


--H" 


-^^ 


g: 


HS„„-K 


<nW 


as 


Hl« -I« 


^+xl 






















CS M IN C^ 


N 


« 


PO PO 


PO 


■rj- Tj- lo to 


to 


o 


"2 .2 ^ P. w 






























T3 O M " 
































« 


■p 






»R 






t-W 


-„ 


Ml-* «|ai -!« MM 


„:, 


t-loo 


mN 


r-W 


on. HH «1C 


o|-« 




c 


z 












'"' 


•^ 


M M l-C M 


•"" 


'"' 


Pt PI 


c^ 


C4 PO CO CO 


CO 


■^ 


A! 


































•0 




H^sBdSSBHS 


sBS; 




SK« 


ss 


"K-CHs 


«» 


s; 


•|S„H. 


ss 


«|co ""Ih «|n Hh 


H« 


H: 


e-i 


















M „ M M 


" 


" 




M 


(N (N <N IN 


<N 


fO 






*n 


T) 


Ch to O w N 


fO 


lO 


O 


o o 


o 


COCO 


o 


o 


o o 


o 


o o o o 


O 


o 




o 




rt 


00 M r- lo 00 


ro 




(N 


^ o 


W) 


PO 00 to o 




to 






o o o o 


O 


o 


.D 


o 


o 
o 


r* 


M fO ^ vO 00 




l-t 


W 


Ov OO 


00 « fO o o 

^ o t^ dv o 




ro H PI 
4 o" m' 


o 

vO 


CO q o_ q 

(N 00 to cT 


to 
d 


q 
o 


O 

8 


M „■ 
C J3 


(ii 


V 














'"' 


^ 


'■^ 


M « 


c^ 


fO CO '<5- to O 


t^ 


































o 


<1> 
,£3 








o 


o 


O 


o o 


o 


o o o o 


O 


o 


o o 


o 


o o o o 


o 


o 






-(J 


■^ po t^ lo r* 


t 


lO 


O* 


H O 


to 


o o o o 


lO 


o 


o o 


o 


o o o o 


o 


o 


+J 


3 


'ri 


ro lO r^ O CO 


r- 


H 


o 


« irt 


a >o Tf rr to 




ro 








o 


o 


rt 


CO 


(i( 


Xi 


M M 


H 


M 


CO 


•* lO 


o 


00 O CS Tf 


o 


Ov 


p^ r- 


^ 


w 0\ 00 I> 


t^ 


CO 


J3 

■a 




















M IH W 


"^ 


*"* 




n 


''t "^ to O 


t^ 


00 


.2 §15 o 
2 o . 4J 


T3 




o 


O 


O 


o o 


o 


o o o o 


o 


O 


o o 


o 


coco 


o 














0> M 


ro 


O ^ ^ to 




Ov 


o ^ 


-o 


o 00 c> a 








N -^ O 0\ M 


'O 


o 


o 


M »0 


o. 


C> to C\ HI 


^ 


^ 


O Cq 


'-' 


W CN O -rt 


^ 


Ov 


tn 


^2l§ 




J_( 


H 


M 


CO 


•* lO 


-o 


00 O M lO 


t^ 


o 


CO O 


r^ 


O rj- to lO 


o 


Ov 




+j 














M M H 


*~' 


<N 


c^ CO 


PO 


■^ lO O t-- 


00 


O^ 




O 


§ 


•d 


o o o o o 


o 


O 


O 


o o 


o 


o o o o 


o 


o 


o o 


r> 


o o o o 


o 


o 










w 


N 


(N 


C?\ w 


PO 


O ^ -t m 




Ov 


O M 


-n 


O 00 Oi 0\ 




m 


.Q 


d 




(M 'd- o a N 




q 


q 

CO 


M lO 

^ lo 




q to a •-. 
00 o' n' to 




o' 


O N 

PO d 


t^ 


vd ■^ to to 


o" 


Ov 


o 
o 


ffi 


< 


'o 














W M W 


^ 


pq 


Cl CO 


ro 


•^t »o o r* 


00 


Ov 




































r- 


J3 






w r- -^ fO ro 


lO 


00 


00 


n ^ 


o 


C^ a M O 


CO 


p< 


O M 


t^ 


O O oo 1-1 


t- 


■^ 






0\ o o o o 


00 


^n 






*:*• 


r^ ^ i> CO 


to 




H O 


00 


0\ 00 to M 


'■t 


vO 




3 


n 


■^ i> H lo 9; 

HI M H 




q 

CO 


't q 00 


ci 


<N 00 O l> 

« Tt t^ d 


q 
4 




M d 


o 

n 


CO O 0> N 

d d N o' 


d 


vq 


CO 


























■* 


to r- 00 Ov 




pt 


A 
































>-* 


M 


M 


_0 § c! 




M lo r- po o 


r^ 


M 


lO 


00 ro 


o 


PO t^ "N M 


w 


CO 


M fO 


to 


ro to w o> 


„ 


-o 


C 


Tl 


r- a. 00 t*3 o 


PO 


n 


00 


PO rt 




C- Tj- rt- M 




lO 


to r- 


fO 


Tj- 00 M O 




t^ 


V 


rt 


rt t^ W O N 


00 


lO 


N 


CO >0 




to Tj- O lO 




00 




o 








i^ 


(A to • +3 


D 


M M <^ 


N 


CO 


lO 


t~ Oi 


pj 


to 00 f^ o 


o 


to 


O N 


to 


O ■* CO M 


w 


^ 




n 












H 


H H c^ C^ 


CO 


PO 


■^ lO o 


00 Ov M fO 


to 


t^ 




H " -- S 


4-> 
























M H 


•^ 


M 




2-s t 




l> lO 00 PO O 


N 


N 


„ 


O. H 


PO 


O T)- Tt to 


to 


Ov 


O w 


vO 


00 Ov Ol 


M 


ro 








o 


o 


o 


H lO 


o\ 


O to O M 


Tj- 


■^ 


O M 


l-l 


C< N o -tt 


Tf 


O 






O O O O M 


t-" 


w 


fO 


■t lo 


•o 


00 O N lO 


r^ 


o 


CO O 


1> 


O '^ to to 


o 


Ov 




*- ° ^ 
















M M M 


M 


M 


W CO 


ro 


rf lo o r^ 


00 


Ov 


<1 


< ^ 




































a r^ o o o 


00 


r-_ 


« 


H to 


^ 


r- to r- 'I' 


to 


HI 


c^ O 


0\ 


O 0( to H 


^ 


so 








Tt r* H lo Oi 


^ 


O 


■* 


O 00 


o> 


P) 00 o t^ 


o 


-<5 




o 


rj- o Ov ri 


^ 


-O 








O O H H M 


N 


fO 


•<t NO t- 


o> 


IN -^ t^ O 


•et 


r^ 


M c^ 


o\ 


Ov O CN O 


o 


to 




o g 


















































M M H « 


<N 


w 


CO CO 


<* 


lO t^ 00 0\ 


^ 


0) 




iJO <« 




































M M 


« 


N 


« 


N N 


PO 


CO CO ^ -^ 


to o 


o t- 


t'. 


00 0» O H 


N 


C4 
































M 


T3 


C o 






HW 


HM 


HiPt 


«Ic* nl-* 




-^^^. ^m 




«K^ 






«W H» ■ Hlei 






t-< 




W IH 


■^ 


•^ 


^ 


H M 


« 


CI W fO PO 


PO 


fO 


CO rh 


t 


-^ t3- ift lO O 


SO 


IH ,« 




O 00 O -^ ro 


c. 


H 


o 


Ch M 


t^ 


t> o o >o 


to 


to 




•^ 


Hit* rt]« hM 


CO 


CO 




Ah C 




N M M H H 


M 


M 


M 
























° "2 

O . ™ 




»0 O r}- -^ O 


^ 


r^ 


o 


M r^ 


o 


lO O ■M- CN 


„ 


-o 


N M 


-o 


o o> o\ o 


00 


r^ 






00 -^ Ov -^ O 


to 


o 


N 


PO ro 


Tj- o O 00 00 


a 




M O 


t^ 


N N t- O 




so 






H N (N ro ■^ 


■t 


»o 


'O 


r^ 00 


o. 


O w N ro 


•^ 


vO 


r- Ov 


•-f 


-:J- O 00 M 


CO 




6 
















H M H 1-1 


M 


M 


M M 


Pt 


C^ N Cfl CO 


PO 


fO 


<! *' 
































































(5 


"o 




_K-C».HS.„ 


•R 




r.. 




H. 


.^^ ni<a .^K* w^ 


«!■* 




^ 


-k. 


«^ ^^ 


«M 
















"-I 


M 


M M M M 


t-i 


^ 


CI N 


PI 


C* PO PO CO 


CO 


^ 








C 



































o 

U 

>t 

i> 
a 

^ .S 

P* ^-|■* 
m|» 

=3 o 






o 



Q 



<3 ° 


Ov 00 r^ Tj- o 

01 C^ Cl C* IN 


Ov oO ro O 00 


t^ Ov to PO 




Exact diam- 
eter bottom 
of thread, ins. 


M o Oi Ov t- 
PO CO CO ■^ to 


0» t-. O 00 -^ 
O O 00 oo Ov 


00 CO M M 

Ov Ov M 

M M N 0) 


to 

M 01 

01 N 




No. of 

threads to 

the inch 


N O O -^ 00 
CO CO rj- N IN 


01 O '^ 00 N 

PO PO 0^ 0* CO 


00 -^ 
ro M 0* 01 


01 
oi CO 


S 

s 


nl3 n^ ;:i: a: ^t 


ng se :21s Si: 1^ 


2!s :i: X nt 


::!:::i: 


d ^ 


O O ^ •yt ^ 


01 00 CO 

MM 10 


PO to M- r^ 

10 ■^ rf -^ ro 


10 Tt 

PO PO 


Exact diam- 
eter bottom 
of thread, ins. 


■^ N 00 ro 00 
O c- t- 00 t- 


to to 

00 Ov Ov 10 

M M M 0* 


t^ 01 
to r^ 00 00 
M 


10 00 






No. of 

threads to 

the inch 


'^t 00 01 O 00 
0* o) CO PO M 


o^ ^ 


'^ 00 01 

'^ ■* '^ PO 




PO Tt 


a 

a 
Q 


HSH^HSHS Hj* 


-<- Hi^ w* -** H2 


-a-^-ii-^^ 


<< 






6 


t- O O O Ov 
lo to to to ■* 


00 M 0> w 
■^ ■^ Tj- CO PO 


-*o o> r- 

ro N 01 M 


to PO 
C< 01 


Exact diam- 
eter bottom 
of thread, ins. 


w 01 r^ 00 M 

■<t -"^ O O t- 

o o o o o 


« PO 00 

t^ Ov Ov Ov M 
M 


■* PO M -<:f 

01 01 PO ■=!- ■* 


t^ 01 

"d- to 




No. of 

threads to 

the inch 


O ■* 00 o o 
O O ** to to 


•=*■ 00 Ol 
^ -^ -^ ro 


--t 00 
CO -^ 0* O) CO 


N 

PO PO 




i 

OJ 

s 


Hh Hh "15 "h "irt 


HS Hw H« ^- HS 


■« -IS "R HS HS 


"p"fi 


t4-l 

o 


M Q :z:w:$: 


•m^^^t:^ 




^^x 


.5 

d 


r+* HS „W HS ««, 


HS^^H. ^ 




M M 01 



212 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 3. — U, S. (Sellers) Standard Screw Thread 

I 




p — pitch = 



No. threads per inch 
Formiilas -I (f = depth =^X. 6495 2 
P 



/=flat = 



Diameter o£ screw, 
ins. 


Threads per in. 


Diameter at root 
of thread, ins. 


Width of flat, 
ins. 


1 


20 


.18S 


.0062 


rs 


18 


•.2403 


. 0069 


1 


16 


.2938 


.0078 


A 


14 


• 3447 


.0089 


\ 


13 


.4001 


.0096 


■h 


12 


■4542 


.0104 


i 


II 


.5069 


.0114 


•H 


10 


• S576 


.0I2S 


1 


10 


.6201 


.0125 


•H 


9 


.6682 


.0139 


i 


9 


■ 7307 


• 0139 


•n 


8 


.7751 


.0156 


I 


8 


.8376 


.0156 


n 


7 


• 9394 


.0179 


li 


7 


I . 0644 


.0179 


ii 


6 


I. 1585 


.0208 


ij 


6 


I •283s 


.0208 


If 


5^ 


1.3888 


.0227 


ii 


S 


I ^4902 


.0250 


i5 


5 


I. 6152 


.0250 


2 


4i 


I.7113 


.0278 


2j 


4^ 


1.8363 


.0278 


2i 


4i 


1.9613 


.0278 


2| 


4 


2.0502 


• 0313 


2i 


4 


2.I7S2 


0313 


2f 


4 


2 .3002 


■ 0313 


2i 


4 


2.4252 


• 0313 


2j 


3^ 


2.5038 


.0357 


3 


3i 


2.6288 


• 0357 


3i 


3i 


2.8788 


.0357 


3J 


3i 


3-1003 


.0385 


3\ 


3 


3.3170 


.0417 


4 


3 


3.5670 


• 0417 


4i 


2j 


3.7982 


• 043S 


4i 


2i 


4.0276 


• 0455 


4J 


2| 


4.2S5r 


.0476 


5 


2\ 


4.4804 


.0500 



*The xi, jf and yf are usually made 11, 10 and 9 threads per 
inch respectively, but under the Sellers' Formula, strictly fol- 
lowed, they should be 10, 9 and 8 respectively. 



Table 4. — British Association Standard Screw Thread 



W-p-H 




Formulas 



i) = pitch 

<i = depth =/)X.6 



r = radius = 



2X^ 



No. 


Diameter, 
mm. 


Pitch, 
mm. ■ 


No. 


Diameter, 
mm. 


Pitch, 
mm. 





6.0 


1 . 00 


7 


2.5 


.48 


I 


5-3 


.90 


8 


2,2 




43 


2 


4.7 


.81 


9 


1.9 




39 


3 


4.1 


■ 73 


10 


!■? 




3S 


4 


3 64 


.66 


12 


13 




28 


5 


3.2 


.59 


14 


I .0 




23 


6 


2.8 


.53 


16 


• 79 




19 




Acme Standard Screw Thread 



/> = pitch = 



No. threads per inch 
d — depth = \p-\-.oio 
/=flat on top of thread = ^X. 3 707 
/' = flat on bottomof thread = />X. 3707- 
.0052 





No. of 


Depth 


Width 


Width at 


Space 


Thickness 


Pitch 


threads per 


of 


at top of 


bottom of 


at top of 


at root of 




inch. 


thread 


thread 


thread 


thread 


thread 


2 


1 


I. DIG 


.7414 


.7362 


1.2586 


1.2637 


ij 


15 


-9475 


.6950 


.6897' 


I. 1799 


I •1850 


ij 


t 


.8850 


.6487 


• 6435 


I . 1012 


I. 1064 


If 


13 


.8225 


.6025 


• 5973 


I .0226 


1.0277 


l^ 


1 


.7600 


.5560 


• 5508 


• 9439 


• 9491 


lA 


23 


.7287 


• 5329 


• 5277 


.9046 


• 9097 


If 


A 


.6975 


• 5097 


• 504s 


.8652 


• 8704 


lA 


n 


.6662 


.4865 


.4813 


• 8259 


.8311 


Ii 


B 


.635 


• 4633 


.4581 


.7866 


• 7918 


lA 


19 


.6037 


.4402 


• 4350 


.7472 


• 7525 


ij 


1 


.5725 


.4170 


.4118 


.7079 


• 7131 


lA 


J? 


• 5412 


.3938 


.3886 


.6686 


•6739 


I 


1 


.510 


.3707 


• 3655 


.6293 


• 634s 


^ 


lis 


.4787 


.3476 


• 3424 


.5898 


• 5950 


i 


I^ 


• 4475 


.3243 


• 3191 


.5506 


• SS58 


ii 


I ft 


.4162 


.3012 


.2960 


.5112 


• 5164 


1 


I5 


• 38s 


.2780 


.2728 


.4720 


•4772 


« 


I A 


.3537 


• 2548 


.2496 


• 4327 


•4379 


5 


n 


• 3433 


.2471 


.2419 


.4194 


• 4246 


f 


It 


• 3225 


.2316 


.2264 


• 3934 


• 3986 


A 


Is 


.2912 


.2085 


• 2033 


• 3539 


•3591 


h 


2 


.260 


• I8S3 


.1801 


• 3147 


•3199 


A 


2} 


.2287 


. 1622 


.1570 


• 2752 


• 2804 


B 


2h 


.210 


.1482 


.1430 


• 2518 


•2570 


1 


2l 


•1975 


• 1390 


.1338 


• 2359 


• 2411 


J 


3 


.1766 


.1235 


.1183 


.2098 


• 2150 


A 


3f. 


.1662 


.1158 


. 1106 


.1966 


.2018 


? 


Z\ 


.1528 


.1059 


. 1007 


.1797 


• 1849 


1 


4 


.1350 


.0927 


.0875 


.1573 


• 1625 


I 


A\ 


. I2II 


.0824 


.0772 


.1398 


• 1450 


E 


5 


. 110 


.0741 


.0689 


• 1259 


■ I311 


A 


5J 


.1037 


.0695 


.0643 


• I179 


.1232 


J 


6 


.0933 


.0617 


• 0565 


• 1049 


■ IIOI 


7 


7 


.0814 


.0530 


.0478 


. 0899 


■ 0951 


i 


8 


.0725 


.0463 


.0411 


.0787 


.0839 


i 


9 


.0655 


.0413 


.0361 


.0699 


.0751 


10 


10 


.060 


• 0371 


.0319 


.0629 


.0681 


A 


16 


.0412 


.0232 


.0180 


.0392 


• 0444 



Table 6. — ^British (Whitworth) Standard Screw Thread 




Formulas 



* =^'*''^ = No. thds. per. in. 
i = depth = PX . 64033 
»- = radiu9 = ^X .1373 



Diam. 
ins. 


No. 
thrds. 
per in. 


Diam. 
ins. 


No. 
thrds. 
per in. 


Diam. 

ins. 


No. 
thrds. 
per in. 


Diam. 
ins. 


No. 
thrds. 
per in. 


1 


20 


5 


9 


2 


4i 


3i 


3i 


A 


I8 


a 


9 


2i 


4^ 


3i 


3i 


1- 


l6 


I 


8 


2} 


4 


3i 


3J 


■u 


14 


Ii 


7 


2i 


4 


3l 


3i 


X 


12 


li 


7 


2i 


4 


3l 


3 


A 


12 


If 


6 


2f 


4 


3J 


3 


i 


II 


li 


6 


2j 


3i 


4 


3 


H 


II 


If 


5 


2j 


3i 






i 


10 


ij 


5 


3 


3i 






H 


10 


il 


4k 


3l 


3i 







BOLTS, NUTS AND SCREWS 



213 



Table 7. — Constants for Finding the Diameter at the Bottom 
OF Screw Threads of U. S. Form (Pratt & Whitney Co.) 

C = constant for number of threads per inch. 
Z? = outside diameter. 
Z)i = diameter at bottom of thread. 

D^ = D-C. 



Threads per inch 


Constant 


1 Threads per inch 


Constant 


64 


.02030 


16 


.08119 


60 


.02165 


14 


.09279 


56 


.02320 


13 


, 09993 


SO 


.02598 


12 


. 10825 


48 


.02706 


Hi 


. I I 296 


44 


.02952 


II 


.11809 


40 


.03248 


10 


.12990 


36 


.03608 


9 


• 14434 


32 


• 04059 


8 


.16238 


30 


• 04330 


7 


.18558 


28 


■ 04639 


6 


.21651 


27 


.04812 


Si 


.23619 


26 


. 04996 


5 


.25981 


24 


■05413 


4^ 


.28868 


22 


•05905 


4 


•32476 


20 


. 06495 


3i 


■3711S 


l8 


.07217 


3 


•43301 



Table 8. — International and French Metric Standard 
Screw Threads 

,U—p — J , . , 

■ \ p = pitch 



Formulas ) d = depth =^X. 64952 
/ = flat =-f 




Diameter of screw, 


Pitch. 


Diameter at root 

of thread, 

mm. 


Width of flat. 


mm. 


mm. 


mm. 


3 


■S 


2. 35 


.06 


4 


• 75 


3 03 


.09 


5 


• 75 


403 


.09 


6 


1 .0 


4^70 


• 13 


7 


1^0 


5^70 


.13 


8 


1^0 


6.70 


.13 


8 


125 


6.38 


.16 


9 


1^0 


7.70 


• 13 


9 


1.25 


7^38 


.16 


10 


I^S 


8.05 


.19 


II 


i.S 


9.05 


.19 


12 


i.S 


10.05 


.19 


12 


1. 75 


9.73 


.22 


14 


2.0 


11,40 


.25 


16 


2.0 


13.40 


.25 


18 


2.5 


14^75 


.31 


20 


2.5 


16.75 


.31 


22 


2.5 


18.75 


.31 


24 


3.0 


20. 10 


.38 


26 


3.0 


22.10 


.38 


27 


30 


23. 10 


.38 


28 


3.0 


24. 10 


.38 


30 


3.5 


25.45 


• 44 


32 


3.5 


27^45 


• 44 


33 


3.5 


28.45 


.44 


34 


3.5 


29^45 


44 


36 


4.0 


30.80 


5 


38 


4.0 


32.80 


5 


39 


4.0 


33.80 


• 5 


40 


4.0 


34 80 


•S 


42 


4.S 


36.15 


• 56 


44 


4-5 


38.15 


.56 


45 


4.S 


39-15 


• 56 


46 


4.5 


40.15 


.56 


48 


S.O 


41^51 


.63 


SO 


S.O 


43.51 


• 63 


52 


5.0 


45 51 


• 63 


56 


5.S 


48.86 


• 69 


60 


$•5 


52.86 


.69 




^BBefers fo 

Borew 

J)Ta Diameter 

d'ClameUi' 



Flat Top 



Table 9. — S. A. E. Screw Standard 

Dimensions. — All dimensions in inches . 

Finish. — All heads and nuts to be semi-finish. 

Material. — For all screws and nuts — steel; tensile strength, not 
less than 100,000 lbs. per square inch; elastic limit, not less than 
60,000 lbs. per square inch. 

Screws are to be left soft. Screw heads are to be left soft. The 
plain nuts are to be left soft. The castle nuts are to be case-hardened. 

It is assumed that where screws are to he used in soft material, such 
as cast-iron, brass, bronze or aluminum, the United States standard 
pitches will he used. 

Tolerance. — The body diameter of the screws shall be one-thou- 
sandth (.001") inch less than the nominal diameter, with a plus 
tolerance of zero and a minus tolerance of two-thousandths (.002") 
inch. 

The nuts shall be a good fit without perceptible shake. 

The tap shall be between two-thousandths (.002") inch and 
three-thousandths (.003") inch large. 



D 


i 


A 


f 


A 


i 


A 


1 


a 


i 


i 


I 


Ii 


Ii 


If 


Ii 


P 


28 


24 


24 


20 


20 


18 


18 


16 


16 


14 


14 


12 


12 


12 


12 


A 


A 


a 


if 


1^- 


A 


fl 


a 


1* 


H 


ft 


I 


lA 


Ii 


lif 


Ii 


Ai 


^ 


H 


ii 


t 


A 


a 


tt 


a 


M 


a 


1 


fj 


lA 


lii 


lA 


B 


A 


h 


A 


f 


i 


i 


a 


I 


lA 


Ii 


lA 


If 


iH 


2 


2A 


C 


A 


A 


1 


i 


A 


A 


1 


i 


i 


1 


•1 


A 


A 


1 


1 


E 


A 


A 


i 


i 


i 


A 


A 


A 


A 


A 


A 


A 


A 


i 


i 


H 


A 


a 


A 


a 


i 


H 


a 


a 


A 


H 


1 


a 


a 


lA 


Ii 


I 


A 


A 


i 


i 


i 


i 


1 


i 


i 


i 


i 


A 


A 


i 


i 


K 


A 


A 


A 


A 


A 


A 


A 


A 


A 


A 


A 


A 


A 


A 


A 


d 


A 


A 


A 


A 


A 


i 


i 


i 


i 


i 


i 


ii 


ii 


a 


ii 



Sizes of Taps 


1 
4 


in.X28 threads 


A 


in. X 24 threads 


1 


n. X 24 threads 


^ 


in. X 20 threads 


i 


in. X 20 threads 


9 

16 


in.XiS threads 


f 


m.Xi8 threads 


H 


m.Xi6 threads 


1 


in.Xi6 threads 


1 


in. X 14 threads 


I 


in.Xi4 threads 


li 


m.Xi2 threads 


Ii 


in. X12 threads 


If 


in.Xi2 threads 


li 


in.Xi2 threads 



Drill Sizes 
j2 in.* 
Hin. 
liin. 
f in. 
A in- 



If in. 
If in. 

Ai in 
6 4 in- 

If in- 

If in. 

iTf in. 

i^in. 

iHin. 

iff in. 



' No. 5 Drill gage. 



.214 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table io.- 


-A. S. M. E. Standard Machine Screws, U. 


S. Standard 


Table 


II. — A. S. M. E. Standard Special Screws, U. S. Standard 


^-^t^ 


Form oi 

— 4 


" Thread 


I 






Form of Thread 




nulas ■ 


/' = pit( 


->i 


■ Basic size 


Outside diam. 


Pitch diam. 


Root diam. 


^^60 


'^ No. 


thds. per in. 
.64952 


No. 


O.D.-T.P.I. 


Min. 


Max. 


Min. 


Max. 


Min. 


Max. 


,^^fcv^ 


d = deptii = px 


I 


.073-64 


.0698 


• 0730 


.0612 


. 0629 


.0494 


• 0527 


^^^ 


^^ 






/=flat = | 




2 
3 

4 

5 


. 086-56 
. 099-48 
.1:2-40 
.112-36 

.125-40 


.0825 
.0952 
.1078 
.1076 

.1208 


.0860 

• 0990 
. 1120 
. 1120 

• 1250 


.0727 
.0836 
.0937 
.0918 

. 1067 




0744 
0855 
0958 
0940 

1088 


.0591 
.0678 
.0747 
.0707 

.0877 


• 0628 

• 0719 


Basic size 


Outside diam. 


Pitch diam. 


Root diam. 


• 0795 
.0759 


No. 


O.D.-T.P.I. 


Min. 


Max. 


Min. 


Max. 


Min. 


Max. 





. 060-80 


.0572 


.0600 


.0505 


.0519 


.0410 


.0438 


• 092s 


I 


.073-72 


.0700 


.0730 


.0625 


.0640 


.0520 


.0550 




.125-36 


.1206 


• 1250 


. 1048 




1070 


.0837 


.0889 


2 


. 086-64 


.0828 


.0860 


■ 0742 


.0759 


.0624 


.0657 


6 


.138-36 


■ 1336 


• 1380 


.1178 




1200 


.0967 


. 1019 


3 


. 099-56 


• 0955 


.0990 


.0857 


.0874 


.0721 


.0758 




•138-32 


• 1333 


• 1380 


• 1154 




1177 


.0917 


• 0974 


4 


.112-48 


. 1082 


. 1120 


.0966 


.0985 


.0808 


.0849 


7 


•151-32 


.1463 


.1510 


■ . 1284 




1307 


.1047 


. 1104 


5 
6 

7 
8 
9 


.125-44 
. 138-40 
.151-36 
.164-36 
.177-32 


. 1210 
.1338 
. 1466 
.1596 
.1723 


.1250 
.1380 
. isio 
.1640 
• 1770 


.1082 
.1197 
.1308 
.1438 
.1544 


.1102 
.1218 
■ 1330 
. 1460 
.1567 


.0910 
.1007 
.1097 
.1227 
.1307 


.0955 
.1055 
.1149 
.1279 
.1364 


8 
9 


.151-30 
. 164-32 
. 164-30 
.177-30 
•177-24 


. 1462 

• 1593 

• 1592 

• 1722 
.1718 


. 1510 
. 1640 
.1640 

.1770 
• 1770 


.1269 

• 1414 

• 1399 
.1529 

• 1473 




1294 
1437 
1423 
1553 
1499 


. 1017 
.1177 
.1147 
.1277 
.1158 


.1077 

• 1234 
.1207 

• 1337 
.1229 


10 

12 

14 
l6 
i8 


. 190-30 
.216-28 
.242-24 
.268-22 
.294-20 


.1852 
.2111 
.2368 
.2626 
.2884 


. 1900 
.2160 
.2420 
.2680 
.2940 


. 1660 

• 1903 
.2123 
.2358 

• 2587 


.1684 
.1928 
.2149 
• 2385 
.2615 


.1407 
.1633 
.1807 
.2013 
.2208 


.1467 
. 1696 
.1879 
. 2090 
.2290 


10 

12 
14 
16 


.190-32 
.190-24 
.216-24 
.242-20 
.268-20 


•I8S3 
.1848 
.2108 
.2364 
.2624 


. 1900 
• 1900 
.2160 
. 2420 
.2680 


.1674 
. 1603 
.1863 
.2067 
• 2327 




1697 
1629 
1889 
2095 
2355 


• 1437 
.1287 
.1547 
.1688 
.1948 


.1494 
• I3S9 
.1619 
.1770 
.2030 


20 
22 
24 
26 
28 


.320-20 
.346-18 
.372-16 
.398-16 

.424-14 


.3144 
• 3402 
.3660 
.3920 
.4178 


.3200 
.3460 
.3720 
.3980 
.4240 


.2847 
.3070 
.3284 
.3544 
.3745 


.2875 
.3099 
.3314 
.3574 
■ 3776 


.2468 
.2649 
.2810 
• 3070 
.3204 


.2550 
.2738 
.2908 
.3168 
.3312 


18 
20 

22 

24 
26 

28 


.294-18 
.320-18 
.346-16 
.372-18 
.398-14 

.424-16 


.2882 

• 3142 

• 3400 

• 3662 

• 3918 

.4180 


• 2940 

• 3200 
.3460 

• 3720 

• 3980 

• 4240 


• 2550 
.2810 
.3024 

• 3330 

• 3485 

• 3804 




2579 
2839 
3054 
3359 
3516 

3834 


.2129 
.2389 

• 2550 
.2909 

• 2944 

• 3330 


.2218 

.2478 
.2648 
.2998 
.3052 

.3428 


30 


.450-14 


.4438 


.4500 


.4005 


.4036 


• 3464 


.3572 


30 


.450-16 


.4440 


.4500 


.4064 




4094 


• 3590 


.3688 



Note. — Maximum sizes are standard. 

There is a fairly widespread feeling that the differences between 

Table 12. — ^Tap Drills For Machine Screw Taps 
(Pratt & Whitney Co.) 



Note. — Maximum sizes are standard. 
the maximum and minimum sizes of the above tables are too large. 

Table 13. — ^Tap Drills for A. S. M. E. 
Standard Machine Screw Taps 



Size of 


No. of 


Size of 


Size of 


No. of 


Size of 


tap 


threads 


drill 


tap 


threads 


drill 


2 


48 


51 


12 


24 


19 


2 


56 


50 


13 


20 


19 


2 


64 


49 


13 


24 


15 


3 


40 


49 


14 


20 


16 


3 


48 


48 


14 


22 


13 


3 , 


56 


44 


14 


24 


9 


4 


32 


48 


15 


18 


13 


4 


36 


45 


IS 


20 


10 


4 


40 


44 


15 


24 


6 


5 


30 


44 


16 


16 


13 


S 


32 


43 


16 


18 


10 


s 


36 


41 


16 


20 


6 


5 


40 


40 


16 


24 


2 


6 


30 


41 


17 


16 


7 


6 


32 


37 


17 


18 


4 


6 


36 


36 


17 


20 


2 


6 


40 


33 


18 


16 


3 


7 


28 


35 


18 


18 


2 


7 


30 


34 


18 


20 


A 


7 


32 


31 


19 


16 


1 


8 


24 


34 


19 


18 


B 


8 


30 


30 


19 


20 


D 


8 


32 


30 


20 


16 


C 


9 


24 


30 


20 


18 


E 


9 


28 


29 


20 


20 


H 


9 


30 


28 


22 


16 


H 


9 


32 


27 


22 


18 


J 


10 


24 


28 


24 


14 


K 


10 


28 


26 


24 


16 


L 


10 


30 


24 


24 


18 


N 


10 


32 


24 


26 


14 


N 


11 


24 


24 


26 


16 





11 


28 


21 


28 


14 





11 


30 


19 


28 


16 


s 


12 


20 


24 


30 


14 


T 


12 


22 


20 


30 


16 


V 





(Pratt & Whitney Co.) 






Size of 


No. of 


Size of 


Size of 


No. of 


Size of 


tap 


threads drill 


tap 


threads 


drill 





80 


.0465 


9 


32 


• 1405 


I 


64 


• 055 


10 


24 


• 140 


I 


72 


.0595 


10 


30 


• 152 


2 


56 


.0670 


10 


32 


• 154 


2 


64 


.070 


12 


24 


• 166 


3 


48 


.076 


12 


28 


• 173 


3 


56 


• 0785 


14 


20 


• 182 


4 


36 


.080 


14 


24 


• 1935 


4 


40 


.082 


16 


20 


. 209 


4 


48 


.089 


16 


22 


.213 


S 


36 


• 0935 


18 


18 


.228 


S 


40 


.098 


18 


20 


.234 


S 


44 


.0995 


20 


18 


• 257 


6 


32 


.1015 


20 


20 


• 261 


6 


36 


.1065 


22 


16 


• 272 


6 


40 


. no 


22 


18 


• 281 


7 


30 


.113 


24 


16 


-295 


7 


32 


.116 


24 


18 


• 302 


7 


36 


.120 


26 


14 


.316 


8 


30 


.1285 


26 


16 


• 323 


8 


32 


.1285 


28 


14 


• 339 


8 


36 


.136 


28 


16 


• 348 


9 


24 


.1285 


30 


14 


• 368 


9 


30 


• 136 


30 


16 


• 377 



The diameter given for each hole to be tapped allows for a practical clear- 
ance at the root of the thread of the screw and will not impose undue 
strain upon the tap in service. 



These drills will give a thread near enough for all practical purposes, but 
not a full thread. 



BOLTS, NUTS AND SCREWS 



215 



Table 14. — A. S. M. E. Standard Proportions of Machine Screw Heads 



KD- 



WSI 






Oval Fillister Head Screws 
A — diameter of body 

B = 1.64/4 — . 009 = diarn. of head and rad. for oval 
C = . 66A — . 002 = height of side 
Z)= .173A + .01S 
£ = jF = depth of slot 
F = . 1 34B + C = height of head 



D I 



A 


B 


C 


D 


E 


F 


.060 


.0894 


.0376 


.025 


.025 


.0496 


• 073 


. 1107 


.0461 


.028 


.030 


.0609 


.086 


• 132 


.0548 


.030 


.036 


.0725 


• 099 


• 153 


• 0633 


.032 


.042 


.0838 


. 112 


■ 1747 


.0719 


.034 


.048 


.0953 


.12.'; 


.196 


.0805 


.037 


• 053 


.1068 


.138 


.217 


.089 


.039 


• 059 


.1180 


.151 


.2386 


.0976 


.041 


.065 


. 1296 


. 164 


.2599 


. 1062 


• 043 


.071 


. 1410 


.177 


.2813 


.1148 


.046 


.076 


.1524 


. 190 


.3026 


• 1234 


.048 


.082 


• 1639 


.216 


• 3452 


.1405 


.052 


-093 


.1868 


.242 


• 3879 


.1577 


.057 


.105 


.2097 


.268 


.4305 


.1748 


.061 


. 116 


.2325 


.294 


.4731 


.192 


.066 


.128 


• 2554 


.320 


.5158 


.2092 


.070 


. 140 


.2783 


•346 


.5584 


.2263 


.075 


.150 


.3011 


• 372 


.601 


• 243s 


.079 


.162 


.3240 


.398 


.6437 


.2606 


.084 


• 173 


• 3469 


•424 


.6863 


.2778 


.088 


.185 


.3698 


■ 450 


.727 


• 295 


.093 


. 201 


.4024 



Flat Fillister Head Screws 
A = diam. of body 
B = 1.64A — .009 = diam. of head 
C = .66 A — . 002 = height of head 
D= . 1734 + .CIS = width of slot 
£; = iC = depth of slot 



A 


B 


c 


D 


E 


.060 


.0894 


.0376 


• 025 


.019 


■ 073 


. 1107 


.0461 


.028 


.023 


.086 


.132 


.0548 


.030 


.027 


.099 


• 153 


.0633 


• 032 


.032 


.112 


.1747 


.0719 


.034 


.036 


.125 


. 196 


.0805 


• 037 


.040 


.138 


.217 


.0890 


039 


.044 


.151 


.2386 


.0976 


.041 


.049 


.164 


• 2599 


. 1062 


• 043 


■ 053 


.177 


.2813 


. 1 148 


..046 


.057 


.190 


.3026 


.1234 


.048 


.062 


.216 


.3452 


.1405 


• 052 


.070 


.242 


.3879 


• 1577 


• 057 


.079 


.268 


•4305 


• 1748 


.061 


.087 


.294 


• 4731 


. 1920 


.066 


.096 


.320 


• 5158 


.2092 


.070 


. 104 


■.346 


• SS84 


.2263 


.075 


.113 


.372 


.601 


.2435 


.079 


.122 


.398 


.6437 


. 2606 


.084 


.130 


.424 


.6863 


.2778 


.088 


.139 


.450 


.727 


• 295 


.093 


•147 



^^h 



,.-82 Deg^, 

—in 



E 



■A-* 



> 



Flat Head Screws 
A = diameter of body 
B = 2A— .008 = diameter o f head 
A -.008 
1.739 

D= . 173A + .015 = width of slot 
£ = 3C = depth of slot 



= depth of head 




Round Head Screws 
A =diam. of body 
B = 1 . 85A — . 005 = diam. of head 
C = . 7A = height of head 
D= . 173A + .015 = width of slot 
E = \C-\- .01 = depth of slot 



A 


B 


C 


D 


£ 


.060 


. 112 


.029 


.025 


.010 


• 073 


.138 


• 037 


.028 


.012 


.086 


. 164 


■ 045 


.030 


.015 


.099 


.190 


.0S2 


.032 


.017 


. 112 


.216 


. 060 


.034 


. 020 


.125 


.242 


.067 


• 037 


.022 


.138 


.262 


.075 


.039 


.025 


.151 


.294 


.082 


.041 


.027 


.164 


.320 


.090 


• 043 


.030 


.177 


• 346 


.097 


.046 


.032 


. 190 


• 372 


.105 


.048 


■ 035 


.216 


• 424 


. 120 


.052 


.040 


.242 


• 472 


• 135 


.057 


.045 


.268 


.528 


.150 


.061 


.050 


.294 


.580 


.164 


.066 


.055 


.320 


.632 


.179 


.070 


.060 


.346 


.682 


• 194 


.075 


.065 


.372 


.732 


.209 


• 079 


.070 


.398 


.788 


.224 


.084 


.075 


.424 


.840 


.239 


.088 


.080 


.450 


.892 


.254 


.093 


.085 



A 


B 


C 


D 


E 


.060 


.106 


.042 


.025 


.031 


.073 


.130 


.051 


.028 


.035 


.086 


.154 


.060 


.030 


.040 


-099 


.178 


.069 


.032 


.044 


.112 


.202 


.078 


.034 


.049 


125 


.226 


.087 


.037 


.053 


.138 


.250 


.096 


.039 


.058 


.151 


.274 


.105 


.041 


.062 


.164 


.298 


.114 


• 043 


.067 


.177 


.322 


.123 


.046 


.071 


.190 


• 346 


.133 


.048 


.076 


.216 


.394 


.151 


.052 


.085 


.242 


•443 


. 169 


• 057 


.094 


.268 


.491 


.187 


.061 


.103 


.294 


.539 


.205 


.066 


. 112 


.320 


.587 


.224 


.070 


. 122 


.346 


.635 


.242 


■ 075 


.131 


.372 


.683 


.260 


.079 


.140 


.398 


■731 


.278 


.084 


■ 149 


.424 


.779 


.296 


.088 


.158 


• 450 


.827 


.315 


.093 


.167 



216 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 15. — S. A. E. Lock Washer Standards 

AUTOMOBILE HEAVY (a. H.) 

For General Use 

Temper. — After compression to flat, reaction shall be sufficient 
to indicate necessary spring power, and on a subsequent compression 
to flat the lock washer shall manifest no appreciable loss in reaction. 

Toughness. — Forty-five per cent, of the lock washer, including 
one end, shall be firmly secured in a vise, and 45 per cent., 
including the other end, shall be secured firmly between parallel 
jaws of a wrench. Movement of the wrench at right angle to 
helical curve shall twist the lock washer through 45 deg. 
without sign of fracture; and shall twist the lock washer entirely 
apart within 135 deg. 

The outside diameters of lock washers shall coincide practically 
with the long diameters of S. A. E. Standard nuts, which are ap- 
proximately the short diameters of United States Standard nuts. 

The inside diameters of the lock washers shall be from ^ in. to 
jj in. larger than bolt diameters. 

All lock washers shall be parallel-faced sections; and bulging or 
malformed ends must be avoided. 



Table 16. — Tensile and Shearing Strengths of S. A. E. Standard 
Bolts and Nuts. By Joseph A. Anglada 



Bolt diameter 



A in. 

i in. 
A in. 

fin. 
j^in. 



TS m. 



Lock washer 
section 



n-Xre in- 
;n.X^ in. 
in. X 5 in. 
!n. X 5 in. 
n.XH in. 



|j in.Xii in. 
H in-XHin. 



Bolt diameter 



Hin. 

f in. 

I in. 
I in. 
li in. 

i\ in. 
if in. 
i4 in. 



Lock washer 
section 



in.Xi in. 
in.Xj in. 
in.XHin. 
in.XA in- 
in.XA in. 

n.Xf in. 
n. X I in. 
n.Xire in. 



automobile light (a. l.) 
For Optional use Against Soft Metal 



Bolt diameter 


Lock washer 


section 


^ 


in. 


i^in.XA 


in. 


1 


in. 


^in-Xi^j 


in. 


A 


in. 


*in.X^ 


in. 


3 

8 


in. 


iin.X^ 


in. 


A 


in. 


Hin-X i 


in. 


4 


in. 


liin.X i 


in. 


9 

16 


in. 


4f in.XA 


in. 


f 


in. 


Hin.XA 


in. 


ii 


in. 


\in.X^ 


in. 


1 


in. 


iin.XA 


in. 


1 


in. 


Hin.XA 


in. 


I 


in. 


Ain.X i 


in. 



Case II. — Elastic Block and Non-elastic Bolt. — Let the block be 
now represented by a spring, as in Fig. 4. As before, let the nut be 
screwed up to produce an initial strain of 5000 lbs., and then let the 
stirrup and weight be added. Obviously the bolt is now loaded with 
a strain of 10,000 lbs., because, unlike the first case, the second 
load has in no way affected the first. 

In actual cases the situation is more involved, as both block and 
bolt are elastic, and the question becomes one of difference of elas- 
ticity; but the conditions and the final effect vary between the two 
cases shown as extremes. It is sometimes possible, but more often 
impossible, to say which extreme is most nearly approximated, but 
when the whole matter hinges on such obscure conditions, the only 
safe course is the conservative one, to regard the initial load as part 
of the final strain. 



Bolt 


Areas 


Tensile strength 


Shearing strength 






■S 






o* 


cV 


d" 


Full 


Bolt 


Bottom of 


U 


J3 


u 


a 




•0 


u 
Q. 




(U 

p. 






thread 










M 




Z 




.a 

4J 


w 


z 


J3 


a 6 


^d 






C 

"6 



ft 
w 
•d 


.2 *^ 


4J 






a 








d . 




q 

M C 






d . 


. 

§ ^ 

M (1) 


. 


"" 

" ID 


•" 
0. 


■A 


H 


Q 


pq 


m 


< 


< 


<; 


< 


< , 


< 


< 


l- 


28 


.2037 


.0491 


• 0325 


651 


814 


977 


737 


1, 105 


488 


731 


A 


24 


.2584 


.0767 


.0525 


1,050 


1,312 


I.S7S 


1,151 


1,726 


788 


1,181 


\ 


24 


.3209 


.1104 


.0808 


1,617 


2,021 


2,426 


1,656 


2,484 


1,212 


1, 818 


A 


20 


.3626 


• 1503 


. 1132 


2,264 


2,830 


3.396 


2,255 


3.382 


1,698 


2,547 


h 


20 


.4351 


• 1963 


. i486 


2,972 


3.726 


4.459 


2,945 


4.417 


2,229 


3.344 


A 


18 


4904 


.2485 


.1888 


3.777 


4.722 


S.666 


3.728 


5,591 


2,832 


4.248 


1 


18 


• 5529 


.3068 


.2400 


4,800 


6,000 


7.200 


4,602 


6,903 


3,600 


5.400 


ii 


16 


.6064 


.3712 


.2888 


S.776 


7,220 


8,664 


5,568 


8,352 


4.332 


6,198 


\ 


16 


.6689 


.4418 


.3514 


7.028 


8,78s 


10,542 


6,627 


9.941 


5.271 


7,907 


i 


14 


.7823 


.6013 


.4816 


9,633 


12,141 


14,449 


9,020 


13,529 


7,224 


10,836 


I 


14 


■ 9073 


.7834 


.6463 


12,926 


i6,is8 


19.389 


11,781 


17.672 


9,695 


14.542 



Table 17. — Tap Drills for Standard Pipe Taps 
(No Reamers Required) 



Nominal 


Thds. per 


Dbl. depth 


Tap drill 


Outside 


size 


in. 


of th'd 


diameter 






inches 






i 


27 


.048 


H 


■ 405 


i 


18 


.072 


n 


.540 


i 


18 


.072 


« 


.675 


i 


14 


.093 


*i 


.840 


i 


14 


093 


n 


1 .050 


I 


Hi 


.113 


lA 


I.3IS 


il 


Hi 


.113 


i« 


1.660 


i^ 


Hi 


.113 


Iff 


1 .900 


2 


Hi 


.113 


2i 


2.375 


25 


8 


.162 


2li 


2.875 


3 


8 


.162 


3A 


3 . 500 


3h 


8 


.162 


3K 


4.000 


4 


8 


.162 


4A 


4. 500 


4i 


8 


.162 


4i4 


5. 000 


S 


8 


.162 


Si 


5. 563 


6 


8 


.162 


6A 


6.62s 


7 


8 


. 162 


7A 


- 7.62s 


8 


8 


.162 


8A 


8.62s 


9 


8 


.162 


9A 


9.62s 


10 


8 


.162 


loA 


10.750 


II 


8 


, 162 


iiH 


12.000 


12 


8 


.162 


12H 


13.000 




6000 lbs. 



5000 lbs. 



Fig. 3. 



Fig. 4. 



Figs. 3 and 4.— Resultant stress on bolts due to initial and applied 

loads. 



BOLTS, NUTS AND SCREWS 



217 



^ 



Taper Bolts 

The taper bolt system oj securing parts together, universally used 
for the exacting conditions of locomotive work, deserves wider use 
in other fields than it has received. Following are the particulars 
of the Baldwin Locomotive Works standard {Amer. Much., May 22, 
1902) which has been in entirely satisfactory use since 1884: 

All bolts in this system are fitted in reamed gages made of cast- 
iron and of suitable length and section; the gages are known as 9, 
12, 18, 24 and 30-in. blocks. A bolt 9 ins. in length — measured from 
under the head to the point— is taken as a starting point. This 
bolt (shown in Fig. 5 at C) is exactly i in. in diameter at the point, 
and, consequently, at a taper of xs in- per foot it is i^ ias. in 
diameter under the head. 



Gag:e Collar 




Fig. s. — The Baldwin Locomotive Works standard taper bolt 

system. 



The diameter at the small end and the angle of taper being the 
same, the amount of taper for the 12-in. bolt is ys, for the i8-in. 
bolt ^, for the 24-in. bolt, i, and for the 30-in. bolt, -j^ in. The 
large end of the hole of the various blocks has the following 
diameters: 

Length Diameter 

9 ins. . i^ ins. 

1 2 ins I P5 ins. 

18 ins ifl ins. 

24 ins 15 ins. 

30 ins i^ ins. 

All bolts 9 ins. and under in length — as B and C — are fitted in 
the 9-in. block, all from 9 to 12 ins. in the iz-in. block, all from 12 
to 18 ins. in the i8-in. block, all from 18 to 24 ins. in the 24-in. 
block, all from 24 to 30 ins. in the 30-in. block. 



A reamer of length suitable to ream a hole for a 30-in. bolt would 
answer for any bolt of lesser length, but would be too clumsy. 

In practice it is found more convenient to have reamers for each 
gage division, and these are known as 9, 12, 18, 24 and 30-in. hand 
and machine reamers. The flutes of all reamers are made long 
enough to allow for 3 ins. wear. 

Gage collars, as shown at A , reamed and counterbored, are driven 
on the upper part of the flutes and under shank or head, and coming 
exactly to the top of block when the reamer is inserted, insure a 
hole in the work the same size as the hole in the block. 

All holes being reamed standard, the allowance for snug driving 
fits is made by fitting aU bolts to stand out of the gage blocks ys 
in., which has been found sufiicient. 

A taper of more than ys i^- P^"^ ^t. offers no advantage, but has 
the fault of making a long bolt too large under the head. 

Thus far only the hexagon and square head bolts have been con- 
sidered. Two other kinds are used, the countersunk and the round 
counterbore head bolts. The countersunk head bolt, shown at F 
and G, is used in places where a hexagon head would interfere. The 
included angle of this head is 60 deg. and the head is | in. thick. 
This style of bolt requires a gage block, E, so made that the standard 
plug gage will stand out f in. The counterbore head bolt is used 
where a very strong concealed head is desired, the head usually 
driving snugly in the counterbore. This style bolt is fitted in the 
hexagon head-bolt gage. The hole for this bolt body when reamed 
is made the size of the bolt under the head. 

The same reamer is used for all bolts, and the same allowance 
for drive, viz., ^g in., is made for all styles. The blocks, as shown 
in the illustration, have cast on the side a descriptive shape which 
aids the workman in finding the one desired, whether hexagon or 
countersunk. 

This system recommends itself in that it contains but few stan- 
dards for each nominal diameter of bolt and provides for a multitude 
of lengths. 

To preserve the sizes, a set of master plugs is kept in the tool- 
room. When the gage blocks are worn they are easily restored to 
standard by re-reaming and facing off the top to suit the plug gage. 

The gage blocks for regular diameters have the following lengths: 



Diameter, 
ins. 



Length, 
ins. 



5} 

s J 



.9 and 12 
.9, 12 and 18 



.9, 12, 18, 24 and 30 



Split Nuts 

The interference oj split nuts with lead screws may be determined 
by the method shown in Fig. 6 by H. S. Fullerton {Mchy. June, 
1912). Parallel with the parting line of the half nuts, draw the line 
AB at a distance K from the center hne such that: 

; 

^ = -^ 7, 

2^ tan 

in which, / = lead of thread, ins., 

^ = angle between side of thread and a perpendicular to 
the axis of the screw. 



218 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



For the Acme thread, in which ^ = 14^ deg., the formula becomes: 

13 

Tangent to the outside and root diameters, draw perpendiculars 
cutting AB at m and n. Draw radial lines to these points cutting 
the outside and root diameters, respectively, at r and s. These are 
the interference points of the respective diameters. Points for in- 
termediate diameters may be located in like manner and a curve 
drawn through them. For all practical purposes a straight line 
drawn through the interference points located on the outside and root 
diameters will be found a sufficiently close approximation. The 
illustration shows in correct proportions an end elevation of a pair 
of nuts for a screw 4 ins. in diameter with i-in. lead single Acme 
threads. The dot-and-dash lines above are the interference curves 
for 2-, 3-, and 4-in. lead Acme threads. 




Fig. 6. — Finding the interference between screws and split nuts. 



also shows that the load produced may be estimated at 16,000 lbs. 
per in. of diameter of bolt, or 

P = 1 6,000c? (a) 

in which P = load due to screwing up, lbs., 

d = nominal (outside) diameter of screw thread, ins. 

This value of P is rather above the average for the tests; but it is 
considerably below the maximum, and it is probably not in excess of 
the load which may reasonably be expected in making a tight joint. 

If P be divided by the cross-sectional area at the root of the thread, 
the intensity of the stress induced is obtained. The diameter at the 
root of the thread of small screws is about o.8d. Dividing equation 

(a) through by , we obtam approximately 

4 

30,000 

in which /> = stress at root of thread, lbs. per sq. in. 

This equation gives a stress of 60,000 lbs. per sq. in. on a ^-in. 
bolt. This conclusion is substantiated by the fact that steel bolts 
of this size were broken during the course of the experiment and it 
also agrees with common experience which forbids the use of screws 
as small as H in- for cases requiring the nuts to be screwed up hard. 

Experiments by J. F. Hobart {Atner. Mack, Nov. 12, igi4) with 
J-^-in. bolts having twelve and thirteen threads, showed that, when 
not lubricated, the friction of the screw and nut (including the face 
friction of the nut) consumed almost exactly 90 per cent, of the 
force applied at the wrench. 

Set Screws 

The safe holding power of set screws according to experiments by 
B. H. D. Pinkney {Amer. Mach., Oct. 15, 1914) is given in Table 18. 
Flat or cup point screws and a flatted shaft are assumed. 



The stresses on holts due to tightening their nuts were experimentally 
investigated by Prof. J. H. Baer. {Amer. Mach., Dec. 17, 1914). 

The sizes of bolts used were 3^, %, i and ij^ ins. One set of 
experiments was made with rough nuts and washers, and another set 
with the nuts and their seats on the washers faced ofi. A bolt was 
placed in a testing machine so that the axial force upon it could be 
weighed after it was screwed up. Each of twelve experienced me- 
chanics was asked to select his own wrench and then to screw up the 
nut as if making a steam-tight joint and the resulting load on the bolt 



Table 18. — Safe Holding Power of Set Screws, Lbs. 



Diameter of set 


Safe holding 


Diameter of 


Safe holding 


screw, ms. 


power, lbs. 


set screw, ins. 


power, lbs. 


H 


100 


% 


850 


Ke 


170 . 


M 


1300 


% 


250 


% 


1800 


Vie 


370 


I 


2500 ■ 


Yi 


500 


1% 


330° 


Me 


650 


iH 


4200 








Fig. 7. Fig. 8. Fig. 9. 

Figs. 7 to 9. — The wire system of measuring screw threads. 




m 



was weighed. Each man repeated the test three times for every size 
of bolt, and each had a helper on the i-in. and i}/i-m. sizes. The 
sizes of wrenches used were 10 or 12 in. on the ^-in. bolts up to 18 
and 22 in. on the i}-i-m. bolts. The results were rather discordant, 
as would be expected; but the loads in the different tests were rather 
more uniform, as well as higher, with the faced nuts and washers. 
The general result indicates that the initial load due to screwing up for 
a tight joint varies about as the diameter of the bolt; that is, a ma- 
chanic will graduate the pull on the wrench in about that ratio. It 



The Wire System of Measuring Screw Threads 

The wire system of measuring screw threads has been worked out 
in the form of Tables 19 and 20 by Walter Cantello {Amer. Mach., 
June 25, 1903). A caution is needed in connection with the method 
shown at x. Fig. 7, which implies that the diameter across the flats 
is correct and that the flats are concentric with the thread sides. 
This concentricity may be tested by measuring as at x and on 
several diameters. 



BOLTS, NUTS AND SCREWS 



219 



Table 19. — Measuring Screw Threads by the Wire System 



|3 = distance from apex of thread angle at root, to center of wire. (See 
Fig. 7.) 
D2 = diameter of cylinder touched by apexes of thread angles. (Fig. 7-) 
X = diameter from top of threads on one side of tap or bolt, to top of wire 

laid in thread groove on opposite side. (Fig. 7.) 
«i = diameter when wires are used as shown in Fig. 8. 
*j = diameter when wires are used as shown in Fig. 9. 

U. S. Standard Thread 

* = lead = -, for single threads. 
n 

.64.05 
d = ^X.649S = ^ 



/=! 

o = from p, max.; to i) X .505, miu. 
^ = — -^sin 30° 

2 2 ' "^ 2 
«i = 7)2 + 2/3+a. 

»;2= •\/(D2 + 2«»+('-^)' + a. 



Let D = outside diameter of thread. 

D\ =root diameter measured in thread groove. 
n = number of threads per inch of length. 
d = depth of thread. 

p = distance from center to center of adjacent threads. 
/= width of fiat on U. S. Standard thread. 
r= radius on Whitworth thread. 
a = diameter of wire used. 

Whitworth Thread 

* = lead = -, for single threads. 
n 

.64033 




V 2 ) 
r=-PX-i373- 
o = ?X.84, max.; to ;^X-4S4, min. 



/3 = — -f-sm 27° 30' = 

„ „ 1.60082s 
D2 = D 

B 

D Di a 

X 1 f-/3+— 

2 2 ' '^ ' 2 

JCi = D2+2/3 + a. 



923 s 



V/lead\ 2 
(D2 + 2^)2+(^-) +0. 



D, 


K 


Oi 


D2 


a. 


/3. 


(f*)' 


X 


XI 


Xi 


ins. 








ins. 


ins. 








H 


20 


. 1867 


. 1742 


.04 


.04 


.000625 


.2721 


.2942 


.29SS 


«6 


18 


.2419 


.2283 


.04 


.04 


.000771 


.3304 


.3483 


■ 3495 


?i 


16 


.2954 


.2803 


.04 


.04 


.000976 


.3876 


.4003 


.4016 


Mo 


14 


.346s 


.3292 


.04 


.04 


.001274 


• 4433 


• 4492 


.4507 


\'i 


13 


.4019 


.3834 


.06 


.06 


. 00 1479 


.5317 


.5634 


■ S647 


91 6 


12 


.4561 


.4362 


.06 • 


.06 


-001735 


.5893 


.6162 


.6177 


Ji 


1 I 


.5089 


.4872 


.06 


.06 


.002065 


.6461 


.6672 


.6681 


iH« 


H 


• 5712 


.5497 


.06 


.06 


.00206s 


.7086 


.7297 


.7312 


Vi 


10 


.622 I 


-5984 


.06 


.06 


.002500 


■ 7643 


.7784 


.7801 


iMe 


10 


.6844 


-6609 


-06 


-06 


.002500 


.8267 


.8409 


.8425 


■'A 


9 


.7327 


.7066 


. 10 


. 10 


. 003086 


.9408 


I . 0066 


1.0083 


'Ms 


9 


.7950 


-7691 


. 10 


. 10 


.003086 


1.0033 


1 . 069 I 


1.0706 


I 


8 


.8399 


.8105 


. 10 


. 10 


.003906 


I-0SS3 


1. 1105 


I. 1124 


iH 


7 


.9421 


-9085 


. 10 


. 10 


.005 102 


I. 1667 


1.2085 


1. 2 107 


iK 


7 


1.0668 


1-0335 


. 10 


- 10 


.005102 


I. 29 17 


1.3335 


1-3355 


M 


6 


1 . 16 14 


I. 1224 


. 10 


. 10 


. 006944 


1.3987 


1.4224 


1-4250 


i\<i 


6 


1.2862 


1.2474 


. 10 


. 10 


- 006944 


I.S237 


I.S474 


1.5497 


1% 


sM2 


1.3917 


1 - 3494 


. IS 


■ 15 


.008263 


I. 7122 


1.7994 


I. 8019 


iM 


s 


1-4935 


1-4469 


. 15 


- 15 


. lOOOO 


1-8234 


1.8969 


1-8997 


M 


s 


1. 6182 


I. 57 19 


. 15 


- IS 


.010000 


1 . 9484 


2 . 02 19 


2.0245 


2 


4H 


1 . 7 149 


1.6632 


- 15 


- IS 


■ 012343 


2.0566 


2. 1 132 


2 . H63 


2M 


4H 


1.8393 


1.7882 


■ IS 


. IS 


.012343 


2. 18 16 


2.2382 


2.2411 


2H 


4H 


1 . 964 I 


1-9132 


. IS 


• IS 


.012343 


2.3066 


2.3632 


2.3667 


2H 


4 


2.0540 


I -996 I 


. IS 


. IS 


.015625 


2.4105 


2 . 446 I 


2.449s 


2H 


4 


2.1787 


2. I2II 


. 15 


• 15 


.015625 


2-5355 


2.5711 


2-5742 


2K 


4 


2.4284 


2-3711 


. IS 


. 15 


.015625 


2.7855 


2.8211 


2.8240 


3 


3^^ 


2.6326 


2.5670 


.20 


.20 


.020392 


3.0835 


3 . 1670 


3- 1704 


3H 


3H 


2.8823 


2.8170 


. 20 


.20 


.020392 


3-3335 


3-4170 


3-4200 


3>i 


3!4 


3 • 104 I 


3-0337 


.20 


.20 


.023654 


3.5668 


3.6337 


3-6368 


3?i 


3 


3-3211 


3 - 2448 


.20 


.20 


-027750 


3.7974 


3 • 8448 


3 - 8486 


4 


3 


3-5708 


3 - 4948 


.20 


.20 


.027750 


4-0474 


4.0948 


4-0983 


4H 


2% 


3 - 80 19 


3-7228 


.20 


.20 


.030240 


4-2864 


4.3228 


4-3264 


4H 


2% 


4-0318 


3-9489 


.20 


. 20 


.033050 


4-5244 


4-5500 


4 -5330 


4% 


254 


4-2592 


4- 1728 


.20 


.20 


.036250 


4.7614 


4-7728 


4-7767 


S 


2}^ 


4.4848 


4.3938 


.20 


. 20 


. 040000 


4.9970 


4-9938 


4 . 9980 


SH 


2\i 


4 - 7346 


4.6438 


.20 


.20 


. 040000 


S.2470 


5-2438 


S-2477 


sh 


2% 


4-9574 


4.8619 


. 20 


. 20 


.044310 


S.4810 


5-4619 


5 - 466 1 


sM 


2% 


5 -2072 


5 - 1 1 19 


.20 


.20 


. 0443 10 


5 -73 10 


5.7 1 19 


5 - 7 160 


6 


2H 


S.4271 


S.3264 


- 20 


.20 


•049373 


5-9632 


5-9264 


5-9307 



D, 

ins. 


H 


Di 


Di 


ins. 


(3. 

ins. 


c^y 


X 


XI 


X2 


Vi 


20 


.1875 


. 1699 


.04 


•04331 


.00062s 


.2733 


.2965 


.2977 


ViO 


18 


.2428 


.2235 


.04 


•04331 


.00077 1 


.3313 


.3501 


■3514 


H 


16 


.2965 


.2749 


.04 


•04331 


.000976 


.3883 


.4015 


.4029 


He 


14 


.3440 


.3231 


.04 


•04331 


.001274 


.4436 


.4497 


•4512 


H 


12 


.3953 


.3666 


.04 


.04331 


•001735 


.4966 


.4932 


.4950 


?i6 


12 


.4576 


• 4291 


.06 


. 06496 


.001735 


.5907 


.6190 


.6204 


H 


H 


.5105 


■4794 


.06 


.06496 


.002065 


.6372 


.6693 


.67 10 


iHe 


II 


.5728 


.5420 


.06 


.06496 


.002065 


.7097 


• 73 19 


•7334 


H 


10 


.6239 


-5899 


.06 


.06496 


.002500 


.7649 


.7798 


• 78 IS 


nu 


10 


.6862 


.6524 


.06 


.06496 


.002500 


.8274 


.8423 


.8438 


% 


9 


.7348 


.6971 


.06 


.06496 


.003086 


.8810 


.8870 


.8882 


iMe 


9 


.7970 


.7596 


.06 


. 06496 


.003086 


.9435 


.9495 


.9512 


I 


8 


.8422 


.7999 


. 10 


. 10839 


.003906 


1.0583 


I. 1167 


I- I185 


m 


7 


-9447 


■8963 


. 10 


. 10839 


.005 102 


I. 1690 


1 . 2 13 I 


I-2153 


iH 


7 


I -0693 


1.02 13 


. 10 


. 10839 


.005102 


1.2940 


1. 3381 


I -3400 


m 


6 


I. 1644 


I. 1082 


. 10 


. 10839 


. 006944 


1.4000 


1.4250 


1.4276 


iH 


6 


1.2892 


1.2332 


. 10 


. 10839 


.006944 


1.5250 


I.SSOO 


1-5523 


m 


5 


1.3726 


1.3048 


• IS 


. 16242 


.0 10000 


1.7023 


1.7796 


1-7826 


m 


S 


I -4970 


1.4298 


.15 


. 16242 


.010000 


1-8273 


1.9046 


1-9074 


m 


4^5 


1-5942 


I ■ 5 193 


. 15 


. 16242 


.012343 


1-9345 


1. 994 I 


1-9973 


2 


4K2 


1.7 18S 


1.6443 


• IS 


. 16242 


.012343 


2.0S9S 


2 . 1 19 I 


2. 122 I 


2H 


4H 


1-8437 


1.7693 


.IS 


. 16242 


.012343 


2 . 184s 


2 . 244 1 


2.2470 


2K 


4 


1-9338 


1.8498 


• IS 


. 16242 


.015625 


2.2873 


2.3246 


2.3280 


2?4 


4 


2.0585 


1.9750 


• 15 


. 16242 


.015625 


2.4123 


2.4498 


2-4530 


2H 


4 


2. 1833 


2. 1000 


. IS 


. 16242 


.015625 


2.5373 


2.5748 


2-5778 


2% 


3K2 


2.3882 


2.2926 


.20 


.21567 


.020392 


2.8370 


2.9240 


2.9276 


3 


3H 


2.6397 


2.5426 


.20 


.21567 


.020392 


3.0870 


3. 1740 


3. 1773 


3H 


3H 


2 . 8600 


2.7574 


.20 


.21567 


.023654 


3-3194 


3.3887 


3-3924 


3^^ 


3H 


3. 1098 


3.0074 


.20 


.21567 


.023654 


3-5694 


3-6387 


3-6420 


3?4 


3 


3-3270 


3 • 2 164 


.20 


.21567 


.027755 


3.7990 


3-8477 


3 -85 IS 


4 


3 


3-5768 


3.4664 


.20 


.21567 


•027755 


4 . 0490 


4.0977 


4. 10 12 


4K 


2% 


3.8080 


3.693 


.20 


.21567 


.030241 


4.2870 


4.3243 


4-3280 


4H 


2% 


4.0582 


3-943 


.20 


.21567 


.030241 


4-5370 


4-5743 


4-5780 


m 


2% 


4.2878 


4- 168 


.20 


.21567 


.033051 


4-7746 


4-7993 


4-8025 


s 


2% 


4-5376 


4.418 


. 20 


.21567 


•033051 


5-0245 


5 - 0493 


5-0524 


5>/4 


254 


4-7658 


4.640 


.20 


.21567 


.036252 


5 . 2607 


S.2713 


5.2750 


5V« 


2=4 


5-0156 


4.890 


.20 


.21567 


.036252 


5-5107 


5-52 13 


5-5248 


S?4 


2V1 


S.2415 


5-110 


.20 


.21567 


.040000 


5-7455 


5-7413 


5 - 7446 


9 


2 1.4 


5-4913 


5-360 


.20 


• 21567 


.040000 


S-99S5 


5-9913 


5 - 9944 



220 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 20. — Acme Standard Threads, Tooth Parts and 

Measurement 

By the \vire system 



Tap Thread 




Ci D1-O.O2. 




Let 



D = diameter of screw. 
Di = diameter of tap or thread plug gage. 
p = pitch. 

d = depth of thread groove, 
a = width of thread groove at top. 
b = width of thread groove at bottom 
c = width of thread at top. 
e = thickness of thread at bottom. 



For Screw Thread 
Then p = -, for single threads. 

<i = - + .0I m. 
a = pX.62g3. 
b = pX.3655- 
c = PX. 3707 
e = pX.634S- 

D = diameter on top of threads. 
x — D + .oi in. 



Rad. of wire section = side opp. = side adj. X tan 37° 45 
= i)X. 24564. 
Diameter of wire = ^X. 4913- 



For Tap Thread 
I 

a = -+.01 m. 

a = pX.6345. 

b = pX.3707. 

c = pX.3655- 

e = pX.62g3. 
Di = D + .02 in. 
xi = Di = D + .02 in. 

634s Xi" 



X.77428 



Threads 






0, 


b. 


c, 


e. 


Diam. 
of wire 


per 


P 


d 


for screw 


for screw 


for screw 


for screw 


inch 






e, for tap 


c, for tap 


6, for tap 


a, for tap 




Ins. 


Ins. 












I 


I. 00 


.5 100 


■ 6293 


• 3655 


• 3707 


■ 634s 


■ 4913 


iM 


• 75 


.3850 


.4720 


.2728 


.2780 


.4772 


■ 368s 


m 


.66666 


■3433 


• 419s 


.2436 


.2471 


.4230 


• 3275 


iM 


.571428 


.2957 


■ 3596 


.2088 


■ 2 1 18 


■ 3626 


.2807 


2 


■ 50 


.2600 


• 3147 


. 180 1 


■ 1853 


■3 199 


■ 2456 


2\i 


.40 


.2 100 


.2517 


. 1462 


. 1482 


■2538 


. 196s 


3 


■33333 


.1767 


.2098 


■ 1183 


■ 1235 


■ 2150 


■ 1637 


4 


.25 


■ 1350 


. 1573 


.0875 


.0927 


■ 162s 


. 1228 


S 


.20 


. 1 100 


.1259 


.0689 


.0741 


.1311 


.0982 


6 


. 16666 


■ 0933 


■ 1049 


.0566 


.0618 


.110 1 


.0819 


7 


. 142857 


.0814 


.0899 


.0478 


.0529 


.0951 


.0702 


8 


.125 


.0725 


■ 0787 


.0411 


.0463 


■ 0839 


.0614 


9 


. mil 


.0655 


.0699 


.0361 


.0413 


.0751 


.0546 


10 


. 10 


.0600 


.0629 


.0319 


■ 0371 


.0681 


.0491 



Much confusion exists regarding the Acme screw thread and the 
Brown and Sharpe worm thread which are alike in their angle of 29 
deg. only. Figs. 10 and 11 give, fuU size, the two threads of i in. 
linear pitch together with the formulas for their tooth parts. 

For constants to be used with the wire system of measuring the 
Brown and Sharpe worm thread, see Index. 





Fig. id. — Brown and Sharpe 
29 deg. work thread. 

Formulas 
P= Linear pitch 
Z> = .6866P 
fl=.66sP 
6=.3iP 

^=•335? 
e=.69P 



Fig. II. — Acme 29 deg. screw 
thread. 

Formulas 
P= Linear pitch 
Z) = .5P + .oi in. 
0= .6293P 

c=.3707P 
e=.6345P 



WIRE AND SHEET METAL GAGES 



The multiplication of wire and sheet metal gages is the source of 
much confusion and has become an intolerable nuisance. The sug- 
gestion has been repeatedly made that the entire gage system be 
abandoned and that sizes be specified by their decimal values. The 
diflGiculty in carrying out this plan is that the gage sizes must be 
adhered to in order to obtain the material from stock and this 
compels constant reference to the gage tables. The following plan, 
due to the Westinghouse Electric and Mfg. Co. (Amer. Mach., Apr. 14, 
1904) while keeping to the gage sizes eliminates all reference to the 
gage numbers. After nine years of use it is found to accomplish 
the desired object. 

The existing standard dimensions of gaged materials are not 
changed, but the gage names, the conflicting and arbitrary gage 
numbers, and the commercial gage plates for identifying materials 
are "discarded. The same actual dimensions as hitherto indicated 
by gage numbers continue in use but are expressed in decimal parts 
of the inch, the unnecessary refinements found in many of the com- 
mercial gage equivalent tables being, however, dropped. 

AU material that has heretofore been known by gage number, as 
No. 20 B. & S. sheet copper, is now known by decimal thickness 
only, as .032 sheet copper. 

Throughout the business of the company, and on all drawings, 
drawing lists, specifications, bills of material and correspondence, 
decimal dimensions are used instead of gage numbers. 

In the shop and storerooms all material that was formerly gaged 
is now measured with the micrometer or Umit gages, and is specified, 
ordered, marked and carried in stock by decimal thicknesses instead 
of by gage numbers. 

Drawings, drawing lists, specifications, bills of material, etc., 
made before the adoption of this plan, and specifying gage numbers 
were not changed except as found convenient by the engineering 
department. 

The extreme refinements shown by the fifth and sixth decimal 
places have been dropped, not more than three significant figures 
being used in specifying sizes. By significant figures is meant all 
figures to the right of ciphers after the decimal point. For example, 
U. S. Standard No. 2 sheet gage is given as . 265625. The Westing- 
house decimal for this gage is .266. 

Materials that were formerly purchased by gage numbers are now 
purchased of the same dimensions expressed in decimals. 

It should be especially noted that, with the exception of twist 
drills, the above changes do not affect finished articles of any kind 
such as are kept in stock by manufacturers and known to the trade 
by gage numbers. 

In Table i will be found a column giving American screw gage 
sizes. This has been inserted for convenience in selecting sizes of 
machine screws and wood screws, and it shoidd be especially noted 
that the gage numbers are retained for these sizes. 

Tables i and 2 are for use in those departments whence specifica- 
tions for material emanate. Columns i to 10, Table 2, are for refer- 
ence in connection with Table i which serves as an index to Table 
2. In each column the decimals coincide with a series of size equiva- 
lents commercially known by gage name and number. Table 2 
ignores all gage names and numbers; the dimensions are not numbered 



in any way, all read from the largest down, and are so arranged with 
respect to one another that the same (approximate) dimensions are 
on the same horizontal Une. This arrangement makes it easy to 
choose in one column a dimension coinciding closely with a dimen- 
sion in any other column. 

The reference numbers in Table i indicate in which column of 
Table 2 to look for commercial diameters or thickness of any given 
material. 

Example. — For commercial diameters of steel spring wire refer 
to Table i opposite Wire, Spring and, under Steel (S) read 3, showing 
that the commercial sizes of steel spring wire are to be found in 
column 3 of Table 2. Similarly Table i shows that the sizes of 
brass or phosphor-bronze spring wire are to be found in column i 
of Table 2. 



The Westinghouse Method of Abandoning Sheet Metal and 

Wire Gages 

Table i. — Index To Column Headlines of Table 2 



Index to 
























Amer. 


columns 






s. 


x> 













PI. 




screw 


I to 10 


0. 



u 

m 




J3 
Oh 


^ 





•0 


5 


< 




c 
t5 


gage 




Commercial .... 


I 


I 


5 


I 


s 


I 




1 


I 


I 


6 


000 


.0310 


_rt 


Planished 






5 




5 














00 


.0442 


u 


Gal vanized 






5 




S 

















• 0573 


4-> 


Tinned 






S 




S 














I 


.0717 


0) 


Terne 






■; 
















7. 


.084s 


J3 


Spring 




I 


2 


I 














3 
4 
S 


.0978 




Bare 


I 
I 


I 


3 


I 


3 


I 






I 
I 


I 


I 


. IIO 




Insulated 


.124 




Galvanized 






3 




3 














6 


.137 


£ 


Tinned* 






3 


















7 


• 150 


t* 








1 


















8 


.163 




Music 






/\ 


















9 


.176 




Resistance 












I 




I 




I 




10 


. 189 




Annealed 






3 




3 














II 


.203 
.216 




Commercial. . . . 


T 


T 




T2 





















Cold rolled 


T 


T 


Ri 


















13 


.229 


Pi 


Drill 






8 


















i4 


.242 


























IS 
t6 


.255 
.268 


V 


Seamless' 


2 


2 




2 




2 






2 












I 




















17 


.282 


H 




























18 
19 


■ 295 




.308 


Twist drills: 9. Coppered steel wire: 3. 


20 


.321 


All cable, lamp cord and fuse wire: i. 


21 
22 


■334 


1 } in. dia. up can be had in fractions. 


.347 


- i in. dia. and larger, in fractions. 


23 


.361 


8 All seamless tubing may be specified by diameters instead of thick- 


24 


•374 


ness of wall. 


2S 


.387 


4 Tinned steel banding wire: special diameters. 


2b 

27 
28 
29 


.400 


Explanatory. — With the exception of twist drills, the following in- 


.413 
.426 
.439 


structions to draftsmen do noi affect ^ni5/(«(i articles such as are 


known to the trade by gage numbers. For example: Machine 


and wood screws will continue to be specified by the American 
screw gage numbers. 


30 

31 


• 4S3 
.466 


Instructions to draftsmen. — When specifying material of "gage" 


32 

33 

34 


• 479 
.492 
•505 


thickness, do not give gage name or number, but specify in deci- 
mals, thus: 


. 036 Sheet copper. . 162 Brass rod. 







221 



222 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 2. — Gage Sizes in Decimals 



I 


2 


3 


4 


5 


6 


7 


8 


9 10 1 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 






■ 490 




.500 


.300 


• SOS 
.492 
.479 






















. 100 
. 106 
.103 


.107 
. 104 




.460 


•454 


.460 
■430 


.460 


.469 
.438 




.466 
.453 
• 439 
.426 








.102 


. 109 


. 106 


. 102 




. 100 


.0973 


.101 
.099 
• 097 
.095 


. 102 
.0995 
.0980 
.0960 


. 100 
.095 


.410 


.42s 


• 394 


.410 


.406 




.413 


• 413 


• 413 




















.0935 


















.404 


.404 




0907 


.095 


.0915 


.090 


.0938 


.090 




.092 


.0890 


.090 














.400 


• 397 


• 397 










.086 








.088 


.0860 
















.387 


.386 


.386 


















.085 


.0820 


.085 














• 374 


.377 


.377 




.0808 


.083 


.0800 


.082 


.0781 


.080 


.0842 


.081 


.0810 


.080 


.36s 


.380 


• 363 


.36s 


• 375 


• 375 


.361 


.368 
.358 


.368 
• 358 










.078 








.079 
.077 


.0785 






■340 


•331 




■ 344 




• 347 


•348 


•348 










• 074 








.075 


.0760 


.075 
















• 339 


.339 




.0720 


.072 


.0720 


.071 


.0703 


.070 


.0710 


.072 


.0730 


.070 














.334 


.332 


• 332 


















.069 


.0700 




.325 






• 32s 


■ 313 


• 


.321 


■ 323 


.323 










.067 








.066 


.0670 


















.316 


.316 




.0641 


.065 


.0625 


.063 


.0625 


.060 




.063 


•0635 


.065 




• 300 


■307 








• 308 


.302 


.302 










.059 










•0595 


.060 
















• 295 


• 29s 




.0571 


.058 


.0540 


• 055 


.0563 


.055 


.0578 


.058 


.0550 


.055 


.289 


.284 


.283 


.289 


.281 




• 295 


.290 


.290 


















.055 


















.282 


.281 
.277 


.281 
• 277 




.0508 


.049 




.051 
.048 


.0500 


.050 




.050 


.0520 


.050 
















. 272 


.272 




.0453 




■ 0475 


.046 


.0438 


.045 


.0447 


• 045 


.0465 


.045 










.266 




.268 


.266 
.261 


.266 

.261 










•044 
.042 








.042 


.0430 
.0420 




.258 


• 259 


.263 


.258 


.250 


.250 


.255 


.257 


.257 


















.041 


.0410 


















• 250 


.250 




• 0403 


.042 


.0410 


.040 




.040 




.040 


.0400 


.040 
















.246 


.246 


















.039 


.0390 








■244 








.242 


.242 
.238 


.242 
.238 


.240 








.038 








.038 
.037 


.0380 
.0370 


















• 234 


.234 




.0359 


.035 


.0348 


.036 


.0375 


.036 




• 036 


.0360 


.036 


.229 


.238 

.220 


.22s 


.229 


• 234 
.219 




.229 


.227 
.219 


.228 

.221 


, 220 








.034 


.0344 






.035 
.033 


.0350 
■ 0330 
















.216 


.212 
.207 


.213 
.209 
.206 




.0320 


.032 


.0318 


.032 
.030 


.0313 


.032 


.0315 


.032 
.031 
.030 


.0320 
.0310 


.332 


.204 


.203 


.207 


. 204 


.203 




.203 


. 204 


. 204 


.200 
















.029 


• 0293 


















. 201 


.201 




.0285 


.028 


.0286 


.028 


.0281 


.028 




.027 


.0280 


.028 
















.199 


.199 










.026 








.026 


.0260 


















.197 


. 196 




.0254 


.025 


.0258 


.024 


.0250 


.024 




.024 


.0250 


.025 
















.194 


.194 




















.0240 








. 192 








. 189 


.191 
.188 


.181 
.189 




.0226 


.022 


.0230 


.022 


.0219 






.023 
.022 


.0225 
.0210 


.022 
















• I8s 


.i8s 




.0201 


.020 


.0204 


.020 


.0188 


,020 




.020 


.0200 


.020 


.182 


.180 


.177 


.182 


.188 




.176 


.182 
. 180 


.182 
. 180 


. 180 


.0179 


.018 


.0181 
.0173 


.018 


.0172 


.018 




.018 


.0180 


.018 
















.178 


.177 




• 0159 


.016 


.0162 


.016 


.0156 


.016 




.016 


.0160 


.016 
















• 175 










.0150 










.015 














.172 






.172 
.168 


• 173 
.170 




.1042 


.014 


.0140 
.0135 


.014 


.0141 


.014 




.014 


.0145 


.014 
















. 164 


.166 




0126 


.013 


.0128 


.013 


.0125 


.012 




.013 


.0135 


.012 


.162 


.i6s 


.162 


.162 


.156 




.163 


.161 

• 157 
.155 
.153 

• 151 


.161 
.159 

• 157 

• 154 

• 152 
.150 


.165 


.0113 
.0100 

.0089 


.012 
.010 

.009 


.0118 
.0104 
• 0095 
.0090 
.0085 


.012 

.on 
.010 

.009 


.0109 
.0102 
.0094 
.0086 


.010 








.010 














.150 


.148 
.146 


.147 




.0080 


.008 


.0080 
.0075 


.0085 


.0078 


.008 








.008 


.144 


.148 


.148 


.144 


.141 




• 137 


• 143 
.139 
•134 


.144 
.141 
.136 


•ISO 
•135 


.Otf7I 
.0063 


.007 


.0070 




.0070 
.0066 
.0063 


.006 








.006 


.129 


• 134 
. 120 


.135 
. 121 


.129 


.125 


• 125 


.124 


.127 

. 120 


.129 

.120 
.116 


.125 


.0056 
.0050 
.0045 


.005 


















.114 






.114 


. 109 




. no 


•115 
. 112 

. no 


.113 
.111 
.110 


. no 


.0040 
• 0035 
.0031 


.004 
















.004 
.002 



The following are the names of the gages to which the dimensions in columns i to 10 agree; in some cases the same gage is known by more than one name. 

I. Brown & Sharpe; American Standard Wire. 2. Birmingham; Stubb's Iron Wire. 3. National; Roebling's; Washburn & Moen; American Steel Wire Co. 
4- Down to .102, Brown & Sharpe; .090 and down, Trenton or Wolff's Music Wire. 5. United States Standard. 6. Zinc. 7. American Screw. 8. Stubb's 
Steel Wire. 9. Morse Twist Drill and Steel Wire. 10. Master Mechanics' Decimal. 



WIRE AND SHEET METAL GAGES 



223 



Table 3. — List of Nine Different Standard Gauges Used in the United States 



American or 

Brown and 

Sharpe iron 

wire 



Birmingham 
or Stubb's 
iron wire 



Washburn 
and Moen 
iron wire 



Imperial 
wire gage 



U. S. Stand- 
ard for plate 
(iron and 
steel) 



Stubb's steel 
wire 



Twist drill and 
steel wire 



Washburn 
and Moen 
music wire 



Wood and 
machine 
screws 



No. of 
gage 



.460 
.410 
.36s 
.325 
.289 
.258 

.229 
.204 
.182 
.162 

.144 
.128 
.114 
.102 
.091 
.081 
.072 
.064 
■057 

.051 

• 04s 
.040 
.036 
.032 
.028 
.02s 
.023 

.020 

.018 

.016 

.0141 

.0126 

.0112 

.010 

.0089 

.0079 

.007 

.0063 

.0056 

.cos 

.0044 

.0039 

• 003 s 
.0031 



.454- 

.42s 

.380 

• 340 

.300 

.284 

.2S9 

.238 

.220 

.203 

.180 

.165 

.148. 

.134 

. 120 

. 109 

■ 095 

.083 

.072 

.065 

.058 

.049 

.042 

.033 

.032 

.028 

.025 

.022 

.020 

.018 

.016 

.014 

.013 

.012 

.010 

.009 

.008 

.007 

.005 

.004 



.394 
.363 
.331 
.307 
.283 
.263 
.244 

.225 
.207 
. 192 
.177 
. 162 
.148 
•135 
. 121 
. 106 
.092 
.080 
.072 
.063 
■054 
.048 
.041 

■ 035 
.032 
.029 
.026 
.023 
.020 
.018 

■ 0173 
.0162 
.015 
.014 
.0132 
.0128 
.oiiS 
.0104 
.0095 
.009 



.464 
432 
.400 
.372 
.348 
.334 
.300 
.276 
.252 
.232 
.212 
. 192 
. 176 
. 160 
.144 
.128 
.116 
. 104 
.092 
.080 
.072 
.064 
.056 
.048 
. 040 
.036 
.032 
.028 
.024 
.022 
.020 
.018 
.0164 
.0149 
.0136 
.0124 
.0116 
.0108 
.010 
.0092 
.0084 
.0076 
.0068 
.006 
.0052 
.0048 



.500 
.469 
.438 
.406 

• 375 
.344 
.313 
.281 
.266 
.250 
.234 
.219 
.203 
.188 
. 172 
.156 
.141 

• 125 
. 109 
.094 
.078 
.070 
.063 
.056 
.050 
.044 
.038 
.034 
.031 
.028 
.025 
.022 
.019 
.0171 
.0156 
.014 

.CI2S 
.0109 
.0101 

.0093 

.0085 

.0078 

.007 

.0066 

.0062 



.227 
.219 

.212 
.207 
.204 
.201 
.199 
.197 
.194 
.191 
.188 
.185 
.182 
.180 

• 178 

• 175 
.172 
.168 
. 164 
.161 
.157 
.155 

■ 153 
.151 
.148 
.146 
.143 

■ 139 

• 134 
. 127 
. 120 

• 115 
. 112 
. no 
.108 
. 106 
.103 

. lOI 

.099 
.097 
.095 
.092 

.088 
.08S 
.081 
.079 
.077 

• 075 
.072 
.069 



This gage 
from one to 
three thous- 
andths larger 
than same Nos. 
of Stubb's steel 
wire gage. 



228 

221 

213 

209 

206 

204 

,201 

199 

196 

194 

191 

189 

18S 

182 

180 

177 

173 

170 

166 

161 

159 

157 

154 

152 

150 

147 

144 

141 

136 

129 

120 

116 

113 

in 

no 

1065 

104 

1015 

0995 

098 

096 

094 

089 

086 

082 

081 

079 

076 

073 

070 



0083 

0087 

0095 

010 

on 

012 

013 

014 

016 

017 

018 

019 

020 

022 

023 

024 

026 

027 

028 

030 

031 

033 

035 

036 

038 

040 

041 

043 

046 

048 

051 

055 

059 

063 

066 

072 

076 

080 



.032 
.045 
.058 
.071 
.084 
.097 
. no 
. 124 
.137 
.ISO 
.163 
. 176 
.189 
.203 
.216 
.229 
. 242 

• 255 
.268 
.282 
.295 
.308 
.321 
.334 
.347 
.360 
.374 
.387 
.400 
.413 
.426 

■ 439 
.453 
.466 

■ 479 
.492 

■ 505 

■ 518 

• 532 

■ 545 

■ 558 
.571 
.584 
.597 
.611 
.624 

■ 637 
.650 
.663 
.676 
.690 
.703 
.716 



8-0 
7-0 
6-0 
5-0 
4-0 
3-0 
2-0 



3 

4 
5 
6 
7 
8 
9 
10 

II 

12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

31 
32 
33 

34 
35 
36 

37 
38 
39 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 



Letter Sizes Stubb's Steel Wire 



A. 
B. 
C. 
D. 
E. 
F. 
G. 
H. 




.339 
.348 
.358 
.368 
.377 
.386 
.397 
.404 
.413 



224 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 4. — Standard Decimal Gage 







Weight per square foot 


Standard 


Thickness in 
fractions 


in pounds, avoirdupois 


decimal gage 


Iron, basis — 480 


Steel, basis — 489 . 6 


in ins. 


of an inch 


lbs. per 


lbs. per 






cu. ft. 


cu. ft. 


.002 


1-500 


.08 


.0816 


.004 


1-250 


.16 


.1632 


.006 


3-500 


.24 


.2448 


.008 


I-I2S 


.32 


.3264 


.010 


i-:oo 


.40 


.4080 


.012 


3-250 


.48 


.4896 


.014 


7-500 


.56 


.S7I2 


.016 


2-I2S(^+) 


.64 


.6528 


.018 


9-500 


• 72 


.7344 


.020 


i-SO 


.80 


.8160 


.022 


11-500 


.88 


.8976 


.025 


1-40 


1. 00 


1.0200 


.028 


7-250 


1 . 12 


I. 1424 


.032 


4-l25(A+) 


1.28 


1.3056 


.036 


9-250 


1.44 


1.4688 


.040 


1-25 


1 .60 ■ 


1.6320 


• 045 


9-200 


1 .80 


1.8360 


.050 


1-20 


2.00 


2 . 0400 


.055 


11-200 


2.20 


2.2440 


.060 


3-50 (fs-) 


2.40 


2.4480 


.06s 


13-200 


2.60 


2.6520 


.070 


7-100 


2.80 


2.8560 


.07s 


3-40 


3 00 


3.0600 


.080 


2-25 


3-20 


3.2640 


.085 


17-200 


3-40 


3.4680 


.090 


9-100 


3 -60 


3-6720 


• 095 


19-200 


3-80 


3.8760 


. 100 


I-IO 


4.00 


4 . 0800 


. no 


II-IOO 


4.40 


4.4880 


.125 


1-8 


S-OO 


S. 1000 


.^35 


27-200 


S-40 


S.S080 


.ISO 


3-20 


6.00 


6. 1200 


.165 


33-200 


6.60 


6.7320 


. 180 


9-50 


7. 20 


7.3440 


.200 


i-S 


8.00 


8. 1600 


. 220 


I I-SO 


8.80 


8.9760 


.240 


6-2S 


9.60 


9.7920 


.250 


1-4 


10.00 


10.2000 



The Standard Decimal Gage has been adopted by the Association 
of American Steel Manufacturers, the American Railway Master 
Mechanics' Association and by about seventy-two of the principal 
railroads of the United States, Canada and Mexico. The decimal 
system of gaging was recommended by the American Institute of 
Mining Engineers in 1877 and by the American Society of Mechanical 
Engineers in 1895. 

Gages Used for Wire 

By the American Steel and Wire Co. 

The gage now known as the steel wire gage is the same as that 
formerly called the Washburn and Moen gage, the American Steel 
and Wire Company's gage, and by other names. It is in practically 
universal use by American steel wire manufacturers, the result being 
that there is really a standard steel wire gage in the United States, 
although this has not been formally recognized. 

Upon the recommendation of the Bureau of Standards at Wash- 
ington, a number of the principal wire manufacturers and important 
consumers have agreed that it would be well to designate this gage 
as the steel wire gage. In cases where it becomes necessary to 
distinguish it from the British standard wire gage, it may be called 
the United States steel wire gage. The name thus adopted has 
ofiScial sanction, although without legal effect. 

{Continued on page 229 first column) 



Table 5. — Sizes of Numbers of the U. S. Standard Gage for 
Sheet and Plate Iron and Steel 

Be it enacted by the Senate and House of Representatives of the United 
States of America in Congress assembled: 

That for the purpose of securing uniformity the following is estab- 
lished as the only gage for sheet and plate iron and steel in the 
United States of America, namely: 





Approximate 


Approximate 


Weight per 


Weight per 


Number 


thickness in 


thickness in 


square foot 


square foot 


of gage 


fractions of 


decimal parts 


in ounces 


in pounds 




an inch 


of an inch 


avoirdupois 


avoirdupois 


0000000 


1-2 


.5 


320 


20.00 


000000 


iS-32 


.46875 


300 


18.7s 


00000 


7-16 


.437s 


280 


17.50 


0000 


13-32 


.40625 


260 


16. 25 


000 


3-8 


■ 375 


240 


15.00 


00 


11-32 


.34375 


220 


13-75 





5-16 


.3125 


200 


12.50 


I 


9-32 


.28125 


180 


11.25 


2 


17-64 


.265625 


170 


10.625 


3 


1-4 


.25 


160 


10.00 


4 


iS-64 


.234375 


ISO 


9-375 


5 


7-32 


.21875 


140 


8.75 


6 


13-64 


.203125 


130 


8. 125 


7 


3-16 


.1875 


120 


7.S 


8 


11-64 


.171875 


no 


6.875 


9 


S-32 


.15625 


100 


6.25 


10 


9-64 


. 140625 


90 


5.625 


II 


1-8 


.125 


80 


5.00 


12 


7-64 


.109375 


70 


4.37s 


13 


3-32 


.0937S 


60 


3.75 


14 


S-64 


.078125 


SO 


3-125 


IS 


9-128 


.0703125 


45 


2.812s 


16 


1-16 


.0625 


40 


2.S 


17 


9-160 


.05625 


36 


2.25 


18 


1-20 


• OS 


32 


2 


19 


7-160 


•0437S 


28 


1.75 


20 


3-80 


.0375 


24 


1.50 


21 


11-320 


.03437S 


22 


1. 375 


22 


1-32 


.03125 


20 


I.2S 


23 


9-320 


.028125 


18 


1. 25 


24 


1-40 


.025 


16 


I . 


25 


7-320 


.021875 


14 


.875 


26 


3-160 


.01875 


12 


.75 


27 


11-640 


.0171875 


II 


.6875 


28 


1-64 


.015625 


10 


.62s 


29 


9-640 


.0140625 


9 


.5625 


30 


1-80 


.0125 


8 


5 


31 


7-640 


•0I0937S 


7 


.4375 


32 


13-1280 


.01015625 


6i 


.40625 


33 


3-320 


.009375 


6 


.375 


34 


11-1280 


.00859375 


5h 


.34375 


35 


S-640 


.007812s 


S 


.312s 


36 


9-1280 


.00703125 


4) 


.28125 


37 


17-2560 


.006640625 


4i 


.265625 


38 


1-160 


.00625 


4 


.25 



"And on and after July first, eighteen hundred and ninety-three, 
the same and no other shall be used in determining duties and taxes 
levied by the United States of America on sheet and plate iron and 
steel. But this act shall not be construed to increase duties upon 
any articles which may be imported. 

"Sec. 3. That in the practical use and application of the standard 
gage hereby established a variation of two and one-half per cent, 
either way may be allowed. 

Approved March 3, 1893." 



WIRE AND SHEET METAL GAGES 



225 



Table 6. — Sizes and Properties of Wire 
By the American Steel and Wire Company- 



Gage numbers 



Steel 
wire 
gage 



Ameri- 
can 
wire 
gage 
(B. & S.) 



Birm- 
ingham 
or Stubs 



British 
Imperial 
Standard 



Diameter 



Ins. 



64ths 



Deci- 
mally 



Milli- 
meters 



Sectional area 



Sq. ins. 



Circular 
mils 



Log. of 
sq. ins. 



Weight, lbs. per ft. 



Copper 



Iron and 
steel 



Aluminum 



Length, ft. per lb. 



Copper 



Iron 
and 
steel 



Alumi- 
num 



6-0 



4-0 



6-0 



6-0 



4-0 



4-0 



5-0 



3-0 



4-0 



63-64 
31-32 
61-64 
15-16 

59-64 
29-32 
57-64 

7-8 
55-64 

27-32 
53-64 
13-16 
5 1-64 
25-32 

49-64 

3-4 
47-64 
23-32 
45-64 

H-16 

43-64 

21-32 

41-64 

S-8 

39-64 
19-32 



37-64 
9-l6 

35-64 
17-32 



33-64 



31-64 
15-32 



29-64 
7-16 



27-64 
13-32 



25-64 



23-64 



1. 0000 

■ 9843 

• 9687 

• 9531 

■ 9375 

.9218 
.9062 
.8906 
.8750 
■8593 

.8437 
.8281 
.8125 
.7968 
.7812 

.7656 
•7500 
•7343 
.7187 
.7031 

.6875 
.6718 
.6562 
.6406 
.6250 

.6093 
•5937 
.5800 
.5781 
.5625 

.5468 

• 5312 
•5165 

• 5156 
.5000 

.4900 

•4843 
.4687 
.4640 
•4615 

.4600 
•4540 
•4531 
•4375 

• 4320 

•4305 
.4250 
.42 18 
.4096 
.4062 

.4000 
•3938 

• 3906 
.3800 
•3750 

• 3720 
.3648 
•362s 

• 3593 

• 3480 



25^40 
25.00 
24.61 
24.21 
23.81 

23.42 
23.02 
22.62 
22.23 
21.83 

21.43 
21.03 
20.64 
20.24 
19.84 

9.45 
9.05 
8.6s 
8.26 
7.86 

7.46 
7.07 
6.67 
6.27 
5.88 

5.48 
5^08 
4^73 
4.68 
4.29 

3.89 
3^49 
3^ 12 
3- 10 

2.70 

2.45 
2.30 
I. 91 
1.79 
1.72 

1.68 
1.53 
I^Sl 
I. II 
0.97 

0.93 
0.80 
o. 72 
0.40 
0.32 

o. 16 

0.00 

9.922 

9.652 

9^525 

9^449 
9.266 
9^208 
9^ 128 
8.839 



78540 


1000000 


76 105 


968994 


73708 


938477 


71349 


908447 


69029 


878906 


66747 


849854 


64504 


821289 


62299 


793213 


60132 


765625 


58004 


738525 


559 14 


7 1 19 14 


53862 


685791 


51848 


660156 


49874 


635010 


47937 


610352 


46039 


586182 


44179 


562500 


42357 


539307 


40574 


516602 


38829 


494385 


37122 


472656 


35454 


45 14 16 


33824 


430664 


32233 


410400 


30680 


39062s 


29165 


371338 


27688 


352539 


2642 I 


336400 


26250 


334229 


24851 


3 16406 


23489 


299072 


22166 


282227 


20952 


266772 


20881 


265869 


1963s 


250000 


18857 


240100 


18427 


234619 


I72S7 


219727 


16909 


2 15296 


16728 


2 12982 


166 19 


2 1 1600 


I6i88 


206 I 16 


16 126 


205322 


15033 


19 1406 


14657 


186624 


14556 


185330 


14186 


180625 


13978 


177979 


13 177 


167772 


12962 


165039 


12566 


160000 


12180 


,155078 


11984 


152588 


H34I 


144400 


II04S 


14062s 


10869 


138384 


10452 


133079 


1032 1 


13 1406 


10143 


129150 


095 1 1 


12 1104 



1.895090 
.881412 
.867514 
•853390 
.839032 

.824434 
.809586 
.794480 
.779 106 
■763456 

• 747518 

• 731282 

• 714736 
•697870 
.680670 

.663 122 

.6452 

.626926 

.608246 

.589156 

. 569636 
.549666 
.529228 
.508298 
.486850 

.464860 
.442298 
.42 1946 
.419134 
•395336 

•370866 
•345688 
•321230 
•319758 
•293030 

I • 275482 
• 265454 
.236972 
.228 126 
•223434 

.220606 
.209202 
.207526 
. 177046 
. 166058 

. 163036 
. 151868 
. 145458 
. 119810 
. I 12676 

.0992 10 

.085642 
.078610 
.054658 
.043152 

.036176 
.019200 
.013706 
.006186 
2,978248 



3 023 
2.930 
2.837 
2.747 
2.657 

2.570 
2.483 
2.398 
2.315 
2.233 

2. 152 
2.073 
1.996 
1.920 
1.845 

1.772 
1.701 
1.631 
1.562 
I 495 

1.429 
1.365 
1.302 
1.241 
I. 181 

I. 123 
1.066 
1.017 
I.OII 

.9566 

.9042 

.8533 
.8066 
.8038 

■ 7559 

.7259 
.7094 
.6643 
.6509 
■6439 

.6398 
.6232 
.6208 

.5787 
•5643 

•5603 

• 5461 
■5381 
•5073 
•4990 

• 4838 
.4689 

■ 46 13 
.4366 
.4252 

.4184 
.4024 
•3973 
•3905 
.3662 



2.667 
2.585 
2.503 
2.423 
2.344 

2 . 267 
2. 191 
2. 116 
2 .042 
1.970 

1.899 
1.829 
I. 761 
1.694 
1.628 

1.563 
1.500 

1-438 
1.378 
1-319 

I. 261 
1.204 
1. 149 
1.095 
1.042 

■9904 
.9403 
.8972 
.8915 
•8439 

•7977 
•7528 
•7 IIS 
•7091 
.6668 

.6404 
.6258 
.5861 

• 5742 

• S68l 

•5644 
•5498 
•5476 
•SIOS 

• 4978 

•4943 
.4818 
•4747 
•4475 
.4402 

.4268 
•4136 
.4070 
•38SI 
.3751 

.3691 
-3550 
•350s 
•3445 

• 3230 



.9076 
.8795 
.8518 
.8245 

• 7977 

• 7713 

• 7454 
.7199 
.6949 
.6703 

.6461 
.6224 
.5992 
.5763 
■5540 

•5320 
•5105 
•4895 
.4689 
.4487 

.4290 
.4097 
•3909 
•3725 
•3545 

•3370 
.3200 
•3053 
•3033 

• 2872 

•2714 

• 2562 
.2421 
.2413 
.2269 

•2179 

.2 129 
. 1994 

. 1954 

• 1933 

. 1920 
. 1871 
. 1864 
. 1737 
. 1694 

. 1682 

• 1639 
. 1615 
. 1523 
. 1498 

• 1452 
. 1408 
. 1385 
.1311 
.1276 

. 1256 
. 1208 

• 1 193 
. 1172 
. 1099 



.3307 


.3749 


.3413 


.3869 


■ 3524 


.3995 


.3641 


.4127 


.3763 


.4266 


.3892 


.4412 


.4027 


•4565 


.4170 


■4727 


.4320 


.4897 


• 4478 


■5077 


• 4646 


.5266 


• 4823 


.5467 


.5010 


• 5679 


.5209 


■ 5904 


.5419 


■ 6143 


.5642 


■ 6396 


.5880 


.6665 


• 6133 


.6952 


.6402 


.7258 


.6690 


.7584 


.6998 


■ 7932 


.7327 


■ 8306 


.7680 


.8706 


.8059 


.9136 


.8467 


.9598 


.8907 


1.010 


.9382 


1.063 


.9832 


I. IIS 


.9896 


1. 122 


1.045 


I. 18s 


1. 106 


1.254 


I. 172 


1.328 


1.240 


1.405 


1.244 


1. 4 10 


I-323 


1.500 


1-378 


1.562 


1. 4 10 


1.598 


I -50s 


1.706 


1-536 


1. 741 


1.553 


1.760 


1.563 


1.772 


1.605 


1.819 


I. 611 


1.826 


1.728 


1-959 


1.772 


2.009 


1.785 


2.023 


I. 83 I 


2.076 


1.858 


2. 107 


I-97I 


2.235 


2.004 


2.272 


2.067 


2.343 


2. 133 


2.418 


2. 168 


2.457 


2. 290 


2.596 


2.352 


2.666 


2.390 


2.709 


2.485 


2.817 


2.517 


2.853 


2.S6I 


2.903 


2-731 


3-096 



I. 102 
I. 137 
I. 174 
1.2 13 

1.254 

1.296 
1.342 
1.389 
1.439 
1.492 

1.548 
1.607 
1.669 
1.735 
1. 80s 

1.880 
1.959 
2.043 

2. 133 
2.229 

2.331 
2.441 
2.558 
2.68s 
2.82 I 

2.967 

3. I2S 
3.275 
3^297 

3^482 

3.684 
3-904 

4. 130 

4. 144 
4.407 

4^589 
4.696 
5.014 
S^ 118 
5^ 173 

5^207 
5 346 
5^366 
5^756 
5^904 

5-945 
6. 100 
6. 191 
6.567 
6.676 

6.886 
7.105 
7.22 I 
7.630 
7-835 

7.962 
8.279 
8.385 
8.531 
9.098 



15 



226 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 6. — Sizes and Properties of Wire — (Continued) 
By the American Steel and Wire Company 



Gage numbers 



Steel 
wire 
gage 



Ameri- 
can 
Wire 
(B. & S.) 



Birm- 
ingham 
or Stubs 



British 
Imperial 
Standard 



Diameter 



Ins. 



64ths 



Deci- 
mally 



Milli- 
meters 



Sectional area 



Sq. ins. 



Circular 
mils 



Log. of 
sq. ins. 



Weight, lbs. per ft. 



Copper 



Iron and 
steel 



Aluminum 



Length, ft. per lb. 



Copper 



Iron 
and 
steel 



11-32 



21-64 



5-16 



19-64 



17-64 



1-4 



15-64 



7-32 



13-64 



3-16 



11-64 



S-32 



9-64 



7-64 



■3437 
.3400 
.3316 
.3281 
.3249 

.3240 
.3125 
.3065 
.3000 
.2968 

■ 2893 
.2840 
.2830 
.2812 
.2760 

.2656 
.2625 

■ 2590 
..2S76 
.2520 

.2500 

• 2437 
.2380 

■ 2343 
.2320 

• 2294 

•22S3 
.2200 
.2187 
.2 120 

.2070 
■2043 
.2031 
.2030 
. 1920 

. 187s 
. 18 19 
. 1800 
. 1770 
. 1760 

. 17 18 

. l6so 
. 1620 
. 1600 
. 1562 

.1483 
. 1480 
. 1443 
. 1440 
. 1406 

• 1350 

■ 1340 
. 128S 
. 1280 
. 1250 

. 120S 
. 1200 
. 1 160 

■ 1 144 
. 1093 



8.731 
8.636 
8.407 
8.334 
8.252 

8.230 
7.938 

7.78s 
7.620 
7.541 

7.348 
7.214 
7. 188 
7. 144 
7.010 

6.747 
6.668 
6.S79 
6.543 
6.401 

6.350 
6. 190 
6.045 
5-953 
S-893 

5.827 
5723 
5.588 
5 556 
5-385 

S.258 
5- 189 
5- 159 
5- 156 
4-877 

4-763 
4.620 
4-572 
4-496 
4.470 

4.366 
4- 191 
4- IIS 
4 -064 
3.969 

3-767 
3-759 
3-665 
3-658 
3-572 

3-429 
3-404 
3-264 
3.251 
3.175 

3.061 
3-048 
2.946 
2.906 

2.778 



.09280 
.09079 
. 08604 
.08456 
.08290 

.08244 
.07669 
.07378 
.07068 
.06922 

.06573 
-06334 
.06290 

.062 12 
.05982 

.05541 
.05411 
.05268 
.05211 

. 04987 

.04908 
. 04664 
. 04448 
-043 14 
.04227 

. 04 133 
.03986 
.03801 

.03758 
■03529 

■03365 
-03278 
.03240 
•03236 
.02895 

.02761 
.02598 

.02544 

.02460 
.02432 

.02320 
.02138 
.02061 

.02010 
.01917 

.01727 
.01720 
.01635 
.01628 
•01553 

.01431 
.01410 
.01296 
.01286 
.01227 

.01140 
.01131 
.01056 
.01027 
.00939 



1 18 164 
115600 
10956 I 
107666 
105560 

104976 
97656 
93942 
90000 
88135 

83694 
80656 
80089 
79 102 

76176 

70557 
68906 
67081 
66358 
63504 

62500 
59390 
56644 
54932 
53824 

52624 
S0760 
48400 
47852 
44944 

42849 
41738 
41260 
41209 
36864 

35156 
33088 
32400 
31329 
30976 

29511 
27225 
26244 
25600 

24414 

21993 
21904 
20822 
20736 
19775 

18225 
17956 
16512 
16384 
15625 

14520 
14400 
13456 
13087 
1 1963 



.967576 
.958048 
-934746 
.927 168 
.918590 

.916180 
.884790 
.867950 
-849332 
.840238 

2.817786 
.801726 
.798662 
•793276 
.776908 

-743628 
-733348 
-72 1690 
-7 16982 
-697892 

-690970 
-668802 
.648244 
-634912 
-626066 

.616276 
.600612 
.579936 
.574986 
.547762 

.527030 
.515626 
. 5 106 16 
.510082 
.461692 

.441092 
.414756 
■405636 
.391036 
.386116 

■ 365516 
•330058 
.314120 
•303330 
•282730 

2 • 237372 

• 2356 14 
.2 13622 
.2118 14 
. 19 12 16 

• 155758 

• 149300 
. 112896 

• 1095 10 
.088910 

.057064 
■053452 
.024006 
. o 1 1942 
3 • 972926 



• 3573 
•3495 

• 3313 
•3255 
.3192 

• 3174 
.2953 

• 2840 
.272 I 
.2665 

-2530 
-2439 
.242 I 
-2392 
-2303 

-2133 
-2083 
.2028 
.2006 

- 1920 

. 1890 

- 1796 

- 17 13 
. 1661 
. 1627 

- 1591 

- 1535 

• 1463 

• 1447 

• 1359 

. 1296 

- 1262 
■ 1247 

- 1246 

- HIS 

- 1063 
. 1000 
.09796 
.09472 
.09366 

-08932 
-0823 1 
-07935 
.07740 
.07382 

. 06649 
.06623 
.06296 
.06269 
-05979 

-05510 
.05429 
. 04992 
■04954 
.04724 

■04390 
■04354 
. 04068 
■03957 

. 036 17 



3152 
3083 
2922 
2872 
2816 

2800 
260s 
2506 
2400 
23s I 

2232 
2151 
2136 

2 110 
2032 

1882 
1838 
1789 
1770 
1694 

1667 
1584 
15 I I 
1465 
1436 

1404 
1354 
129 I 
1276 
1199 

I 143 
I I 13 
110 1 
1099 
09832 

09377 
08825 
08642 
08356 
08262 

07879 
0726 I 
07000 
06828 
o6s'i2 

05866 
05842 
05554 
05531 
0527s 

04861 
04789 
04404 
04370 
04168 

03873 
03841 
03589 

03491 
03191 



. 10720 
. 10490 
.09944 
.09772 
.09581 

.09528 
.08863 
.08526 
.08168 
.07999 

■07596 
■ 07320 
.07269 
.07179 
. 069 14 

.06404 
.06254 
.06088 

- 06023 
-05764 

-05673 
05390 

- OS 14 I 

- 04986 
.04885 

.04776 
. 04607 
-04393 

- 04343 
-04079 

-03889 
-03788 
-03745 
03740 
-03346 

- 03 19 1 
03003 

.02941 
.02843 
.02811 

- 02681 
-02471 
.02382 
.02323 
.022 16 

. o 1996 
.01988 
.01890 
.01882 
-01795 

01654 
.01630 
.01499 
.01487 
.01418 

.01318 
.01307 

.01221 
.01188 
.01086 



2.799 
2.861 
3.019 
3.072 
3. 133 

3. 151 
3-387 
3.521 
3-675 
3-753 

3-952 

4. 101 
4- 130 
4. 181 
4-342 

4-688 
4-800 
4-931 
4-984 
S-208 

S-292 
5.569 
5-839 
6.02 I 
6. 14s 

6.285 
6.516 
6.834 
6.912 
7.359 

7.719 
7.924 
8.016 
8.026 
8.972 

9.408 
9.996 
10.2 1 
10.56 
10.68 

11.20 

12. IS 
12.60 
12.92 

13. 55 

15-04 
15- 10 
15-88 
15-95 
16-73 

18. IS 
18.42 
20.03 
20. 19 
2 1. 17 

22 . 78 
22.97 
24.58 
25-27 
27-65 



3- 173 
3-243 
3-422 
3-482 
3.552 

3-572 
3.839 
3-991 

4- 166 

4-254 

4-480 
4.648 
4.681 
4-740 
4.922 

5.314 
5-441 
5- 589 
5.6SO 
5-904 

5-999 
6-313 
6.6 19 
6-825 
6.966 

7. 12s 
7.386 
7-746 
7-835 
8-342 

8-750 
8.983 
9-087 
9.098 
10. 170 

10.66 
11-33 
11-57 
11-97 

12. 10 
12.69 

13-77 
14.29 
14-65 

IS -36 

17-05 
17. 12 
18.01 
18.08 
18,96 

20.57 
20.88 
22.7 1 

22.88 

24.00 

25-82 
26.04 
27.86 
28.65 
31-34 



WIRE AND SHEET METAL GAGES 



227 



Table 6. — Sizes and Properties of Wire — {Continued) 
By the American Steel and Wire Company- 



Gage numbers 



Ameri- 
can 
wire 
gage 
(B. & S.) 



Birm- 
ingham 
or Stubs 



British 
Imperial 
Standard 



Diameter 



Ins. 



64ths 



Deci- 
mally 



Milli- 
meters 



Sectional area 



Sq. ins. 



Circular 
mils 



Log. of 
sq. ins. 



Weight, lbs. per ft. 



Copper 



Iron and 
steel 



Aluminum 



Length, ft. per lb. 



Copper 



Iron 
and 
steel 



Alumi- 
num 



14 



16 



17 



18 



19 



23 



24 



25 



26 



18 



3-32 



5-64 



I- 16 



18 



19 



24 



26 



19 



24 



25 



26 



3-64 



1-64 



. logo 
.1055 
. 1040 
. 10 19 
.0950 

.0937 
.0920 
.0915 
.0907 
.0830 

.0808 
.0800 
.0781 
.0720 
.0650 

.064 I 
.0640 
.0625 
.0580 

• 0571 

.0360 
.0540 
.OS08 
.0490 
.0480 

.0475 
.0468 

• 0453 
.0420 
.0410 

.0403 
.0400 
.0360 

• 0359 

• 0350 

.0348 
.0320 
■ 0317 
.03125 
.0286 

.0285 
.0280 
.0258 
.0253 
.0250 

.0240 
.0230 
.0226 
.0220 
.0204 

.0201 
.0200 
.0181 
.0180 
.0179 

.0173 
.0164 
.0162 
.0160 

• 0159 

• 0156 
.0150 



2.769 
2^68o 
2.642 
2.588 
2.413 

2.381 
2.337 
2.324 
2.304 
2. 108 

2.052 
2.032 
1.984 

1.829 
1.651 

1.628 
1.626 
1.588 

1-473 
1.450 

1 .422 
1.372 
1 .290 
1.245 
1. 219 

1 .207 
1 . 191 
1. 151 
1 .067 
1 .041 

1 .024 
1 .016 
.9144 
.9119 



.8839 
.8128 
.8052 
.7938 
.7264 

.7239 
.7112 

• 6553 
.6426 
.6350 

.6096 

.5842 
.5740 
.5588 
.5182 

.5105 
.5080 
.4597 

• 4572 
.4547 

.4394 
.4166 

■ 411S 
.4064 

■ 4039 

■ 3969 
.3810 



•00933 

.00874 
.00849 
.00815 
.00708 

• 00690 
.00664 
.00657 
.00646 

• 0054 I 

.00512 
.00502 
•00479 
.00407 
.00331 

.00322 
.0032 1 
.00306 
.00264 
.00256 

.002463 
.002290 
.002026 
.001885 
.001809 

.001772 
.001725 
.001611 
.001385 
.001320 

.001275 
.001256 
.001017 
.001012 
. 000962 

.000951 
.000804 
.000789 
.000766 
.000642 

.000637 
.000615 
.000522 
.000502 
. 000490 

.000452 
.000415 
.000401 
.000380 
.000326 

.000317 
.000314 
.000257 
.000254 
.000251 

.000235 

.000211 
.000206 
.000201 
.000198 
.000191 
.000176 



ii88r.o 
11130.0 
108 16.0 
10384.0 
9025.0 

8789. 1 
8464.0 
8372.3 
8226.5 



6528^6 
6400.0 
6 103. 5 

5184.0 
4225.0 

4108.8 
4096.0 

3906.3 
3364-0 
3260.4 

3I36-0 
2916.0 
2580.6 

2401 .0 
2304.0 

2256.3 
2197.3 
2052.1 
1764.0 
I68I.0 

1624. 1 
1600.0 
I296-0 
1288.8 

I22S.0 

I2II.0 
1024.0 
1004.9 
976.56 
817.96 

812.25 
784.00 
665.64 
640.09 
625.00 

576.00 
529-00 

SI0.76 
484.00 
416.16 

404.01 
400.00 
327 -61 
324.00 
320.41 

299.29 
268.96 
262 .44 
256.00 
252.81 
244 • 14 

225 .00 



.969942 
.941594 
.929156 
.911438 
.850538 

.839032 
.822666 
.817932 
.810304 
•733246 

. 7099 12 
.701270 
.680670 
-609754 
.520916 

.508806 
-507450 
.486850 
.421946 
-408362 

3.391466 
-359878 
.306818 
.275482 
.257572 

.248478 
.236972 
.207286 
.141588 
.120658 

.105700 
.099210 
.007696 
.005278 
4.983226 

.978248 
.905390 
.897208 
.884790 
.807822 

.804780 
.789406 
.718330 
.701332 
.690970 

-655512 
.618546 
.603306 
-579936 
-514350 

.501482 
.497150 
.410448 
.405636 
.400796 

4-37II 
-324778 
.314120 
•303330 
-29788J 
.282730 
.247272 



■03592 
-0336s 
.03270 
-03139 
.02729 

.02657 
■02559 
■02531 
.02487 
.02083 

■ o 1974 

■ 01935 

■ 0184 s 

■ 01567 
.01277 

.01242 
.01238 
.01181 
.0 10 17 
.009858 

.009482 
.008816 
.007802 
.007259 
.006966 

.006822 
.006643 
.006204 
005333 
.005082 

.004910 
.004838 
.003918 
.003897 
.003704 

.003662 
.003096 
.003038 

.002953 

.002473 

.002456 
.002370 
.002013 
•001935 

.001890 

.001742 
.001599 
.001544 

- 001463 
.001258 

.001222 

.001209 

.0009905 

.0009796 

.0009688 

.0009049 
.0008132 
-0007935 
.0007740 
.0007644 

- 0007382 
.0006803 



•03169 
.02969 
.02885 
■02770 
.02407 

•02344 
.02258 
■02233 
.02 194 
.01837 

.01741 
.01707 

• 01628 

• 01383 
.01127 

.01096 

.01092 

.01042 

.008972 

.008696 

.008364 
.007778 
.006883 
.006404 
.00614s 

.006018 
.005861 
•005473 
.004705 
.004484 

.004332 
.004268 
•003457 
.003438 
.003267 

.003230 
.002731 
.002680 
.00260s 
.002182 

.002166 
.002091 
■OOI77S 
.001707 
.001667 

.001536 
.001411 
.001362 
.001291 
.001110 

.001078 

.001067 

.0008738 

.0008642 

.0008546 

.0007983 
.0007174 
- 0007000 
.0006828 
.0006743 
.0006512 
.0006001 



.01078 
.01010 

.009817 
- 009424 
.008191 

-007977 
■007682 
.007599 
.007466 
.006252 

-005925 
.005809 
.005540 
.004705 
•003835 

.003729 
.007318 
•003545 
•003053 
-002959 

.002846 
.002647 
.002342 
.002179 
.002091 

.002048 
.001994 
.001862 
.001601 
.001526 

.001474 
.001452 
.001176 
.001170 

.001112 

- 001099 

.0009294 

.0009120 

.0008863 

.0007424 

.0007372 
.0007116 
.0006041 
.0005810 
.0005673 

.0005228 
.0004801 
.0004636 
0004393 
.0003777 

.0003667 
.0003630 
.0002973 
.0002941 
.0002908 

.0002716 
.0002441 
.0002382 
.0002323 
.0002295 
.0002216 
.0002042 



27.84 



29- 

30. 
31- 

36 ■ 



37^63 
39.08 
39-51 

40- 21 
48.01 

50-66 
51-68 
54- 19 
63-80 

78.28 

80.50 
80.75 
84-67 
98.32 
101.4 

105.5 
113-4 
128.2 
137-8 
143-6 

146.6 
150-5 
161 .2 
187^5 
196.8 

203.7 
206.7 
255-2 

256.6 
270.0 

273-1 
323-0 
329-1 
338.7 
404.4 

407-2 
421.9 
496.9 
SI6.7 
529.2 

574-2 

625 .2 

647.6 
683.4 

794-8 

818.7 

826.9 

1010. O 

1021.0 

1032.0 

1105.0 
1230.0 
1260.0 
1292 .0 
1308.0 
I3S5-0 
1470.0 



31-56 
33 69 
34-66 
36-11 
41-54 

42.66 
44-30 
44.78 
45-58 
54-42 

57-43 
58.58 
61.43 
72-32 
88.74 

91-25 
91-53 
95-98 

111. 5 
115 -0 

119. 6 
128.6 
145-3 
156.2 
162 .7 

166.2 
170.6 
182.7 
212.5 
223.0 

230.9 
234-3 
289-3 
290.9 
306.1 

309-6 
366.1 
373-1 
383-9 

458.4 

461 .6 
478.2 
563.3 
58s. 7 
599 9 

650.9 
708.7 
734-1 
774^6 
900.9 

928.0 
937.3 

1144.0 
1157. O 
1170.0 

I253^0 
I394.0 
1429.0 
1465.0 
1483.0 
1536.0 
1666.0 



92.74 
98.99 

101.9 

106. I 

122. I 

125.4 
130^2 
131.6 
133-9 
159.9 

168.8 

172.2 
180. S 

2 12.5 

260.8 

268.2 
269.0 
282. 1 
327 -s 
337.9 

351-3 

377.8 
426.9 
458.9 ' 
478.2 

488.3 
SOI./? 
536.?> 
624.6 
655-4 

678.4 
688.6 
850.2 
854-9 
899.4 

909.8 
1076 
1096 
1128 
1347 

1356 
1405 ■ 
1655 
1721 
1763 

1913 
2083 
2157 
2276 

2648 

2727 
2755 
3363 
3401 
3439 

3681 
4097 
4198 
4304 
4358 
4513 
4897 



228 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table.. 6. — Sizes and Properties of Wire — {Concluded) 
By the American Steel and Wire Company 



Gage numbers 



Ameri- 
can 
wire 
gage 
(B. & S.) 



Bir- 
ming- 
ham or 
Stubs 



British 
Imperial 
Standard 



Diameter 



Ins. 



64ths 



Deci- 
mally 



Milli- 
meters 



Sectional area 



Sq. ins. 



Circular 
mils 



Log. of 
sq. ins. 



Weight, lbs. per ft. 



Copper 



Iron and 
steel 



Aluminum 



Length, ft. per lb. 



Copper 



Iron 
and 
steel 



z8 



31 



34 



36 



38 



40 

41 

42 



44 
AS 



46 
47 
48 



49 
SO 



30 



32 



33 



34 



35 



33 



36 



38 



40 



36 



41 
42 



44 



4S 



46 



47 



49 
SO 



0148 
0142 
0140 
0136 
0132 

0130 
0128 
0126 
0124 
.0120 

.0118 
,0116 
.0113 
,0108 
,0104 

.0100 
0095 
.0092 
.0090 
.00893 

.008s 

.0084 

.0080 

.00795 

.0076 

0075 
.00708 
.0070 
.0068 
.0066 

.0063 
.0062 
.0060 

.oos8 
. 00S61 

.0055 
.0052 
.0050 
.0048 
.0046 

• 00445 

.0044 

.0040 

.00396 

.0036 

■00353 

.0032 

.00314 

.00280 

.00249 

.00240 
.00222 
.00200 
.00198 
.00176 

.00160 

.00157 

.00140 

.00124 

.00120 

.00100 

.000986 

.000878 



.3759 
.3607 
.3556 
•3454 
.3353 

.3302 
.3251 
.3200 
.3150 
.3048 

.2997 
.2946 
.2870 
.2743 
.2642 

.2540 
.2413 
.2337 
.2286 
.2268 

.2159 
.2134 
.2032 
.2019 
.1930 

.190S 
.1798 

.1778 
.1727 
.1676 

.1600 

.1575 
.1524 
.1473 
.1425 

.1397 
.1321 
.1270 
.1219 
.1168 

.1130 
.1118 
.1016 
.1006 
.09144 

.08966 
.08128 
.07976 
.07112 
.06325 

.06096 
.05639 
.05080 
.05029 
.04470 

. 04064 
.03988 
.03556 
.03150 
.03048 
.02540 
.02504 
.02230 



000172 
000158 
000153 
000145 
000136 

000132 
000128 
000124 
000120 
000113 

,000109 
.000105 
OOOIOO 

.000091 
.000084 

.000078 
.000070 
.000066 
.000063 
.000062 

.000056 
.000055 
.000050 
. 000049 
,000045 

. 000044 

.000039 

.000038 

,0000363 

,0000342 

,0000311 
,0000301 
,0000282 
,0000264 
.0000247 

.0000237 

.0000212 
.0000196 
.0000180 
.0000166 

.0000155 
.0000152 
.0000125 
.0000123 
.0000101 

.0000097 
.0000080 
.0000077 
.0000061 
.0000048 

.0000045 
.0000038 
.0000031 
.0000030 
.0000024 

.0000020 
.0000019 
.0000015 
.0000012 
,0000011 
.0000007 
.0000007 
.0000006 



219.04 
201 .64 
196.00 
184.96 
174-24 

169.00 

163.84 
158.76 
153-76 
144.00 

139.24 
134-56 
127.69 
116.64 
108.16 

100.00 

90.2s 
84.64 

81 .00 

79.74 

72.2s 
70.56 

64.00 
63.20 
57.76 

56.2s 

so. 13 

49.00 
46.24 

43.56 

39.69 

38.44 
36.00 
33.64 
31.47 

30.25 
27.04 
25.00 
23.04 

21 .16 

19.80 
19.36 
16.00 
15.68 
12.96 

12.46 
10.24 
9.860 
7.840 
6.200 

S.760 

4.928 
4.000 
3.920 
3.098 

2.560 
2.465 
1 .960 

1.538 
1.440 

I .000 

.9722 
.7709 



.235614 
. 199666 

.187346 
.162168 
.136238 

.122976 
.109510 
.095832 
.081934 
.053452 

.038854 

.024006 

.001246 

S. 961938 

.929156 

.895090 
.850538 
.822666 
.803576 
.796792 

.753928 
. 743648 

.701270 
.695824 
.656718 

.645212 

.595156 
_ .585286 
5.560108 

.534178 

.493772 
.479874 
.451392 

.421946 

.393016 
.375816 

.327096 
.293030 
.257572 
.220606 

.191810 
.181996 
.099210 
. 090480 
.007696 

6.990640 
.905390 
.888950 
.789406 
.687488 

.655512 
.587796 
.497150 
.488420 
.386116 

.303330 
.286890 
.187346 
.081934 
,.053452 
7.895090 
.882844 
.782080 



0006623 
0006097 
0005926 
0005592 
0005268 

0005110 
0004954 
0004800 
0004649 
.0004354 

.0004210 
. 000406& 
.0003861 
.0003527 
,0003270 

,0003023 
,0002729 
,0002559 
,0002449 
,0002411 

,0002184 
,0002133 
.0001935 
.0001911 
.0001746 

.0001701 
,0001516 
.0001481 
.0001398 
.0001317 

,0001200 
,0001162 
.0001088 
,0001017 
.00009515 

.00009146 
.00008175 
.00007559 
.00006966 
.00006398 

.00005987 
.00005853 
.00004838 
.00004741 
.00003918 

.00003768 
.00003096 
.00002981 
.00002370 
.00001875 

,00001742 
,00001490 
,00001209 
,0000118s 
,000009366 

,000007740 
.000007453 
.000005926 
. 000004649 
.000004354 
.000003023 
.000002939 
.000002331 



0005842 
0005378 
0005228 
0004933 
0004647 

.0004508 
.0004370 
.0004234 
.0004101 
.0003841 

,0003714 
.0003589 
. 0003406 
,0003111 
,000288s 

,0002667 
,0002407 
,0002258 
,0002160 
.0002127 

,0001927 
,0001882 
,0001707 
,0001686 
.0001541 

,0001500 
.0001337 
.0001307 
.0001233 
.0001162 

.0001059 

.0001025 

.00009602 

.00008972 

.00008394 

.00008068 
.00007212 
.00006668 
.00006145 
.00005644 

.00005282 
.00005164 
.00004268 
.00004183 
.00003457 

.00003314 
.00002731 
.00002630 
.00002091 
,00001654 

,00001536 
,00001315 
.00001067 
.00001046 
.000008262 

,000006828 
,000006574 
.000005228 
.000004101 
.000003841 
.000002667 
.000002593 
.000002056 



.0001988 
.0001830 
.0001779 
.0001679 
.0001581 

.0001534 
.0001487 
.0001441 
.0001396 
.0001307 

.0001264 
.0001221 
.0001159 
.0001059 
.00009817 

.00009076 
.00008191 
.00007682 
.OOQO7352 
.00007238 

.00006557 
. 00006404 
.00005809 
.00005736 
.00005242 

.00005103 
.00004549 

.00004447 
.00004197 
.00003954 

.00003602 
.00003489 
.00003267 
.00003053 
.00002856 

.00002746 
.00002454 
.00002269 
.00002091 
.00001920 

.00001797 
.00001757 
.00001452 
.00001423 
.00001176 

.00001131 

.000009294 

.000008949 

.000007116 

.000005627 

.000005228 
.000004473 
.000003630 
.000003558 
.000002811 

.000002323 

.000002237 

.000001779 

.000001396 

.000001307 

.0000009076 

.0000008824 

.0000006997 



1510 
1640 
1687 
1788 



I9S7 
2019 
2083 
2151 
2297 

2375 
2458 
2590 
2836 
3058 

3307 
3665 
3908 
4083 
4148 

4S78 
4687 
5 168 
5233 
5726 

5880 
6598 
6750 
7153 
7593 

8333 
8604 
9187 
9832 
10509 

10934 
12232 
13230 
14355 
15631 

16702 
17084 
20672 
21091 
25520 

26543 
32299 
33546 
42187 
53345 

57421 
67110 
82687 
8436s 
106775 

129198 
134182 
168748 
215105 
229685 
330746 
34020s 
429047 



1712 
1859 
1913 
2027 
2152 

2218 
2288 
2362 
2438 
2604 

2693 
2786 
2936 
3214 
3466 

3749 
4IS4 
4430 
4629 
4702 

S189 
5314 
5858 
5932 
6491 

666s 

7480 
7652 
8108 
8607 

9446 

9753 

1041S 

11145 

11913 

12394 
13866 
14997 
16273 
17718 

18933 
19366 
23433 
23909 
28929 

30088 
36614 
38026 
47822 
60471 

65091 
76074 
93731 
95634 
121036 

I 464s S 
152105 
191288 
243837 
260364 
374924 
385646 
486354 



WIRE AND SHEET METAL GAGES 



229 



The only wire gage which has been recognized in Acts of Congress 
is the Birmingham gage. The Treasury Department has for many 
years used this gage in connection with importations of wire, and the 
adoption of succeeding tariff acts with provisions for the assessment 
of duty according to gage numbers gives legislative sanction to the 
gage. 

Until certain provisions of the tariii act are amended, the Treasury 
Department probably cannot discontinue the use of the Birmingham 
gage. It should, however, be abandoned by all other users, since 
the gage itself is radically defective and it is nearly obsolete, both in 
the United States and in Great Britain, where it originated. 

For copper wires and wires of other metals the gage universally 
recognized in the United States is the American wire gage, also 
known as the Brown and Sharpe. No confusion need arise between 
the steel wire gage and the American wire gage, because the fields 
covered by the two gages are distinct and definite. 

The piano wire gage now designated as music wire gage is the 
same as heretofore employed under the name American Steel and 
Wire Co.'s music wire gage and is adopted as standard for piano 
wire upon recommendation of the United States Bureau of Standards. 

Table 7. — ^Ultimate Tensile Strength of Wire 
From J. Bucknell Smith's Treatise on Wire 

Lbs. per 
sq. in. 

Black or annealed iron 56,000 

Bright hard-drawn iron 78,400 

Bessemer steel 89,600 

Mild Siemens-Martin steel 134,000 

High carbon Siemens-Martin steel 179,200 

Crucible cast steel 224,000 

Plow steel 268,800 

Piano wire 315,000 

Hard-drawn copper 63,000 

Annealed copper 34,000 

Hard-drawn brass (depending on com- 
position) 4S>ooo to 90,000 



Table 8. — Loads Carried by Wires when Stressed to 100,000 

Lbs. per Sq. In. 

Loads for other stresses in direct proportion 

By the American Steel & Wire Co. 



Steel wire 


Diam., 


Load, 


Steel wire 


Diam., 


Load, 


gage no. 


ins. 


lbs. 


gage no. 


ins. 


lbs. 


8 


.1620 


2061 


IS 


.0720 


407 


H 


•1551 


1889 


H 


.0696 


380 


9 


.1483 


1727 


H 


.0672 


355 


M 


.1416 


IS7S 


M 


.0649 


331 


10 


■1350 


1431 


16 


.0625 


307 


H 


.1277 


1281 


M 


.0604 


286 


II 


.1205 


1 140 


M 


.0582 


266 


M 


.1167 


1070 


M 


.0561 


247 


M 


.1130 


1003 


17 


.0540 


229 


^ 


.1092 


937 


yi 


■0524 


216 


12 


•loss 


874 


y2 


.0507 


202 


H 


. 1020 


817 


M 


.0491 


189 


H 


.0985 


762 


18 


■047s 


177 


M 


.0950 


709 


y^ 


•0459 


165 


13 


•091S 


658 


H 


.0442 


153 


y^ 


.0886 


616 


M 


.0426 


143 


H 


•0857 


577 


19 


.0410 


132 


^ 


.0829 


S40 


M 


•0394 


122 


14 


.0800 


503 


M 


• 0379 


113 


M 


.0780 


478 


M 


■ 0363 


103 ■ 


H 


.0760 


454 


20 


.0348 


95 


y^ 


.0740 


430 









HYDRAULICS AND HYDRAULIC MACHINERY 



For tabulated values of barometric pressures at various altitudes 
in ins. of mercury, lbs. per sq. in., and ft. of water see Barometric 
Pressure. 

For the relations of British and American measxires of capacity see 
Weights and Measures. 

Table i. — Hydraulic Constants 
Weight, Volume and Pressure of Water 



I cu. in. 






= 0.03608 lb. 


I cu. ft. 






= 62.35s lbs. 


I cu. ft. 






= 7.481 U., S. gals. 


1 U. S. gal. 






= 231. cu. ins. 


I U. S. gal. 






= 8.34 lbs. 


I U. S. gal 






= 0.1337 cu. ft. 


lib. 






= 0.016037 cu. ft. 


lib. 






= 27.712 cu. ins. 


lib. 






= 0.1199 U. S. gal. 


loo ft. head of water 




= 43.31 lbs. per sq. in 


100 lbs. per sq. 


in. 




= 230.9 ft. head. 


Value of I 


Atmosphere of Pressure 


Lbs. per 




Ft 


head Ins. of 


sq. in. 




of 


water mercury 


14.7 




33-947 30 



Temperature 62 deg. Fahr. 
Table 2. — Pressure per Square Inch Converted into Feet Head 
OF Water at 62° Fahr. 

Read the tens at the left and the units at the heads of the columns. Thus for 
30 lbs. per sq. in., read 69.2841 ft. head, and for 34 lbs. read 78.522 ft. For pres- 
sures greater than those shown, move the decimal point. Thus the head for 26 lbs. 
is 60.046 ft., and for 260 lbs., 600.46 ft. 



4) t, 

3 p,,f 



30 

40 

50 
60 
70 
80 
90 



23.0947 
46. 1894 
69.2841 
92.3788 

115.473s 
138.5682 
r6i .6629 
184.7576 
207.8523 



2.309 

25.404 
48.499 
71.594 
94.688 

117.78 
140.88 
163.97 
187.07 
210. 16 



4.619 
27.714 
50.808 
73.903 
96.998 

120.09 

143.19 
166.28 
189.38 
212.47 



6.928 
30.023 
S3. 118 
76.213 
99.307 

112.40 
145.50 
168.59 
191 .69 
214.78 



Heads, ft. 

11-547 
34-642 
57-737 
80.831 
103-93 

127.02 
150.12 
173-21 
196.31 
219.40 



9. 


238 


32. 


333 


55. 


427 


78. 


522 


101 


.62 


124 


-71 


147 


.81 


170 


-90 


194 


.00 


217 


09 



13.857 
36-952 

60.046 

83.141 

106.24 

129.33 

152.42 
175-52 
198.61 

221 .71 



16.166 
39.261 
62.356 
85.450 
108.55 

131.64 

154-73 
177-83 
200.92 
224,02 



18.476 
41.570 
64.665 
87.760 
110.8s 

133-95 

157-04 
180-O4 
203.23 
226.33 



20.78s 
43-880 
66-975 
90 . 069 
113-16 

136.26 
159-35 
182.45 
205.54 
228.64 



Table 3. — Feet Head of Water Converted into Pressure per 
Square Inch at 62° Fahr. 

Read the tens at the left and the units at the heads of the columns. Thus for 
30 ft. head, read 12.990 lbs. per sq. in., and for 34 ft. read 14.722 lbs. For heads 
greater than those shown move the decimal point. Thus the pressure for 26 ft. is 
11.258 lbs., and for 260 ft., 112.58 lbs. a 



a J 





I 


2 


3 


4 


S 


6 


7 


8 


9 










Pressure, lbs. per sq. in. 













0.433 


0.866 


1.299 


1-732 


2.165 


2.598 


3-031 


3.464 


3.897 


10 


4-330 


4763 


S.196 


5.629 


6.062 


6.495 


6.928 


7-361 


7-794 


8.227 


20 


8.660 


9 093 


9.526 


9.959 


10.392 


10.825 


11.258 


II .691 


12-I24 


12.557 


30 


12 .990 


13-423 


13.856 


14.289 


14.722 


15.155 


15.588 


16 -02 1 


16.454 


16.887 


40 


17.320 


17-753 


18.186 


18.619 


19.052 


19.485 


19.918 


20.351 


20.784 


21.217 


SO 


21.650 


22.083 


22.516 


22.949 


23.382 


23-81S 


24.248 


24.681 


25.114 


25.547 


60 


25 .980 


26.413 


26.846 


27.279 27.712 28.145 


28.578 


29.011 


29-444 


29.877 


70 


30.310 


30 . 743 


3I-I76 


31.609 32.042 32.475 


32.908 


33.341 


33-774 


34-207 


80 


34-640 


35.073 


35.506 


3S.939I36.372 36.805 


37.238 


37.671 


38. 104 


38.537 


90 


38.970 


39.403 


39.836U0.269I40.702I41.135 


41.568 


42 .001 


42.436 


42.867 



The Capacity of Cylindrical Tanks 

The capacity of horizontal cylindrical tanks with flat ends, full and 
partly full, may be obtained from Tables 4 and 5, the latter by 
Robert Mawson (Amer. Mach., Nov. 20, 1913) giving the capacity 
per foot of length for depths of liquid up to the center line and for 
full tanks. For tanks more than one-half full find the vacant 
volume from the table and subtract it from the volume of the full 
tank. If the tank is more than one-eighth full, intermediate values 
may be found by direct interpolation with small error; 

The capacity of vertical cylindrical tanks may be obtained from 
Table 6. 

The capacity of rectangular tanks may be obtained from Table 7. 

The Flow of Water 

The fundamental formula for the flow of water under the action 
of gravity is the same as that for the law of falling bodies, viz: 



v = \/64^4h 
= 8 \/h nearly 

in which v = velocity of efiBux, ft. per sec, 
A = pressure head, ft. 

The theoretical volume of water discharged is equal to the velocity 
multiplied by the area of the orifice. The actual volume is equal 
to the theoretical volume multiplied by a coefiicient of discharge 
which varies with the nature of the orifice. The following values of 
this coefiicient are from Clark's Manula of Rules, Tables and Data: 

Nature of orifice CoefiScient 

of discharge. 

Thin plate 0.62 

Cylinder at least 2 diameters in length 0.82 

Converging cone, length = 25 diameters 0.95 

Contracted vein, length = | diameter of orifice, i.oo 

smallest diameter = .785 diameter of orifice. 
Diverging cone, length = 9 diameters 1.46 

The spouting velocity, discharge and horse-power of water jets, to- 
gether with the proper diameters of impulse water-wheels, may be 
obtained from Fig. i, by R. A. Bruce {Amer. Mach., Jan. 5, 1899). 
The use of the chart is shown by the example below it. 

The assumption is made that the speed of an impulse wheel should 
be 50 per cent, of the speed of the jet. This is, of course, sometimes 
departed from for various practical reasons. 

The velocities given by the actual velocity curve are 95 per cent, 
of those given by the theoretical velocity curve. If the reader pre- 
fers to make his own allowances from theoretical results, it is only 
necessary to trace the theoretical velocity curve and then proceed 
as in the example. 

Exact results must not be expected in calculations of the flow of 
water in pipes. In addition to the different results given by the 
different formulas is the ever present question of the condition of 
the pipe, which is scarcely capable of exact expression and which 
renders unimportant the differences between the results given by 
different formulas. The formulas are intended to apply directly 
to a standard condition of smoothness and cleanness and, since pipe- 
lines are almost certain to become foul in time, the calculated di- 
ameters should be increased. An addition of about 15 percent, to 
the calculated diameter will provide for an extreme condition of 
roughness. 

(.Continued on page 234, first column) <, 



230 



HYDRAULICS AND HYDRAULIC MACHINERY 



231 



Revs, per Min. 



Revs, per Min. 

BOO M 80 70 6045049 30 20 1040090 80 70 6035040 30 20 10 300 90 80 70 60 25040 30 20 10 200 90 80 70 60I5O40 30 20 10 100 9o' 

Lbs. pex Sec. 
" • - "" 30 40 SO 60 70 80 90 100 10 20 30 40 150 60 70 80 00 200 10 20 30 40 250 60 70 80 90 300 10 20 30 40 350 60 70 80 90 400 10 

■ U^J.^.I I/I !.<!/ I |i N I lei 1/ I IS.I IMI 1.1 I/I I I I.V.I Mill l>\N/1l'.>l I M I K< M I I IM m I M \\\ M IX'^ 



10 20 



) 70 60 SO 40 30 20 10 
20 30 40 450 60 70 80 90 500 




20 40 60 80100 20 40 60 80 200 20 40 60 



1300 20 40 60 80 400 20 40 60 80 600 20 40 60 80 600 20 40 60 80JOO20 40 60 80 800 20 40 60 80 900 20 40 60 80 1000 

Head in Feet or Horse Power of Jet 



Assuming a head of 440 ft., trace vertically from A to E, thence horizontally to H, where read 159 ft. per sec, actual velocity, or trace 
to F and thence to G where read 168 ft. per sec. theoretical velocity. Trace horizontally from £ to X on 2i-in. diagonal, thence vertically 
to M and read 274 lbs. per sec. discharge of 2^-in. nozzle under 440 ft. head. Trace vertically from ^ to 5, thence horizontally to C on 
2i-in. diagonal, thence down to D, where read 220 horse-power for a 2i-in. nozzle under 440 ft. head. Trace from ^ to £ to iV on 
i8-ft. diameter line, thence vertically and read 85 r.p.m. for an 85 -ft. impulse wheel under 440 ft. head. 

Fig. I. — Spouting velocity, discharge and horse-power of water jets. 




Taiik-|-full y=.860 
Tank-^full y = .748 



Tank -|- full ^ = .626 



-^Tank -^tall :p =.500 



Tank 4 iuH -^=.374 



Tank ■\- full ^ = .140 
Tank-i full^ = .052 



Tank empty 



Tank divided into lo parts o£ equal height. 





10 ^^,^ .^ 




w \^^ 


/ 


^ X.^--"^ 




^ X^--^ 




V 




10 


\ 


10 / 


\ 


To /^~'~^--^ 


\ 


10 / ^v^ 




1 ^^-^ "^ 



• Tanlt full 



• Tank ^ full -^ = 
Tank 4" full 4 = 



Tank-TT full ~ = . 



Tank 4- full -g- = 



-►Tank i-full-^- 



>.Tank empty 
Tank divided into lO parts of equal volume. 



V = total volume of tank, 

II = volume occupied by liquid, 

D = diameter of tank, 

h = height of liquid in tank. 

Table 4. — Capacity of Horizontal Cylindrical Tanks 



232 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 5. — Capacity of Horizontal Cylindrical Tanks per Foot of Length in Cubic Inches, Cubic Feet and U. S. Gallons 


Depth of 


Volume 


Depth of 


Volume 


Depth of 


Volume 


Depth of 


Volume 


liquid, 




liquid, 
ins. 




liquid, 
ins. 




liquid, 
ins. 




ins. 


Cu. ins. 1 Gals. 


Cu. ft. 1 Gals. 


Cu. ft. Gals. 


Cu. ft. 1 Gals. 


Tank 6 ins. diameter 


Tank 16 ins. diameter 


Tank 27 ins. diameter 


Tank 3 ft. 3 ins. diameter 


I 


36.9552 


.160 


I 


.0359 


.268 


12 


1. 70s 


12.723 


12 


2 . 161 


16. 127 


2 


98.8666 


■ 423 


2 


. 100 


.746 


13 


1. 891 


14. Ill 


13 


2.414 


17.988 


3 


169.6464 


• 734 


3 


. 180 


1.343 


13J2 


1.988 


14-835 


14 


2.688 


20.000 


Full 


339.2928 


1.469 


4 


.273 


2.037 


Full 


3.976 


29.671 


15 


2.933 


21.888 




5 


.372 


2.776 


Tank -^0 ins. diameter 


16 


3.202 


23.895 


Ta 

I 


nk 7 ms. diame 
40.5273 


ter 

• 175 


6 

7 


.478 
.586 


3.567 
4./73 


I 
2 


.049 
.138 


.365 
1 .029 


17 
18 


3.410 
3.72s 


25.447 
27.798 


2 


109 .0299 


.472 


8 


.698 


5 .208 


3 


.255 


1 .905 


19 


4.010 


29.925 


3 


189.299s 


.819 


Full 


1.396 


10.417 


4 


• 387 


2.888 


19M 


4.147 


30.951 


3H 


230.9076 


.999 


Tank 18 ins. diameter 


5 


■534 


3.985 


Pull 


8.295 


61 .902 


Full 


461 .8152 


1.999 


I 


.038 


.283 


6 


.696 


5.194 


Tank 3 ft. 6 ins. diameter 


Tank 8 ins. diameter 


2 


.107 


.799 


7 


.868 


6.477 


I 


.056 


.417 


1 




3 


.192 


1.432 


8 


1.047 


7.813 


2 


.164 


1.223 


I 


43.5179 


.188 


4 


.292 


2.179 


9 


1.238 


9.238 


3 


.302 


2.253 


2 


116.7233 


.506 


5 


.399 


2.977 


10 


1.430 


10.671 


4 


.464 


3.462 


3 


206 .6027 


.894 


6 


.515 


3.843 


II 


1 .627 


12. 141 


5 


.646 


4.820 


4 


301.5936 


1.305 


7 


.635 


4.738 


12 


1.835 


13.679 


6 


.852 


6.358 


Pull 


603.1872 


2 .610 


8 


.757 


5.649 


13 


2.036 


15.194 


7 


1.075 


8,022 


Tank g ins. diameter 


9 


.883 


6.592 


14 


2.242 


16.732 


8 


I .272 


9.492 








Full 


1.767 


13.185 


15 


2.454 


18.313 


9 


1 .509 


II .261 


I 


46 . 2992 


.200 


Tank 20 ins. diameter 


Full 


4.908 


36.626 


10 


1.754 


13.089 


2 
3 


126.1393 
222.4499 


.5^6 
.963 


I 


.040 


.298 


Tank 33 ins. diam 


ster 


II 

12 


1.998 

2 .260 


14.911 
16.865 


4 


327.3851 


1.417 


2 


.113 


.843 


I 


.051 


.380 


13 


2.540 


18.95s 


4K2 


381.7044 


1.652 


3 


.205 


1.529 


2 


.145 


1 .082 


14 


2.803 


20 .917 


Full 


763.4088 


3.304 


4 


.310 


2.313 


3 


.269 


2 .007 


IS 


3.082 


23 .000 




S 


.426 


3.179 


4 


.408 


3.044 


16 


3.366 




Tank 10 ins. diameter 


6 


.550 


4. 104 


5 


.564 


4.208 


25.119 


I 


49.0500 


.212 


7 


.680 


5. 074 


6 


.732 


5.462 


17 
18 


3.641 - 
3.931 


27.171 
29.33s 


2 


134.1888 


.580 


8 


.814 


6.075 


7 


.919 


6.858 


19 


4.223 


31 .514 


3 


237 .8016 


1 .029 


9 


.952 


7.104 


8 


1 .109 


8.276 


20 


4.516 


33.701 


4 


352.0440 


1.524 


10 


1 .090 


8.138 


9 


1.307 


9.753 


21 


4.846 


36 . 164 


5 


471 .2400 


2 .040 


Full 


2. 181 


16.276 


10 


1. 519 


11.335 


Full 


0.602 


72 .328 


Full 


942.4800 


4.080 


Ta 


nk 22 ins. diam 


ster 


II 
12 


1.732 
1.946 


12 .920 

14.522 


Tank 4 ft. diameter 


Tat 

I 


ik 11 ins. diamc 
51 .6636 


ter 

.223 


I 
2 
3 


.042 
.117 
.215 


■ 311 

.873 

1 .604 


13 

14 


2.166 
2.397 


16. 164 
17.888 


2 

4 


•174 
.497 


I .298 
3.708 


2 
3 


141 .8244 
252.3000 


.614 
1 .092 


4 

s 


.325 

.452 


2 ..42s 
3.373 


15 
16 


2 .622 
2.848 


19.567 

21.253 


6 
8 


.906 
1.368 


6.761 
10.208 


4 


373.8000 


1. 618 


6 


.581 


4-335 


i6^i 


2.969 


22. 160 


10 


1.892 


14. 119 


S 


503 4960 


2.179 


7 


.721 


5.380 


Full 


5939 


44-323 


12 


2.456 


18.328 


sH 


570. 1980 


2.468 


8 


.865 


6.455 


Tank 3 ft. diameter 


14 


3.038 


22.677 


Full 


1 140. 3960 


4.936 


9 


1 .015 


7. 573 


I 


.052 


.388 


16 


3.361 


25.082 
















18 


4.304 


32 . 119 


Tank 12 ins. diameter 


10 


1.165 


8.694 


2 


.152 


I -134 


20 


4.961 


37 .022 




II 


1. 319 


9.850 


3 


.279 


2 .082 


























22 


5 .611 


41 .873 


I 


53.7012 


.232 


Full 


2.639 


19.700 


4 


.428 


3.194 


24 


6.283 


46.888 


2 


147.8220 


• 639 


Tank 24 ins. diameter 


5 


.595 


4.440 


Pull 


12.566 


03.776 


3 


265 .3274 


1 .148 




6 


.769 


5. 738 






4 


395.4660 


I. 712 


I 


.044 


.328 


7 


.963 


7.186 


Tank 4 ft. 6 ins. diameter 


5 


534-1176 


2.312 


2 


.124 


.925 


8 


1 .167 


8.708 


2 


.190 


I. 417 


6 


678.5880 


2.924 


3 


.226 


1.686 


9 


1. 381 


10. 30s 


4 


.531 


3.962 


Full 


1357.1760 


5.848 


4 


.342 


2.553 


10 


1.595 


II .902 


6 


.964 


7.194 




5 


.473 


3.529 


II 


1.824 


13. 611 


8 


1 .466 


10.940 


Tank 14 ins. diameter 


6 


.614 


4.582 


12 


2.059 


15.365 


10 


2.056 


15.343 


I 


0336 


.250 


7 


.759 


5.664 


13 


2.299 


17.156 


12 


2.627 


19.604 


2 


.093 


■ 693 


8 


.915 


6.777 


14 


2.534 


18.910 


14 


3.268 


24.388 


3 


.167 


1 .246 
1.873 
2.552 


9 


1.076 


8.029 


15 


2.781 


20.753 


16 


3.901 


29. Ill 


4 
5 


.251 
• 342 


10 

II 


I .222 
1.402 


9. 119 

10.462 


16 
17 


3.031 
3.282 


22 .619 

24.492 


18 
20 


4.634 
5.349 


34582 
39.917 


6 


.436 


3 .253 


12 


1. 570 


II .720 


18 


3.534 


26.373 


22 


6.079 


45.365 


7 


• 534 


3.984 


Full 


3. 141 


23.441 


Full 


7.068 


52.746 


24 


6.820 


50.895 


Full 


1 .067 


7.968 


Tank 27 ins. diameter | 


Tank 3 ft. 3 ins. diameter j 


26 


7.567 


56.470 














27 


7.952 


59. 343 


Tank 15 ins. diameter 


I 


■ 047 


.351 


I 


.055 


.414 


Full 


I S . 004 


118.686 




2 


.132 


.985 


2 


.159 


1. 186 




I 


.0346 


.258 


3 


.241 


1.798 


3 


.293 


2.186 


Tank 5 ft. diameter 


2 


.096 


.724 


4 


.366 


3.733 


4 


.444 


3.313 


2 


.197 


1.470 


3 


.175 


1.305 


5 


.506 


3.776 


S 


.619 


4.619 


4 


553 


4.126 


4 


.261 


1.947 


6 


.656 


4.895 


6 


.8io 


6.044 


6 


I .021 


7.619 


5 


.357 


2.664 


7 


.817 


6.097 


7 


1 .015 


7.574 


8 


1.555 


II .604 


6 


.458 


3.417 


8 


.984 


7.343 


8 


1. 231 


9. 186 


10 


2.138 


15.955 


7 


.567 


4.231 


9 


1. 158 


8.642 


9 


1. 441 


10.753 


12 


2.795 


20.858 


7^^ 


.613 


4.578 


10 


1.337 


9.977 


10 


1.676 


12.507 


14 


3.474 


25.92s 


Full 


1 .227 


9.156 


II 


1.539 


10.738 


11 


1 .920 


14.328 


16 


4.188 


31.253 



HYDRAULICS AND HYDRAULIC MACHINERY 



233 



Table 5. — Capacity of Horizontal Cylindrical Tanks per Foot of Length in Cubic Inches, Cubic Feet and U. S. Gallons- 

(Continued) 



Depth of 
liquid. 


Volume 


Depth of 

liquid, 

ins. 


Volume 


Depth of 

liquid, 

ins. 


Volume 


Depth of 

liquid, 

ins. 


Volume 


ins. 


Cu. ft. Gals. 


Cu. ft. 


Gals. 


Cu. ft. Gals. 


Cu. ft. 


Gals. 


Tank 5 ft. diameter 


Tank 6 ft. 6 ins. dia 


meter 


Tank 8 ft. diameter 


Tank 9 ft. diameter 


18 


4-954 


36.970 


36 


14.944 


III .522 


26 


10.951 


81.723 


52 


30.270 


225 .896 


20 


S.72I 


42 . 694 


38 


16.083 


120 .022 


28 


12.159 


90.738 


54 


31.808 


237.376 


22 


6.507 


48.559 


39 


16.591 


123.817 


30 


13.388 


99.5^10 


Full 


63.617 


474-753 


24 
26 
28 


7.334 
8.147 
8.965 


54-731 
60.798 
66 . 902 


• Full 33.183 1 247.634 
Tank 7 ft. diameter 


32 
34 
36 


14-645 
15-923 
17.152 


109.290 
118.828 
128 .000 


Tank 
2 


9 ft. 6 ins. diameter 

.289 2.156 


30 


9.817 


73 ■ 264 


2 


.241 


1.798 


38 


18.458 


137.746 


4 


.779 


5-813 


Full 


19.635 


146.529 


4 


.656 


4-895 


40 


19-777 


147-589 


6 


1 .460 


10.890 


Tank 5 ft. 6 ins. diameter 


6 
8 


1.209 
1. 857 


9-023 
13-858 


42 
44 


21 .III 
22.444 


157-545 
167 .492 


8 
10 


2.184 
3.008 


16. 290 
22.433 


2 


.207 


1-544 


10 


2.584 


19.283 


46 


23.791 


177.544 


12 


3.962 


29.567 


4 


.581 


4-335 


12 


3-377 


25.200 


48 


25-132 


187.558 


14 


4-995 


37.276 


6 


1.076 


8.029 


14 


4-191 


31.276 


Full 


50-265 


375.116 


16 


6.220 


46.417 


8 
ID 


1.634 
2.256 


1^.194 
16.835 


16 
18 


5.090 
6.034 


37.980 
45 . 029 


Tank 8 ft. 6 ins. diameter 


18 
20 


6.992 
8.333 


52.179 
62.186 


12 


2.931 


21.873 


20 


7.018 


52.373 


2 


.270 


2.014 


22 


9.590 


71.567 


14 


3.675 


27-425 


22 


7.869 


58. 723 


4 


.730 


5. 446 


24 


10.819 


80.738 


16 


4.281 


31-947 


24 


9.041 


67.470 


6 


1.348 


10.059 


26 


12.159 


90.738 


18 


5.229 


39.022 


26 


10. IIS 


75.48s 


8 


2.056 


15-343 


28 


13-465 


100.480 


20 


6.077 


45.355 


28 


II .215 


83 - 694 


10 


2 .850 


21 .268 


30 


14.881 


III. 053 


22 


6.922 


51-656 


30 


12.333 


92.037 


12 


3.716 


27.731 


32 


16 . 243 


121 .216 


24 


7.786 


S8. 104 


32 


13-465 


100.485 


14 


4.679 


34-917 


34 


17.715 


132 .200 


26 


8.666 


64.671 


34 


14.618 


109.087 


16 


5. 702 


4*- 552 


36 


19131 


142 .768 


28 


9.500 


76.896 


36 


15.729 


117 .380 


18 


6.724 


SO. 179 


38 


20.652 


154 .110 


30 


10.486 


78.253 


38 


16.89s 


126 .082 


- 20 


7.848 


58.567 


40 


22.134 


165.17s 


32 


II .361 


84 - 783 


40 


18.069 


134-843 


22 


8.958 


66.850 


42 


23.666 


176 .611 


33 


11.879 


88 . 648 


42 


19-242 


I43-S97 


24 


10.166 


75.86s 


44 


25-243 


188.380 


Full 


23.758 


177.297 


Full 


38.484 


287.194 


26 


11.409 


85.141 


46 


26.743 


199.570 


Tank 6 ft. diameter 


Tank 7 ft. 6 ins. dia 


meter 


28 
30 


12.618 
13 -920 


94 - 164 
103,880 


48 
50 


28.340 
29.861 


211 ,493 
222 .844 


2 


.211 


1.570 


2 


.243 


1. 813 


32 


IS -180 


113-283 


52 


31.562 


235.530 


4 


.608 


4-537 


4 


.682 


5.089 


34 


16.534 


123.388 


54 


33 097 


246 . 993 


6 


1. 118 


8-343 


6 


1.274 


9.S07 


36 


17.882 


133.447 


56 


34-625 


258.396 


8 


1. 714 


12.791 


8 


1.936 


14.447 


38 


19-223 


143.455 


57 


35-442 


264 .486 


10 


2.356 


17.582 


10 


2.679 


19.992 


40 


20.701 


154 -480 . 


Full 


70.883 


528.973 


12 

14 


3.079 
3.853 


22.977 
28.753 


12 
14 


3.488 
4-398 


26.029 
32.820 


42 
44 


21.972 
23.402 


163.970 
174-641 


Tank 10 ft. diameter 


16 


4.671 


34-858 


16 


5-321 


39.708 


46 


24-833 


185 .320 


2 


.297 


2.216 


18 


S.527 


41 . 246 


18 


6.290 


46 . 940 


48 


26.208 


195 580 


4 


.790 


5.895 


20 


6.384 


57-641 


20 


7.298 


54-462 


50 


27.722 


206 . 883 


6 


1.470 


10.970 


22 


7.299 


54-470 


22 


8.347 


62 .291 


Full 


56.74s 


423.470 


8 


2.215 


16.529 


24 
26 


8.236 
9.194 


61 .462 
68.611 


24 
26 


9.472 
10.S83 


70.686 
78.977 




Tank 9 ft. diameter 


10 
12 


3.107 
4.087 


23. 186 
30 . 500 


28 


10. 138 


75.655 


28 


II-7IS 


87.440 


2 


.280 


2.089 


14 


S.080 


37.910 


30 


II. 125 


83.302 


30 


12.875 


96 . 082 


4 


.760 


5-671 


16 


6.403 


47.780 


32 


12. 125 


90.480 


32 


14. 104 


105.253 


6 


1 .406 


10.492 


18 


7.387 


55. 126 


34 


13.125 


97 • 947 


34 


15.291 


114. Ill 


8 


2.125 


15.856 


20 


8.556 


63.850 


36 


14.137 


105.500 


36 


16.500 


123.134 


10 


2.929 


21.858 


22 


9.916 


74 . 000 


Full 


28.274 


211 .000 


38 


17.722 


132.241 


12 


3.858 


28.790 


24 


11.180 


83.433 


Tank 6 ft. 6 ins. diameter 


40 
42 


18.944 
20.236 


141-373 
151 ,000 


14 
16 


4.80s 
S.868 


35.858 
43 . 790 


26 
28 


12.473 
13-896 


93 082 
103 .700 


2 


.221 


1.649 


44 


21.472 


160.238 


18 


6.929 


SI. 708 


30 


15-354 


114-582 


4 


.638 


4-761 


45 


22.089 


164.840 


20 


8.034 


59-955 


32 


16.750 


125 000 


6 


1 .152 


8-597 


Full 


44.178 


329.680 


22 


9.250 


69.029 


34 


18.270 


136-343 


8 
10 


1.777 
2.450 


13 -261 
18.283 


Tank 8 ft. diame 


ter 


24 
26 


10.506 
11.736 


78 . 400 
87.580 


36 
38 


19.819 
21 .291 


147.902 
IS8.891 


12 


3.i8i 


23-738 


2 


-257 


1.923 


28 


13.069 


97.529 


40 


22.882 


170.761 


14 


4.029 


30 - 067 


4 


.699 


5.216 


30 


14.361 


107 .170 


42 


24.500 


182.840 


16 


4.894 


36.522 


6 


1 .292 


9.641 


32 


15.680 


117.014 


44 


26.027 


194 231 


18 


5. 765 


43 022 


8 


1.988 


14-835 


34 


17-097 


127.589 


46 


27.673 


206.514 


20 


6.707 


50.052 


10 


2.770 


20.671 


36 


18-534 


138.313 


48 


29.333 


218.156 


22 


7.563 


56 . 440 


12 


3.626 


27.059 


38 


19916 


148.626 


SO 


30.908 


230.656 


24 


8.64s 


64.514 


14 


4.500 


33.580 


40 


21-395 


159-664 


52 


32.590 


243 . 209 


26 


9.666 


72.134 


16 


5. 475 


40.858 


42 


22.812 


170.238 


54 


34-277 


255-790 


28 


10.673 


79 . 649 


18 


6.5^9 


48.874 


44 


24.319 


181 .480 


56 


35-868 


267 .671 


30 


11.736 


87 . 582 


20 


7.46s 


55 -708 


46 


25.833 


192.783 


58 


37-569 


280.360 


32 


12.805 


95-557 


22 


8.680 


64.776 


48 


27.277 


203. 559 


60 


39-269 


293-056 


34 


13.722 


102 .403 


24 


9.826 


73.328 


SO 


29. OSS 


216.828 


Full 


78.539 


586 -112 



234 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 6. — Capacity of Vertical Cylindrical Tanks in Cubic 

Feet and U. S. Gallons, from the National Tube Go's. 

Book of Standards 



Diam- 


Cu. ft. 


Gals. 


Diam- 


Cu. ft. 


Gals. 


Diam- 


Cu. ft. 


Gals. 


eter, 


per ft. 


per ft. 


eter, 


per ft. 


per ft. 


eter, 


per ft. 


per ft. 


ft. ins. 


depth 


depth 


ft. ins. 


depth 


depth 


ft. ins. 


depth 


depth 


I 


.785 


5. 87 


5 


8 


25.22 


188.66 


19 





283.53 


2120.9 


I I 


.922 


6.89 


5 


9 


25-97 


194-25 


19 


3 


291 .04 


2177.1 


I 2 


1.069 


8.00 


S 


10 


26.73 


199 ■ 92 


19 


6 


298.65 


2234.0 


I 3 


1.227 


9.18 


S 


11 


27.49 


205.67 


19 


9 


306.35 


2291.7 


I 4 


1-396 


10.44 


6 





28.27 


211.51 


20 





314-16 


2350.1 


I S 


1-576 


11.79 


6 


3 


30.68 


229.50 


20 


3 


322.06 


2409 . 2 


I 6 


1.767 


13-22 


6 


6 


33.18 


248.23 


20 


6 


330.06 


2469 . I 


I 7 


1.969 


14.73 


' 6 


9 


35-78 


267 .69 


20 


9 


338.16 


2529.6 


I 8 


2.182 


16.32 


7 





38.48 


287.88 


21 





346.36 


2591.0 


I 9 


2.405 


17-99 


7 


3 


41.28 


308.81 


21 


3 


354-66 


2653.0 


I 10 


2.640 


19-75 


7 


6 


44.18 


330.48 


21 


6 


363.05 


2715.8 


I 11 


2.88s 


21.58 


7 


9 


47.17 


352.88 


21 


9 


371-54 


2779.3 


2 


3-142 


23 -SO 


8 





50.27 


376.01 


22 





380.13 


2843.6 


2 I 


3-409 


25-50 


8 


3 


53.46 


399.88 


22 


3 


388.82 


2908.6 


2 2 


3.687 


27-58 


8 


"6 


56.75 


424.48 


22 


6 


397.61 


2974-3 


2 3 


3.976 


29-74 


8 


9 


60. 13 


449.82 


22 


9 


406 . 49 


3040.8 


2 4 


4-276 


31-99 


9 





63.62 


475.89 


23 





415.48 


3108.0 


2 S 


4.587 


34-31 


9 


3 


67.20 


502.70 


23 


3- 


424-56 


3175.9 


2 6 


4-909 


36.72 


9 


6 


70.88 


530.24 


23 


6 


433.74 


3244.6 


2 7 


5-241 


39-21 


9 


9 


74.66 


558.51 


23 


9 


443-01 


3314.0 


2 8 


5.585 


41.78 


10 





78.54 


587.52 


24 





452-39 


3384-1 


2 9 


5 -940 


44-43 


10 


3 


82.52 


617.26 


24 


3 


461 .86 


3455.0 


2 10 


6 -30s 


47-16 


10. 


6 


86.59 


647.74 


24 


6 


471'- 44 


3526.6 


2 II 


6.681 


49-98 


10 


9 


90.76 


678.9s 


24 


9 


481 . 11 


3598.9 


3 


7.069 


52.88 


II 





95.03 


710.90 


25 





490.87 


3672.0 


3 I 


7-467 


55.86 


II 


3 


99.40 


743.58 


25 


3 


500.74 


3745-8 


3 2 


7.876 


58.92 


II 


6 


103.87 


776.99 


•25 


6 


510.71 


3820.3 


3 3 


8.296 


62.06 


II 


9 


108.43 


811 . 14 


25 


9 


520.77 


3895-6 


3 4 


8.727 


65.28 


12 





113. 10 


846 . 03 


26 





530.93 


3971.6 


3 5 


9.168 


68.58 


12 


3 


117.86 


881.65 


26 


3 


541 -19 


4048 . 4 


3 6 


9.621 


71.97 


12 


6 


122.72 


918.00 


26 


6 


551-55 


4125.9 


3 7 


10.085 


75.44 


12 


9 


127.68 


955-09 


26 


9 


562.00 


4204.1 


3 8 


10.559 


78.99 


13 





132.73 


992.91 


27 





572.56 


4283.0 


3 9 


11.045 


82.62 


13 


3 


137.89 


103I.S 


27 


3 


583.21 


4362.7 


3 10 


11-541 


86.33 


13 


6 


143.14 


1070.8 


27 


6 


593-96 


4443 . I 


3 II 


12.048 


90.13 


13 


9 


148.49 


1110.8 


27 


9 


604.81 


4524.3 


4 


12.566 


94.00 


14 





153-94 


11515 


28 





615.75 


4606.2 


4 I 


13 095 


97.96 


14 


3 


159.48 


1193-0 


28 


3 


626.80 


4688.8 


4 2 


13-635 


102.00 


14 


6 


165.13 


1235-3 


28 


6 


637.94 


4772.1 


4 3 


14-186 


1 06 . 1 2 


14 


9 


170.87 


1278.2 


28 


9 


649. 18 


4856.2 


4 4 


14-748 


110.32 


15 





176.71 


1321.9 


29 





660.52 


4941 . 


4 5 


15.321 


114.61 


IS 


3 


182.65 


1366.4 


29 


3 


671 .96 


5026.6 


4 6 


15-90 


118.97 


IS 


6 


188.69 


141I-S 


29 


6 


683 . 49 


5112.9 


4 7 


16.50 


123.42 


15 


9 


194.83 


I4S7-4 


29 


9 


695.13 


5199.9 


4 8 


17- 10 


127.9s 


16 





201.06 


1504.1 


30 





706.86 


5287.7 


4 9 


17-72 


132.56 


16 


3 


207.30 


ISSI.4 


30 


3 


718.69 


5376.2 


4 10 


18-35 


137-25 


16 


6 


213.82 


1599- 5 


30 


6 


730.62 


5465.4 


4 IJ 


18-99 


142.02 


16 


9 


220.35 


1648.4 


30 


9 


742.64 


5555-4 


S 


19-63 


146.88 


17 





226.98 


1697-9 


31 





754.77 


5646.1 


5 I 


20.29 


151.82 


17 


3 


233.71 


1748-2 


31 


3 


766.99 


5737. S 


S 2 


20.97 


156.83 


17 


6 


240.53 


1799-3 


31 


6 


779.31 


S829.7 


S 3 


21 .65 


161.93 


17 


9 


247.4s 


185I-I 


31 


9 


791.73 


5922.6 


S 4 


22.34 


167. 12 


18 





254-47 


1903-6 


32 





804.25 


6016.2 


5 S 


23-04 


172.38 


18 


3 


261.59 


1956.8 


32 


3 


816.86 


6110.6 


5 6 


23.76 


177.72 


18 


6 


268.80 


2010. 8 


32 


6 


829.58 


6205.7 


S 7 


24.48 


183.15 


18 


9 


276, 12 


2065.5 


32 


9 


842.30 


6301.5 



The formulas proposed by Wm. Cox, {Amer. Mach., Oct. 4, 25, 
1894) give, quite closely, the same results as the more cumbersome 
formulas by Weisbach. Mr. Cox's formulas are as follows: 
Let £> = discharge, cu. ft. per min., 

(i = diameter of pipe, ins., 

F = velocity of discharge, ft. per sec, 

i = length of pipe, ft., 

Z^ = head required to produce velocity V, ft., 



Table 8. — Theoretical Spouting Velocity and Actual Dis- 
charge OF Water Through a Clean Square-edged Orifice 
of I Sq. In. Area, the Latter Calculated for a Coefficient 
of Discharge of .6 



Head, 
ft. 


"0 
_o 


Discharge of ori- 
fice of I sq.in.area, 
cu. ft. per. min. 


Head, 

ft. 


+^ 

. 
■3 " 
> Si 

a . 

1! 


Discharge of ori- 
fice of I sq.in.area, 
cu. ft. per. min. 


Head, 
ft. 


>. 

'0 


13 . 
> 
Si 

t i 
g » 


Discharge of ori- 
fice of 1 sq. in. area, 
cu. ft. per. min. 


i 


2.835 


.709 


9 


24.061 


6.01 


51 


57-31 


14-33 


i 


4.010 


1.003 


9^ 


24.720 


6.18 


52 


57.87 


14-47 


1 


4.911 


1.228 


10 


25.362 


6.34 


S3 


58.42 


14.60 


i 


5.671 


1.417 


10^ 


25.988 


6.50 


54 


58.97 


14.74 


1 


6.340 


1-59 


11 


26.600 


6.65 


55 


59-52 


14- 83 


i 


6.946 


1-737 


Hi 


27.198 


.6.80 


56 


60.05 


15.01 


i 


7.502 


1.89 


12 


27.783 


6.94 


57 


60.59 


15.14 


I 


8.020 


2.00 


I2J 


28.356 


7.08 


58 


61.12 


15.28 


ij 


8.507 


2. 126 


13 


28.917 


7.23 


59 


61.64 


IS. 41 


li 


8.967 


2.242 


I3i 


29.468 


7.367 


60 


62.16 


IS. 54 


ij 


9.404 


2.351 


14 


30 . 009 


7 -SO 


61 


62.68 


15.69 


a 


9.823 


2.46 


I4i 


30.540 


7.64 


62 


63.19 


15.80 


If 


10.224 


2.556 


15 


31.062 


7.77 


63 


63.70 


15-92 


If 


10.610 


2.65 


^5\ 


31.576 


7.88 


64 


,64. 20 


16.05 


ij 


10.982 


2.75 


16 


32.081 


8.02 


65 


64.70 


16.18 


2 


11.342 


2.83 


l6i 


32.579 


8. 14 


66 


65.20 


16.30 


2j 


II .691 


2.92 


17 


33-068 


8.26 


67 


65.69 


16.42 


2i 


12.030 


3. 


18 


34-027 


8.50 


68 


66.18 


16.54 


2| 


12.360 


3 09 


19 


34.959 


8.74 


69 


66.66 


16.67 


2h 


12.681 


3-17 


20 


35-89 


8.96 


70 


67.15 


16.79 


2f 


12.994 


3-249 


21 


36.78 


. 9.18 


71 


67.62 


16.91 


2f 


13. 30c 


3-325 


22 


37.64 


9.40 


72 


68.10 


17.02 


2i 


13 399 


3-40 


23 


38.49 


9.61 


73 


68.57 


17.14 


3 


13 -90 


3-47 


24 


39 32 


9.82 


74 


69.03 


17.26 


3i 


14.177 


3-54 


25 


40.13 


10.02 


75 


69.50 


17.38 


3i 


14-459 


3-614 


26 


40.92 


10. 22 


76 


69.96 


17.49 


3i 


14-734 


3-69 


27 


41.70 


10.42 


77 


70.42 


17.60 


3i 


IS -004 


3-75 


28 


42.47 


10.62 


78 


70.88 


17.72 


3f 


iS-270 


3-82 


29 


43 ■«2 


10. 80 


79 


71.33 


17-83 


35 


15-531 


3-88 


30 


43 96 


10.98 


80 


71.78 


17.95 


3i 


15-783 


3-94 


31 


44-68 


11.17 


81 


72.23 


18.05 


4 


16.040 


4.01 


32 


45.40 


11-35 


82 


72.67 


18.17 


4i 


16.534 


4-13 


33 


46. 10 


11.57 


83 


73.11 


18.28 


4i 


17.013 


4-25 


34 


46.79 


II . 70 


84 


73-55 


18.38 


4i 


17.480 


4-37 


35 


47.48 


11.86 


85 


73-99 


18.50 


5 


17.934 


4-48 


36 


48. IS 


12.04 


86 


74-42 


18.61 


Si 


18.377 


4-59 


37 


48.81 


12.20 


87 


74-85 


18.71 


Si 


18.809 


4.70 


38 


49.47 


12.36 


88 


75-28 


18.82 


5i 


19.232 


4.81 


39 


SO. 12 


12.53 


89 


75-91 


18.93 


6 


19-645 


4.91 


40 


50.76 


12.69 


90 


76-13 


19.03 


6i 


20.050 


S-Oi 


41 


51.39 


12.85 


91 


76-56 


19.13 


6i 


20.448 


5-11 


42 


52.01 


13.00 


92 


76-97 


19.24 


6i 


20.837 


5-21 


43 


52.62 


13. 16 


93 


77.39 


19.35 


7 


21 . 219 


5. 30 


44 


53-23 


13-31 


94 


77.81 


19-45 


7i 


21.595 


5-4° 


45 


53-84 


13-46 


95 


78.21 


19-55 


7i 


2 I . 964 


5-50 


46 


54-43 


13-61 


96 


78.63 


19.66 


7i 


22.327 


5. 58 


47 


55-02 


13-75 


97 


79.04 


19.76 


8 


22.685 


S.67 


48 


55- 60 


13-90 


98 


79.24 


19.86 


8i 


23 . 036 


5-75 


49 


56.18 


14-05 


99 


79.85 


19.96 


8i 


23.383 


5. 84 


SO 


56.75 


14. 18 


100 


80.25 


20.06 



Then D = .^2TsVd^ 
whence 



\.327s7 



and 
Also 



7 = - 



•32751 
D 



.327 sd^ 



4F2+sF-2 = 



i2oodH 



(i) 
(la) 

(i&) 

(2) 



HYDRAULICS AND HYDRAULIC MACHINERY 



235 



Putting 



whence 



and 



47*+5F-2=2s:, (2) becomes, 
„ 1200 dH 



I\. — 


" L 


d = 


KL 


1200 H 


H = 


KL 

1200d 


Z,= 


1 20odH 
K 



i2a) 

{2b) 
(2C) 
{2d) 



Formulas {2)-{2d) apply to clean smooth cast-iron pipes. To 
make them and the tables applicable to lapped and rivetted pipes, 
1000 must be used instead of 1200 in formula (2) and its transposi- 



Table 9 gives the discharge in cu. ft. per min. from pipes of i to 
48 ins. diameter for a uniform velocity of i ft. per sec. and from it 
the discharge for any other velocity may be obtained by simple 
proportion thus: 

What diameter of pipe will discharge 500 cu. ft. per min. with a 
velocity of 4 ft. per sec ? 

The proportionate discharge for a velocity of i ft. per sec. would 

125 cu. ft., and from Table 9 we see that this would require 

a pipe 20 ins. diameter. 

Table 10 gives values of Z' = 4F^+sF— 2 corresponding to veloc- 
ities V from I to 20 ft. per sec. Its use is best shown by examples: 



be^ 
4 











Table 7.- 


-Capacity or Rectangular Tanks Per Foot of 


Depth in U. S. Gallons. 








Width of 
tank 


Length of tank, ft. 


Ft. 


Ins. 


2^ 


3 


3\i 


4 


4}^ 


5 


5'/^ 


6 


6H 


7 1 7^^ 1 8 


8!.^ 1 9 


9\i 1 10 


lQ'/2 


11 


ii!-^ 


12 


2 
2 
3 
3 

4 
4 
5 
5 

6 


6 


37.40 
46.75 


44.88 

56. 10 

67.32 


52.36 

65.45 
78.54 
91.64 


59.84 

74.80 

89.77 

104.73 

119.69 


67.32 

84.16 

100.99 

117.82 

134.65 
151.48 


74-81 
93-51 

112 .21 
130.91 

149.61 
168.31 
187.01 


82.29 
102.86 
123-43 
144 - 00 

164.57 
185.14 
205.71 
226.28 


89.77 
112 .21 
134.6s 
157.09 

179.53 
201 .97 

224.41 
246 . 86 

269.30 


97-25 
121 .56 
145.87 
170.18 

194-49 
218.80 
243.11 
267.43 

291.74 
316.05 


104.73 
130.91 
157.09 
183.27 

209.45 
235.63 
261.82 
288.00 

314.18 
340.36 
366.54 


112. 21 
140.26 
168.31 
196.36 

224.41 
252.47 
280.52 
308.57 

336.62 
364.67 
392.72 
420.78 


119.69 
149.61 
179.53 
209.45 

239.37 
269.30 
299.22 
329.14 

359.06 
388.98 
418.91 
448.83 

478.75 


127-17 
158.96 
190.75 
222.54 

254.34 
286.13 
317-92 
349-71 

381.50 
413.30 
445-09 
476.88 

508.67 
540.46 


134.65 
168.31 
202.97 
235.63 

269.30 
302.96 
336.62 
370.28 

403.94 
437.60 
471.27 
504.93 

538.59 

572.25 
605.92 


142.13 
177.66 
213.19 
248.73 

284.26 
319.79 
355.32 
390.8s 

426.39 
461 .92 
497.45 
532.98 

568.51 
604 . 06 
639.58 
67s. 11 


149.61 
187.01 
224.41 
261 .82 

299.22 
336.62 
374.03 
411.43 

448.83 
486.23 
523.64 
561 .04 

598.44 
635.84 
673.25 
710.65 

748.05 


157.09 
196.36 
235.63 
274-90 

314-18 

353-45 
392-72 
432.00 

471.27 
510.54 
549.81 
589.08 

628.36 
667.63 
706.90 

746.17 

785.45 
824.73 


164.57 
205.71 
246.86 
288.00 

329.14 
370.28 
411.43 
452.57 

493.71 
534.85 
575.99 
617.14 

658.28 
699.42 
740.56 
781.71 

822.86 
864 . 00 

90s . 14 


172. 05 

2 1 5 . 06 
258.07 
301 .09 

344.10 
387.11 
430.13 
473.14 

516.15 
559.16 
602 .18 
645.19 

688.20 
731.21 
774-23 
817-24 

860.26 
903.26 
946.27 
989.29 


179.53 
224.41 
269.30 


6 




314. 18 






359.06 


6 








403 .94 










448 . 83 


6 












493 -71 














538 .59 


6 


6 
















583 .47 


7 
7 

g 


















628.36 


6 




















673.24 
718.12 






















g 


6 


























9 

9 


























807 .89 


6 




























852 .77 






























897 .66 




6 
































942-56 
987.43 

103 . 23 




































1 1 


6 










































































107 72 
















































Table 9. — Discharge From Pipes in Cu. Ft. 
PER Min. With Velocity = i Ft. per Sec. 



Table 10. — Values of K = /^V^+sV—2 



Diam., 


Cubic 


Diam., 


Cubic 


Diam., 


Cubic 


ins. 


ft. 


ins. 


ft. 


ins. 


ft. 


I 


0.3272s 


17 


94.575 


33 


356.37 


2 


I .3090 


18 


106.03 


34 


378.30 


3 


2.9452 


19 


118. 14 


35 


400.88 


4 


S.2360 


20 


130.90 


36 


424.11 


S 


8.1812 


21 


144.32 


37 


448 . 00 


6 


II. 781 


22 


158.39 


38 


472.55 


7 


16.03s 


23 


173. II 


39 


497.75 


8 


20.944 


24 


188.50 


40 


523.60 


9 


26.507 


25 


204.53 


41 


550.11 


10 


32.72s 


26 


221.22 


42 


577.27 


II 


39.597 


27 


238.56 


43 


605 . 09 


12 


47.124 


28 


256.56 


44 


633 56 


13 


55 - 30S 


29 


275.22 


45 


662.68 


14 


64 . 141 


30 


294.52 


46 


692 .46 


15 


73.631 


31 


314.49 


47 


722.90 


16 


83.776 


32 


335.10 


48 


753-98 



All other velocities in strict proportion. 



0.0 



1 .0 
2.0 
30 
4.0 
5-0 

6.0 
7.0 
8.0 
9.0 
10. o 

II. o 
12.0 
13-0 
14.0 

is.o 

16.0 
17.0 
18.0 
19.0 

20. O 



7.00 

24.00 

49-00 

82.00 

123.00 

172.00 
229. 00 
294. 00 

367.00 

448 . 00 
537-00 

634.00 

739 00 
852 .00 
973.00 

1102.00 
1239.00 
1384.00 
1537.00 
1698.00 



I 0.2 I 0.3 



8.34 

26.14 

51-94 

85.74 

127.54 

177.34 
235-14 
300.94 
374.74 
456.54 

546.34 
644.14 
749 . 94 
863 . 74 
985.54 

1115-34 
1253-14 
1398.94 
1552.74 
1714.54 



9-76 

28.36 

54 96 

89.56 

132.16 

182.76 
241 .36 
307.96 
382.56 
465.16 

555.76 
65436 
760.96 
875.56 
998.16 

1128.76 
1267.36 
1413.96 
1568.56 
1731.16 



11.26 
30.66 
58.06 
93-46 
136.86 

188.26 
247 .66 
315 06 
390.46 
473-86 

565.26 
664 . 66 
772.06 
887.46 
1010.86 

1142.26 
1281.66 
1429.06 
1584.46 
1747-86 



0.4 



0.64 
12.84 
33-04 

61.24 

97 . 44 

141.64 

193.84 
254-04 
322.24 
398 . 44 
482.64 

574.84 
675.04 
783.24 
899 . 44 
1023 . 64 

1155.84 
1296.04 
1444.24 
1600.44 
1764.64 



0.5 



I. SO 

14- SO 

35-50 

64.50 

101.50 

146.50 

199.50 
260.50 
329-50 
406.50 
491.50 

584-50 
685.50 
794.50 
911.50 
1036.50 

1169.50 
1310.50 
1459.50 
16x6. so 
1781.50 



0.6 I 0.7 



0.8 



2.44 

16. 24 

38.04 

67.84 

105.64 

151-44 

205.24 
267.04 
336.84 
414.64 
500.44 

594-24 
696 . 04 
805.84 
923.64 
1049 . 44 

1183.24 
1325.04 
1474.84 
1632.64 
1798.44 



3.46 

18.06 

40.66 

71 .26 

109.86 

156.46 

211 .06 
273.66 
344-26 
422. 86 
509.46 

604 . 06 
706.66 
817.26 
935.86 
1062.46 

1197.06 
1349.66 
1490.26 
1648.86 
1815.46 



4.56 
19.96 
43.36 

74.76 
114. 16 
161.56 

216.96 
280.36 
351.76 
431.16 
518.56 

613.96 
717.36 
-828.76 
948. 16 
1075.56 

1210.96 
1354.36 
1505-76 
1665. 16 
1832.56 



0.9 



5.74 

21.94 

46.14 

78.34 

118.54 

166.74 

222.94 
287 . 14 
359-34 
439-54 
527-74 

623.94 
728. 14 
840.34 
960.54 
1088.74 

1224.94 
1369.14 
1521.34 
1681.S4 
1849.74 



0.0 

I .0 
2.0 
3-0 

4.0 

5-0 

6.0 

7.0 

8.0 

9-0 

10. o 

II. o 
12.0 
13.0 
14.0 

IS.O 

16.0 

17.0 
18.0 
19.0 
20.0 



tions, while for seamless wrought-iron pipes with flush joints the 
constant 1500 should be used. 

The velocities in pipe-lines seldom exceed 6 ft. per sec. in low and 
medium head-water power plants. In high-head plants the velocity 
sometimes reaches 13 ft. per sec. For the mere delivery of water 
no such restriction holds. 



Given a pipe 12 ins. diameter, 3000 ft. long and 20 //. head; what 
will be the velocity of discharge and the discharge? 
By equation (2a) we have 



K = 



12X20X1200 
3000 



= 96, 



236 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



which, according to Table lo, corresponds to a velocity of 4.4 ft. per 
sec, nearly. Now, from Table 9, we have 

Discharge = 47. 124X4-4 = 207.3 cu. ft. per min. 

Given a pipe 2400 ft. long, with a head of 80 //., what must be its 
diameter to produce a velocity of discharge of 5 ft. ? 

Taking from Table 10 the value of K corresponding to a velocity 

of 5 ft., and inserting it in equation (26), we have 

, 123X2400 

d= o .. =3 ms. 

80X1200 ^ 

Given a 24-in. pipe 1800 ft. long, what head is required to produce 
a velocity of discharge of &ft. per sec? 

Taking from Table 10 the value of K corresponding to a velocity 

of 8 ft., and inserting it in equation (2c), we have 

^^ , 294X1800 „ ^ 

Head = ^^^^r; = 18.4 ft. 

24X1200 

What diameter of pipe 4500 //. long will discharge 4020 cu. ft. per 
min., with a head of 24 ft.? 

In this problem neither the velocity nor the diameter of the pipe 
are given, so that we must proceed by trial. We will assume, there- 
fore, a trial diameter of 40 ins., and inserting this in equation (2a), 
we have 

40X24X1200 



K=- 



4500 



^256 



which, according to Table 10, corresponds to a velocity of 7.4 ft. per 
sec. nearly. Now by Table 9 the discharge of a 40-in. pipe with 
this velocity is 

Z) = S23.6X7-4 = 3874-64 cu. ft. 

This diameter is, therefore, clearly not enough, as there is a short- 
age of 4020 — 3874.64 = 145.36 cu. ft. From Table 7 we now see 
that a 41-in. pipe will discharge 26.5 cu. ft. per .unit of velocity 
more than a 40-in. pipe; therefore, with the same velocity of 7.4 ft., 
we have 

26.5X7.4 = 196.1 cu. ft., 

which is more than the previous shortage, so that a 41-in. pipe is 
amply large enough to satisfy the requirements of the problem. 

This problem is probably the one whose solution is most frequently 
called for and is the most tedious to solve. The solution here given 
is believed to be the simplest that has been offered. 

What head is required to discharge 1000 cu. ft. per. min. from a 30- 
in. pipe 3000 ft. long? 

From Table 9 we find that the velocity of discharge must be 

= 3.4 ft. per sec. Taking from Table 10 the value of K corre- 
sponding to this velocity and inserting it in equation (2c), we have 

61.24X3000 f. 

Head= — = 5.1 ft. 

30X1200 

To make the calculations in U. S. gallons instead of cubic feet, 

formula (i) becomes: 

G = 2.4S Vd^ . (3) 

in which G = discharge, gals, per min., 

F = velocity of discharge, ft. per sec, 
(^ = diameter of pipe, ins. 

Table 11 gives this discharge for pipes from 1 to 48 ins. dimaeter 
for a uniform velocity unit of i ft. per sec. It is used in precisely 
the same manner as Table 9. 

To find the velocity head, that is, the head required to produce any 
given velocity of discharge, use the formula: 

Fa = .oi55 V' (4) 

in which Fa = the head in ft. required to produce the velocity V. 
Similarly, to find the pressure head, that is, the pressure required to 
produce any given velocity of discharge, use the formula: 

Fp = . 00673 V^ ' is) 



in which Vp- 
velocity V. 

Table 11.- 



• the pressure in lbs. per sq. in. required to produce the 

-Discharge From Pipes in U. S. Gals, per Min. With 
Velocity = I Ft. pee Sec. 



Diam., 


Gallons 


Diam., 


Gallons 


Diam,, 


Gallons 


ins. 




ins. 




ms. 




I 


2.448 


17 


707.47 


33 


2665.9 


2 


9.792 


18 


793-15 


34 


2829.9 


3 


22.032 


19 


883.73 


35 


2998.8 


4 


39.168 


20 


979-20 


36 


3172.6 


S 


61.200 


21 


1079.6 


37 


3351-3 


6 


88.128 


22 


I184.8 


38 


3534.9 


7 


119-95 


23 


1295-0 


39 


3723.4 


8 


156.67 


24 


1410.0 


40 


3916.8 


9 


198.29 


25 


1530.0 


41 


4115-I 


10 


244.80 


26 


1654-8 


42 


4318.3 


II 


296.21 


27 


1784.6 


43 


4526.3 


12 


352.51 


28 


1919.2 


44 


4739.3 


13 


413.71 


29 


2058.8 


45 


4957.2 


14 


479-81 


30 


2203.2 


46 


5180.0 


15 


550.80 


31 


2352.5 


47 


5407-6 


16 


626.69 


32 


2506.7 


48 


5640.2 



All other velocities in strict proportion. 

Table 12 gives a list of heads in ins. and ft. with their equivalent 
pressures in lbs. per sq. in. and also the corresponding velocities 
in ft. per sec. produced by these heads or pressures. 

It is often more convenient to base all calculations upon pressure 
in lbs. instead of hea'ds in ft. To reduce heads in ft. to pressures in 
lbs. per sq. in., multiply the head by .4331 or consult Table 3. So 
likewise equation {2a) becomes: 

2768 dP 



K = - 



(6a) 



in which /*= pressure, lbs. per sq. in. corresponding to head H of (2a), 
the remaining notation being as in (2a). 

Table 13 for the loss of head due to friction in lapped and rivetted 
pipe has been calculated by the Pelton Water Wheel Co. from Cox's 
formulas, except that the factor 1200 is replaced by 1000 as directed 
by Mr. Cox. 

Table 12. — Velocities With Corresponding Heads and 
Pressures 



Vel. head. 

ins. 


Vel. pressure, 

lbs., per 

sq. in. 


Vel., 
ft. per sec. 


Vel. head, 
ft. 


Vel. pressure, 
lbs. per 
sq. in. 


Vel., 
ft. per sec. 


.186 


.0067 


1. 00 


I.2SS 


-5451 


9.00 


.744 


.0269 


2.00 


1-550 


-6730 


10.00 


I. 000 


.0361 


2.32 


2.000 


.8670 


11.35 


1.674 


.0606 


3.00 


2.307 


I . 0000 


12.19 


2.000 


.0722 


3-27 


3.000 


1.3005 


13.90 


2.976 


.1077 


4-00 


4.000 


1.7340 


16.05 


3.000 


.1084 


4.01 


4.614 


2 . 0000 


17-24 


4.000 


-1445 


4.63 


5.000 


2.167s 


17-94 


4.650 


.1682 


5.00 


6.000 


2.6010 


19.66 


S-Ooo 


. 1806 


S.I8 


6.920 


3 . 0000 


21.12 


6.000 


.2167 


5.67 


7.000 


3-0345 


21.23 


6.696 


.2423 


6.00 


8.000 


3.4680 


22.70 


7.000 


.2529 


6.13 


9.000 


3-90IS 


24-07 


8.000 


.2890 


6.55 


9.227 


4.0000 


24-38 


9.000 


.3251 


6.95 


10.000 


4-3350 


25-38 


9. 114 


.3298 


7.00 


H-S34 


5 . 0000 


27.26 


10.000 


.3612 


7-32 


13.841 


6.0000 


29.86 


1 I . 000 


.3974 


7.6s 


16.148 


7 . 0000 


32.26 


11.904 


.4307 


8.00 


18.454 


8 . 0000 


34-48 


12 


.4335 


8.02 


20.761 


9.0000 


36.58 



A graphical method of making pipe-line calculations for the flow of 
water is given in Fig. 2, by Walter R. Clark, Mech. Engr. of the 
(.Continued on page 238 first column) 



HYDRAULICS AND HYDRAULIC MACHINERY 



237 













Table 13.- 


-Loss or 


Head 


BY Friction Per 100 Ft 


Length oe Pipe 








y 




Diam. 

ins. 


6 


7 


8 


9 


10 


II 


12 


13 


I 


t 


IS 


I 


5 


I 


5 


Vel. 


Loss 
of 


4-1 


Loss 
of' 


<u .5 

s 


Loss 
of 




Loss 
of 




Loss 
of 


5J c 


Loss 
of 




Loss 
of 


1 d 


Loss 
of 




Loss 
of 


+5 

S d 


Loss 
of 


m C 

M-, •- 


Loss 
of 


•1^ 

QJ . 

>2 c 


Loss 
of 




in ft. 
per 
sec. 


head 
in 
feet 


2 S 
0. 


head 

in 

feet 


head 
in 
feet 


H 


head 
in 
feet 


b 
6 ^ 


head 

in 

feet 


h 
IS h 

ft 


head 
in 
feet 


H 
^ 


head 
in 
feet 


a 
ft 


head 
in 
feet 


head 
in 
feet 


s 
ft 


head 
in 
feet 


B 

1 ^ 
ft 


head 
in 
feet 


■a u 
a 


head 
in 
feet 


a 
6 ft 


2.0 


• 39 


23 5 


■ 33 


32.0 


• 30 


41.9 


.26 


53.0 


• 23 


65^4 


. 21 


79 


.198 


94 


.183 


no 


. 169 


128 


• 158 


147 


• 147 


167 


.132 


212 


2. 2 


.46 


25^9 


.40 


35.3 


• 35 


46.1 


■ 31 


58.3 


.28 


72. 


.25 


87 


• 234 


103 


.216 


121 


.200 


141 


.187 


162 


• 175 


184 


.156 


233 


2.4 


• 54 


28.2 


.46 


38.5 


• 41 


50.2 


.36 


63.6 


.32 


78.5 


■ 29 


95 


.273 


113 


.252 


133 


• 234 


154 


.218 


176 


• 205 


201 


.182 


254 


2.6 


.63 


30.6 


.54 


41 -7 


• 47 


54.4 


.42 


68.9 


.37 


85. 1 


• 34 


103 


.31S 


122 


.290 


144 


.270 


167 


.252 


191 


.236 


218 


. 210 


275 


2.8 


.72 


32^9 


.61 


44.9 


•54 


58.6 


.48 


74.2 


• 43 


91 .6 


• 39 


III 


.360 


132 


• 332 


156 


.308 


179 


.288 


206 


.270 


234 


.240 


297 


3-0 


.81 


35^3 


.69 


48.1 


.61 


62.8 


• 54 


79-5 


• 48 


98.2 


• 44 


119 


• 407 


141 


.375 


166 


• 349 


192 


• 32s 


221 


• 306 


251 


.271 


318 


3-2 


• 91 


37.7 


• 78 


51.3 


.68 


67.0 


.60 


84.8 


• 54 


I OS 


.49 


127 


.457 


151 


• 422 


177 


• 392 


205 


• 366 


23s 


.343 


268 


.305 


339 


3-4 


1 , 02 


40.0 


• 87 


S4.S 


.76 


71.2 


.68 


90. 1 


.61 


III 


• 55 


134 


Sio 


160 


• 471 


188 


.438 


218 


.408 


250 


• 383 


284 


.339 


360 


3.6 


I. 13 


42.4 


• 96 


57. 7 


.84 


75.4 


.75 


95 4 


.67 


118 


.61 


142 


.566 


169 


• 522 


199 


■ 485 


231 


• 452 


26s 


• 425 


301 


• 377 


382 


3.8 


I^2S 


44-7 


1.07 


60.9 


.93 


79.6 


.83 


lOI 


•74 


124 


• 68 


ISO 


.624 


179 


• 576 


210 


■ 535 


243 


• 499 


280 


.468 


318 


.416 


403 


4.0 


l^37 


47.1 


1^17 


64. 1 


1.02 


83.7 


• 91 


106 


.82 


131 


• 74 


158 


.685 


188 


• 632 


221 


■ 587 


256 


• 548 


294 


.513 


335 


.456 


424 


4.2 


I^49 


49. 5 


1.28 


67.3 


I. 12 


87.9 


.99 


III 


.89 


137 


.81 


166 


■ 749 


198 


.691 


232 


• 641 


269 


• 598 


309 


.561 


352 


.499 


445 


4-4 


1.62 


51.8 


1.39 


70-5 


1.22 


92. 1 


1.08 


116 


• 97 


144 


.88 


174 


.815 


207 


• 751 


243 


.698 


282 


• 651 


324 


.611 


368 


• 542 


466 


4.6 


1.76 


54-1 


I. SI 


73.7 


1.32 


963 


1. 17 


122 


1^05 


ISO 


.96 


182 


.883 


217 


• 815 


254" 


.757 


295 


• 707 


339 


.662 


385 


.588 


488 


4.8 


1 .90 


56. S 


1.63 


76.9 


1.43 


TOO. 


1.27 


127 


1. 14 


157 


1.04 


190 


■ 954 


226 


.881 


265 


.818 


308 


• 763 


353 


• 715 


402 


.636 


509 


SO 


2.05 


S8.9 


1.76 


80.2 


1^54 


105 


1.37 


132 


1.23 


163 


1 . 12 


198 


1.028 


23s 


■ 949 


276 


• 881 


321 


• 822 


368 


• 770 


419 


.685 


530 


S.2 


2. 21 


61.2 


1.89 


83-3 


1^65 


109 


1.47 


138 


1.32 


170 


1.20 


206 


1. 104 


245 


1.020 


287 


• 947 


333 


• 883 


383 


• 828 


435 


• 736 


551 


■ S.4- 


2.37 


63.6 


2.03 


86.6 


1^77 


113 


1.57 


143 


1. 41 


177 


1.28 


214 


1. 183 


254 


1 .092 


298 


1 .014 


346 


• 947 


397 


.888 


452 


• 788 


572 


S.6 


2.53 


65.9 


2.17 


89.8 


1.89 


117 


1.68 


148 


LSI 


183 


1.37 


222 


1 . 26 


264 


1 . 167 


309 


1.083 


359 


I. on 


412 


•949 


469 


• 843 


594 


S.8 


2.70 


68.3 


2.31 


93 


2.01 


121 


1 .80 


154 


I. 61 


190 


1.46 


229 


1-34 


273 


1.245 


321 


I-ISS 


372 


1.078 


427 


1 .011 


486 


• 899 


61S 


6.0 


2.87 


70.7 


2.46 


96.2 


2. IS 


I2S 


1.92 


159 


I.7I 


196 


1.56 


237 


1.43 


283 


1.325 


332 


1.229 


385 


1. 148 


442 


1.076 


502 


• 957 


636 


7.0 


3.81 


82.4 


3.26 


112. 


2.85 


146 


2.52 


18S 


2.28 


229 


2.07 


277 


1. 91 


330 


1.75 


387 


1 .630 


449 


1.520 


51s 


1-430 


586 


1 .270 


742 



Diam. 
ins. 


20 


22 


24 


2 


6 


28 


30 


33 


36 


39 


42 


45 


48 


Vel. 
in ft. 
per 
sec. 


Loss 
of 




Loss 
of 


6 ft 


Loss 
of 


c3 ft 


Loss 
of 


■♦J 

1 S3. 

6 ft 


Loss 
of 


"Z e 
6 ft 


Loss 
of 


cj ft 


Loss 
of 


■w 

Ji d 

S ft 


Loss 
of 


^ d 
CJ ft 


Loss 
of 


ft 


Loss 
of 


^■s 

B 

c3 ft 


Loss 
of 


'.S u 

aft 


Loss 
of 


■;; B 

a ft 


head 
in 

feet 


head 
in 
feet 


head 
in 
feet 


head 

in 

feet 


head 

in 

feet 


head 
in 
feet 


head 
in 
feet 


head 
in 
feet 


head 
in 
feet 


head 
in 
feet 


head 
in 
feet 


head 
in 
feet 


2.0 


.119 


262- 


• 108 


316 


.098 


377 


.091 


442 


.084 


513 


• 079 


589 


• 073 


712 


.066 


848 


.061 


995 


• 057 


nSS 


• 053 


1325 


.050 


1508 


2. 2 


• 140 


288 


• 127 


348 


.116 


414 


.108 


486 


• 099 


564 


■ 093 


648 


• 085 


78s 


.078 


933 


.072 


1094 


• 067 


1270 


• 063 


1456 


.059 


1658 


2.4 


. 164 


314 


.149 


380 


.136 


452 


.126 


531 


.116 


616 


. 109 


707 


.100 


85S 


.091 


1018 


.084 


1194 


.079 


1385 


• 073 


1590 


• 069 


1809 


2.6 


.189 


340 


.171 


412 


• 157 


490 


• 145 


575 


.134 


667 


.126 


766 


• H5 


927 


. 104 


IIOO 


• 097 


1294 


.090 


1500 


.084 


1721 


.079 


i960 


2,8 


.2X6 


366 


• 195 


443 


.180 


528 


.i6s 


619 


• 153 


718 


.144 


824 


• 131 


1000 


.119 


II88 


• III 


1394 


.103 


1617 


.096 


1855 


.090 


2110 


30 


.245 


393 


• 222 


475 


.204 


565 


.188 


663 


• 174 


770 


.163 


883 


• 148 


1070 


• 135 


1273 


• 125 


1442 


.117 


1730 


.109 


1987 


. 102 


2260 


3-2 


• 275 


419 


• 249 


507 


.229 


603 


.211 


708 


• 195 


821 


.182 


942 


.167 


II40 


• 152 


1367 


• 141 


1591 


• 131 


1845 


. 122 


2120 


• 115 


2410 


3.4 


.306 


445 


• 278 


538 


.255 


641 


.235 


752 


• 218 


872 


.204 


lOOI 


.186 


I2I0 


. 169 


1442 


• 157 


1690 


.146 


1 96 1 


.136 


2250 


.128 


2560 


3.6 


■ 339 


471 


• 308 


570 


.283 


678 


.261 


796 


• 242 


923 


.226 


1060 


.206 


1282 


• 188 


1527 


• 174 


1790 


.162 


2079 


.151 


2382 


.142 


2715 


3^8 


•374 


497 


• 340 


601 


.312 


716 


.288 


840 


• 267 


974 


.249 


I1I9 


.226 


1355 


.207 


I6I2 


.191 


1891 


.178 


2190 


.166 


251S 


.156 


286s 


4^0 


.410 


523 


• 373 


633 


■ 342 


754 


• 315 


88s 


• 293 


1026 


.273 


II78 


.248 


1425 


.228 


1697 


.210 


1990 


.195 


2310 


.182 


2650 


.171 


3016 


4.2 


•449 


550 


• 408 


665 


.374 


791 


• 345 


929 


• 320 


1077 


.299 


1237 


.270 


1495 


.249 


1782 


.229 


2091 


• 213 


2422 


.198 


2780 


.186 


316s 


4^4 


• 488 


576 


•444 


697 


.407 


829 


.375 


973 


.348 


1129 


• 325 


1296 


• 295 


1568 


.271 


1866 


.250 


2190 


.232 


2540 


■ .216 


2910 


.203 


3318 


4^6 


•529 


602 


• 482 


728 


.441 


867 


.407 


1017 


• 378 


1180 


■ 353 


1355 


.321 


1640 


.294 


1951 


.271 


229c 


• 252 


2658 


.235 


304s 


.220 


3470 


4^8 


• 572 


628 


• 521 


760 


.476 


90s 


.440 


1062 


• 409 


1231 


.381 


I4I4 


.346 


1710 


.318 


2036 


• 293 


2389 


.270 


2770 


• 254 


3180 


■ 238 


3619 


50 


.617 


654 


.561 


792 


• S13 


942 


■ 474 


II06 


■ 440 


1283 


.411 


1472 


• 374 


1780 


■ 342 


2I2I 


• 316 


2490 


• 294 


2885 


• 273 


3310 


.256 


3770 


5^2 


.662 


680 


.602 


823 


■ 552 


980 


■ 510 


IISO 


.473 


1334 


■441 


1531 


• 403 


1852 


.368 


2206 


• 342 


2590 


• 317 


3000 


.296 


3442 


.278 


3920 


5^4 


.710 


707 


.645 


855 


• 591 


1018 


■ 546 


1 194 


■ 507 


138s 


■473 


1590 


• 430 


1922 


•394 


2291 


• 364 


2689 


.338 


3115 


■ 315 


3578 


.295 


4071 


5.6 


.758 


733 


.690 


887 


■ 632 


1055 


.583 


1239 


■ 542 


1437 


.506 


1649 


• 453 


1995 


• 421 


2376 


•393 


2790 


■ 374 


3230 


• 340 


3710 


.319 


4222 


5.8 


.809 


759 


■ 735 


918 


• 674 


1093 


.622 


1283 


.578 


1488 


• 540 


1708 


• 495 


2065 


•450 


2460 


• 419 


2886 


.389 


3348 


■ 363 


3840 


.340 


4373 


6.0 


.861 


785 


.782 


950 


.717 


II3I 


.662 


1327 


.615 


1539 


• 574 


1767 


• 520 


2140 


• 479 


2545 


• 441 


2986 


.408 


3461 


• 382 


3970 


.358 


4524 


7.0 


1 .143 


916 


1.040 


1 1 09 


.953 


1319 


■ 879 


1548 


.817 


1796 


• 762 


2061 


• 693 


2495 


• 636 


2968 


• 586 


3484 


• 545 


4030 


• 509 


4638 


■ 476 


5277 



238 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Cu. Ft. per Min. Flow 




Gals, per Min. Flow 

Example : 200 gals, per min. are to be transmitted 500 ft. with an allowable loss of pressure of 50 lbs. per sq. in. From 200 gals, on the 
base line trace vertically to the intersection with the line for a pressure loss of 10 lbs. per sq. in. The intersection is found near the diagonal 
for 2 1/2 in. pipe which is nearest the required size. The intersection also falls near the diagonal for 12 ft. per sec. velocity showing the 
velocity to be about that figure. If preferred read cu. ft. on the top scale in place of gals, on the bottom. Full lines refer to pipe of which 
the nominal and actual diameters are the same ; dotted lines refer to standard pipe. 

Fig. 2. — Flow of water in pipes. - 



Bridgeport Brass Co. {Atner. Mach., July 8, igog). Large pressure 
losses are included in the scale in order to adapt the chart to pipe- 
lines for high-pressure hydraulic machine service, in which much 
larger losses are permissible than in others because, under the high 
pressures, a large absolute loss is still a small percentage loss. 

The chart represents the following formulas which were deduced 
from Ellis and Howland's tables 

e = 2.4S VD^ 

in which Q = discharge, gals, per min., 

F = velocity, ft. per sec, 

Z) = inside diameter, ins., 

i^ = friction loss, lbs. per sq. in. for each 100 ft. of clean 
straight iron pipe, the values given being approxi- 
mately true for V greater than 3. 

The use of the chart is shown by the example below it. 



The full lines refer to pipe of which the nominal and actual diam- 
eters are the same, while the dotted lines refer to standard pipe. 
To use the chart for extra and double extra strong pipe, determine 
the actual diameter from the full lines and then refer to Tables 14 
and 15 for the pipe having its diameter nearest to that given by the 
chart. 

The friction losses given by the chart do not include those due to 
water flowing into the pipe from another vessel or vice versa, nor 
to ells or tees, for which latter see below. 

The chart may also be used in other ways. Thus if the pipe 
diameter and velocity are given, we find the intersection of the two 
diagonals representing pipe size and velocity and read down from 
their intersection to get gals, per min.; up to get cu. ft. per min.; 
and to the right to get lbs. pressure drop per 100 ft. of pipe. 

For the equation of main and branch lines of commercial pipe, 
including standard, extra and double extra strong by their actual 
inside diameters, see Index. 

The resistance of pipe fittings to the flow of water formed the subject 



HYDRAULICS AND HYDRAULIC MACHINERY 



239 



of experiments by Prof. F. E. Giesecke (Domestic Engineering, Nov. 
2, 1912). The information sought was for use in the design of 
apparatus for warming buildings by hot water and hence the fittings 
tested did not exceed 2 ins. nominal diameter and the observed 
velocities of flow did not exceed i ft. per sec. In spite of these limita- 
tions the experiments are the best of which the author has knowledge 
— the more so as the concordance of the results indicates that they 
may be applied materially beyond the limits of the observations 
without sensible error. 

Table 14. — Actual Internal Diameters oe Extra Strong Pipe 



Nominal size, ins. 


Internal diam- 
meter, ins. 


Nominal size, ins. 


Internal diameter, 
ins. 


i 


• 21S 


4 


3.826 


i 


.302 


4i 


4.290 


i 


.423 


5 


4.813 


i 


.546 


6 


5.761 


I 


.742 


7 


6.62s 


t 


.957 


8 


7.625 


li 


1.278 


9 


8.625 


li 


1. 500 


10 


9 750 


2 


1.939 


II 


10.750 


2i 


2.323 


12 


II. 750 


3 


2.900 


13 


13 000 


3i 


3.364 


14 


14.000 



The resulting data, as related to elbows, are given in graphic form 
in Fig. 3, while the accompanying table gives ratios for other fittings. 

In laying pipe-lines for flow due to gravity it should not be forgotten 
that no part of the line must rise above the hydraulic grade line. 
That is to say, referring to Fig. 4, if ABC represents a pipe-line, then 

Table 15. — Actual Internal Diameters of Double Extra 
Strong Pipe 



Nominal size, ins. 


Internal diam- 
eter, ins. 


Nominal size, ins. 


Internal diameter, 
ins. 


i 


.252 


3j 


2.728 


i 


.434 


4 


3.152 


I 


• 599 


4i 


3.580 


li 


.896 


5 


4.063 


14 


I. 100 


6 


4.897 


2 


I.S03 


7 


5.875 


2i 


1. 771 


8 


6.875 


3 


2.300 







its hydraulic grade is the straight line AC joining its two extremities, 
and no part of the pipe-line must rise above this grade. Therefore, 
before laying pipes through rough country, it is necessary to have a 
profile of the ground so as to be sure that the pipe, when laid, shall 
conform to this requirement. Neglect of this indispensable condition 
wiU lead to an interrupted or diminished flow. 

. Should the pipe at any point rise above the hydraulic grade line, 
it becomes in effect a siphon and subject to the uncertainties of a 
siphon. 

The Discharge of Water over Weirs 

By the De Laval Steam Turbine Company (except Table 17). 

There are two forms of weirs used for measuring water quantities, 
the rectangular and the triangular or V-notch weir. The rectangular 
weir is generally used for measuring large water quantities while the 
V-notch weir is more suitable for smaller quantities. 

In order to obtain good results with the rectangular weir, the up- 
stream edge of the crest of the weir should be made straight, sharp 
and smooth. This is usually accomplished by constructing it with 
an iron edge beveled sharply, which edge should also form the sides 
of the weir. In order to eliminate the velocity of approach in esti- 
mating the flow over the weir, the depth of the water below the crest 
of the weir should not be less than one-third the width of the weir. 
There must be free access of air under the sheet flowing over the weir. 
The height of water over the crest should be measured by means of 



a hook gage placed several feet from the weir, where the water is 
quiet and the surface level. 

The J. B. Francis formula for rectangular weirs with two end con- 
tractions is: 

Q = KV2'gXlH'^'' 
in which ^ = discharge, cu. ft. per sec, 
g = acceleration of gravity, 
Z = length of weir, ft., 
/? = effective head, ft. 

C = a constant depending on head and length of weir as 
given in Table 16. 

Table 16. — Coefficients C in Formula for Discharge of 
Water over Rectangular Weirs 



Head 








Length (0, ft. 








H, ft. 


.66 


I 


2 


3 


5 


7 


10 


19 


. I 


.632 


• 639 


.646 


.652 


.653 


.654 


■65s 


• 656 


■ IS 


.6 19 


.62s 


■634 


.638 


.640 


.640 


.641 


.642 


.2 


.611 


.618 


.626 


.630 


.631 


.632 


• 633 


•634 


.25 


.60s 


.612 


.62 I 


.624 


.626 


.627 


.628 


.629 


• 3 


.60 I 


.608 


.616 


.619 


.621 


.623 


• 624 


• 62s 


■ 4 


■595 


.601 


.609 


.613 


.615 


.617 


.618 


.620 


■ 5 


■590 


• 596 


.60s 


.608 


.6u 


.613 


■ 61S 


• 617 


.6 


■587 


-593 


.601 


.60s 


.608 


.611 


• 613 


• 615 


.7 


■58s 


■ 590 


■598 


.603 


.606 


.609 


.612 


.614 


.8 






■595 


.600 


.604 


.607 


• 611 


• 613 


•9 






■ 592 


■ 598 


.603 


.606 


• 609 


.612 


I.O 






■590 


• 595 


.601 


.604 


.608 


.611 


1. 2 






■585 


■ 591 


• 597 


.601 


■ 605 


.6 10 


1.4 






.580 


.587 


.594 


.598 


.602 


.609 


1.6 








■ 582 


■ 591 


• 595 


.600 


.607 



Table 17 is more convenient but less accurate and may be used 
for gaging the flow of streams. When test results are aimed at 
Table 16 should be used. 

Table 17. — Discharge of Water over Rectangular Weirs 

The figures in the body of the table give the discharge, cu. ft. per min., for 
each inch of width and for the depths given for whole inches at the left and 
for added fractions at the heads of the columns. The depth is to be measured 
upstream above the beginning of the contraction of depth. 



Inch , 


H 


H 


H 


M 


% 


H 


% 


I 


.40 


■ 47 


■ ss 


-65 


-74 


.83 


■ 93 


1.03 


2 


I. 14 


1.24 


1.36 


1-47 


1-59 


I. 71 


1.83 


1.96 


3 


2.09 


2.23 


2.36 


2.50 


2.63 


2.78 


2.92 


3-07 


4 


3.22 


3.37 


3-52 


3.68 


3-83 


3-99 


4. 16 


4-32 


5 


4^50 


4.67 


4.84 


5-01 


S-18 


S-36 


5-54 


5-72 


6 


5^90 


6.09 


6.28 


6-47 


6.65 


6.8s 


7-05 


7.25 


7 


7.44 


7.64 


7-84 


8.05 


8.25 


8-45 


8.66 


8.86 


8 


9^ 10 


9.31 


952 


9-74 


9-96 


10. 18 


10-40 


10.62 


9 


10.86 


11.08 


n.31 


1 1- 54 


11-77 


12.00 


12.23 


12,47 


10 


12.71 


12.95 


13. 19 


13-43 


13-67 


13-93 


14. 16 


14.42 


II 


14.67 


14.92 


15. 18 


15-43 


IS -67 


IS -96 


16.20 


16.46 


12 


16.73 


16.99 


17.26 


17.52 


17-78 


18.05 


18.32 


18.58 


13 


18.87 


19 • 14 


19.42 


19-69 


19.97 


20. 24 


20-52 


20.80 


14 


2 1.09 


2^37 


21.65 


2 1-94 


22.22 


22.51 


22.79 


23.08 


IS 


23 38 


23.67 


23.97 


24.26 


24.56 


24.86 


25. 16 


25.46 


16 


25^76 


26.06 


26.36 


26.66 


26.97 


27.27 


27-58 


27.89 


17 


28.20 


28.51 


28.82 


29- 14 


29-45 


29-76 


30-08 


30.39 


18 


30.70 


3 102 


31-34 


31-66 


31-98 


32-31 


32-63 


32.96 


19 


33-29 


33-61 


33-94 


34-27 


34-60 


34-94 


35-27 


35^60 


20 


35-94 


36-27 


36.60 


36.94 


37-28 


37-62 


37-96 


38.31 


21 


38-65 


39-00 


39-34 


39 69 


40.04 


40.39 


40-73 


41.09 


22 


41-43 


41-78 


42- 13 


42-49 


42.84 


43-20 


43 56 


43-92 


23 


-,4-28 


44-64 


45-00 


45-38 


45-71 


46.08 


46-43 


46.81 


24 


47- 18 


47-55 


47-91 


48. 28 


48.65 


49.02 


49-39 


49.76 



For a right-angled triangular weir 90° notch the formula for dis- 
charge is: 

0=2.544^5^^ 



240 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Notation as before, 
formula. 



Table i8 has been computed from this 





Table i 


B. — Discharge of 


Water 


OVER V- 


NOTCH Weirs 


Head, ft. 


Discharge, 
gals, per min. 


Head, ft. 


Discharge, 
gals, per min. 


■25 


36.0 


.65 


389 




3° 


56.2 




70 


468 




35 


83.0 




75 


555 




40 


115. 




80 


655 




45 


1550 




85 


760 




50 


202.0 




90 


879 




55 


256.0 




95 


1000 




60 


318.0 


I 


00 


1 140 



trace upward to the diagonal line for Barlow's formula and then to 
the left, where read 5 for the value of ^- The value of R being 10 



gives ~ = Y = 5 or T = 2 ms. 



Or, using the line for Lame's formula, 



. , R , . i? 10 

we find ji = S.S, that is, y = -y =5.5 or, r=i.82 ins. 

Professor Jenkins recommends a fiber stress of 6000 lbs. per sq. in. 
for cast-iron and 13,000 to 18,000 for steel cylinders. The former 
figure seems rather high unless air-furnace iron be used, which, 
indeed, it should be to avoid the porosity of cupola iron. 

The common construction by which the radius of the closed end 
of a hydraulic cylinder is made equal to the diameter of the cylinder, 
is a mistaken application of the fact that the stress per sq. in. due to 
internal pressure in a sphere is one-half the longitudinal stress in a 



Table 19. — Effective Fire Streams 

Using 100 ft. oi 2^4, in. ordinary best quality rubber-lined hose between nozzle and hydrant or pump 

By J. R. Freeman, C. E. 



Smooth nozzle, size 



?4 in. 



Jiin. 



Pressure at hydrant, lbs 

Pressure at nozzle, lbs 

Pressure lost in 100 ft. 2V^-in. hose. 

Vertical height, ft 

Horizontal distance, ft 

Gallons discharged per min 



Smooth nozzle, size 



Pressure at hydrant, lbs 

Pressure at nozzle, lbs 

Pressure lost in 100 ft. z^i-in. hose. 

Vertical height, ft 

Horizontal distance, ft , 

Gallons discharged per rain 



32 
30 
2 
48 
37 
90 



43 


54 


6S 


75 


86 


34 


46 


57 


69 


80 


91 


37 


SO 


62 


75 


87 


40 


SO 


60 


70 


80 


30 


40 


50 


60 


70 


80 


30 


40 


SO 


60 


70 


3 


4 


S 


5 


6 


4 


6 


7 


9 


10 


II 


7 


10 


12 


15 


17 


60 


67 


72 


76 


79 


49 


62 


71 


77 


81 


85 


51 


64 


73 


79 


85 


44 


50 


54 


S8 


62 


42 


49 


55 


61 


66 


70 


47 


55 


61 


67 


72 


104 


116 


127 


137 


147 


123 


142 


IS9 


174 


188 


201 


161 


186 


208 


228 


246 



100 

80 

20 

89 

76 

263 



iH in. 



iH in. 



206 238 266 291 314 



112 
80 
32 
92 
81 

336 



49 

30 

9 

53 

54 
256 



6S 
40 
25 
67 
63 
296 



81 
50 
31 
77 
70 
331 



97 
60 
37 
85 
76 
363 



113 
70 
43 
91 
81 

392 



129 
80 
49 
95 
85 

419 



S8 
30 
28 
55 
56 
315 



77 
40 
37 
69 
66 
363 



96 


116 


135 


154 


50 


60 


70 


80 


46 


56 


65 


74 


79 


87 


92 


97 


73 


79 


84 


88 


to6 


445 


480 


514 



Hydraulic Press Cylinders and Rams 

A reaction has taken place against the high pressures (5000 to 
6000 lbs. per sq. in.) which were favored at one time for hydraulic 
machinery. Such pressures are now used only when necessary and 
are obtained locally by intensifiers from a lower service pressure. 
When subject to free choice, the general service pressures used range 
between 1000 and 1500 lbs. per sq. in. for which, so far as possible, 
the operating machines are designed. 

The thickness of hydraulic press cylinders may be determined from 
Fig. 5 by Prof. A. L. Jenkins {Amer. Mack., Mar. 31, 1910) which 
plots Barlow's and Lame's formulas. Experiments by Professor 
Goodman, of Leeds, England, confirm the substantial correctness of 
Barlow's formula, which is to be preferred. The extensive use of 
Lame's formula leads the author to include its chart line for those 
who, using it, prefer to continue to do so, and for those who wish to 
make comparisons. It should be said also that Merriman's formula 
gives identical results with Barlow's and that Burr's and Lanza's 
formulas give identical results with Lame's. Following are Barlow's 
and Lame's formulas: 

S R 



P~T 



+ 1 



•-(VS-) 



Barlow. 



Lame. 



in both of which 5 = fiber stress, lbs. per sq. in., 

P = hydraulic pressure, lbs. per sq. in 
/? = inside radius of cylinder, ins. 
r = thickness of cylinder, ins. 

The use of the chart is best shown by an example. Required 
the thickness of a cylinder 20 ins. internal diameter, subjected to a 
pressure of 1000 lbs. per sq. in., the fiber stress on the material being 

, I, ... 5 6000 T,. 1 . , , 1. 

6000 lbs. per sq. in., giving 5^ = = 6. Find 6 m the base line, 

P 1000 



30 



20 



^.t 



En H 

•" C 
o o 



c 2. 



























s 






/, 


'1 

/ 
















0. 


M 






f 
















<D P- 


" 


// 




















s § 




'// 























// 


/ 




















M ; 


< 


■// 


/ 


















,!; 
S 

-d 




/a 


■ 




















0. 


H 


A 


y/ 




















a 

M 


i 


f 


7 












— 






J 



















.6 . .7 .8 .9 1.0 



5 6 J 8_9 10 



Diameter of pipe, ins. 

I- 45 deg. elbow equals \- 90 deg. elbow 

I- open return bend equals i- 90 deg. elbow 

I- gate valve equals 5- 90 deg. elbow 

I- globe valve equals 1 2- 90 deg. elbows 

I- tee equals 2- 90 deg. elbows 

Fig. 3. — Resistance of pipe fittings to the flow of water. 



HYDRAULICS AND HYDRAULIC MACHINERY 



241 



cylinder of the same diameter. The theory of such stresses is based 
on the supposition of a complete hemisphere and is not true for lesser 



I 




Fig. 4. — Precaution in pipe laying. 



20 



o^h 



2 10 

j2 









































/ 






































/ 


y 




































/ 


y 




































/ 


y 


































^/. 


y 


































(?C 


<*^ 




































^yyi^ 
































\ 


4z 


f 


































/ 


y\^ 


































A 


'4 


\° 


































/ 


y 


































/ 


y 




































/ 


y 




































/ 


y 




































/ 


y 




































/ 


y 




































/ 


y 




































/ 


y 




































/ 


y 






































^ 


































_ 





10 



Values of -!5- 
P 



15 



20 



Fig. 5. — Thickness of hydraulic press cylinders. 

segments. This application of the theory leads to bending stresses 
at the junction of the end with the barrel. Such stresses can be 



avoided only by making the end of the cylinder a complete hemisphere 
as shown in Fig. 12. When necessary to save room, some designers 
use an inside radius of three-fourths, a fillet of one-fourth the diam- 
eter of the cylinder and an end thickness equal to that of the cylinder 
walls. 

The thickness of hydraulic press rams, centrally and eccentrically 
loaded, may be determined from Fig. 6, by Professor Jenkins 
(Amey. Mach., Dec. 8, 1910). For centrally loaded rams, the relation 
is expressed by the formula: 



■aJi-i. 



75^ 



in which i? = outside radius of ram, ins., 
7" = thickness of ram, ins., 
P = pressure on ram, lbs. per sq. in., 
5 = stress on ram, lbs. per sq. in. 

This equation is plotted in the curve AB, Fig. 6. Enter the chart 
p 
by the assumed value of ^) trace upward to the ciurve AB and then 

to the left for the value of j. when, R being known, T is quickly found. 

For pressures less than 2000 lbs. per sq. in., the chart gives values 
for T that are small compared with those used in practice, due to the 
fact that the assumption of central loading can seldom be made. 
It is, therefore, best to assume an eccentric load for which the formula 
is: 



aK 



in which 



K- 



eccentricity of load, ins. 



R 



the remaining notation being as in the last formula, while the 
plus sign relates to the compressive and the negative to the tensile 

stress. 



i 





— 


1^ 


— 






'^1 


t 




tr 


./b 


— 




— 




— 


In- 


7\ 


1 , 


T\ 


— 


— 


— 








— 1 


— 


— 


— 


— 


— 


— 


7\ 





-0. 




■2 










^ 1 


i 


/ 


# 




? 












.f^ 




s 


/ 




























/ 






.? 










f,7 


^/ 




li 




h' 












fi 


i' 


/ 


// 
























• n 


-> 


/ 








H 


j 


a, 










-i 


i 


1 


/ 


^ 




//S 










1 


^°; 


/ 


\ 


y 
























r^ 












T 


A 








^ 


V 


^ 


' 


1^ 




/ 


-s' 












N 


-/ 




" 1 


/ 






















n^ 


^/ 














,«, 


r^ 










ylv 


1 


i^ 


? 


/* 










■fu 


"/ 




t 


p 






















V 


'> 
















'«1 


T ■ 










k 


/ 


11) 


'■? 


/ 














/ 


/ 




/ 




















\ 


< 


/ 




















V 










/ 


/ 


/-i 




l\ 


' 












/ 




/ 






















A 


y 


































/ 


h 


\ 


1 














/ 




1 


/ 




















/ 


/ 




































V 


„■> 




/ 












/ 






/ 




















/ 








































^ 




1 












/ 


/ 




/ 




















/ 














































1 












/ 




A 




















/ 






























' 
















1 












/ 






/ 


















/ 
















































1 




, 






/ 






/ 


















/ 
















































1 












/ 




1 


/ 














/ 


/ 












































; 
















/ 






/ 














/ 


/ 














































/ 






1 








/ 






j 














/ 


^ 
















































// 














/ 




















/ 


















































' 












/ 








1 












/ 
































































/ 






j 












/ 














































1 


















1 


r 






1 










/ 
















































1 


















1 








I 








1 


1 
















































1 


















/ 








f 












































































' 
















1 




















































A 


















































































Votle: 


both 'Curve 


s for 


K 


= % are Compre 


ssi 


on 


b 


;ca 


us 


\ with 














































go small a'n Ecc 


en 


triiity, the|Tenslo4 due lo fee 


iding 


is 


no 


t 


































_ 


_ 




enoughl to] ov,er( 


onie the' direk fcomptessioli, ijf 


,he likm 


Tbad 































10 



11 12 13 14 15 16 17 18 19 20 21 



Values of -^ 



16 



Fig. 6. — Thickness of hydraulic press rams. 



242 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



m 



Fig. 7. 
Cup Packing 



Fig. 9. 
Hat Packing 



Ti2 



% vmm////m //,. 



Fig. 8. 
U Packing 

M//WM V/M/////!. 



Fig. 10. 
Failure of Packing 



^ 



S' 







2 to 4 




Fig. 13. 

Cup and Hat 

Packing 



Fig. 14. 
Leather ring plunger packing. 

Figs. 7 to 16 



Fig. 16. 

Leather ring flange packing. 



Diameter of Rubber 
Ring ^16 in. 



Fig. is 

Rapieff flange packing 



-High-pressure hydraulic packings. 



The use of the formula is best shown by an example: 

Let F = 2000 lbs. per sq. in. 
R = S ins., 
2"= if ins., 
eccentricity of load=io ins., 

givingiT = — = 2 

Enter the chart with ^ = -^ 

=3 with suflicient accuracy, and trace 

to the right for the curves for K=2 and then down, finding p =8.2 

for tension and 11.75 for compression, or, 5 = 8.2X2000 = 16,400 
for tension and 11.75X2000=23,500 for compression. 

Hydraulic Packing 

The present tendency, for work of a rugged character, is to adopt 
thef stuffing box in place of all forms of leather packing in hydraulic 
machinery. The chief reason lies in the easier renewal and the 
reduced liability of stufiing-box packing to injury from rusty and 
scored rams and cylinders. Security against injury from these 






'^^" • 




— - '; 






$$$SM-sd 


w 



Fig. 18. 



i 



7~-i 



EiG. 19- Fig. 20. 

Figs. 17 to 20.— Molds for making hydraulic cupped leathers 



causes makes necessary the use of brass or copper linings with leather 
packings, a precaution that is not necessary with the stuffing box. 
Leather, however, still has, and probably always will have, appli- 
cation in packings for valves and in places where it can be protected 
from injury. 



Figs. 7, 8 and g show the cup, U and hat forms of hydraulicleathers, 
Figs. II, 12 and 13 showing applications. Fig. 10 shows the manner 
in which the leathers fail at the bend A. In order to reduce this 
tendency it is important to shape the part against which the bend 
of the leather fits to the shape of the leather as shown in Figs. 11, 12 
and 13. The depth of the packing has no effect on its tightness or 
friction and, to save unnecessary stress on the leather when forming 
it, a moderate depth is best. Prof. A. Lewis Jenkins {^Amer. 
Mach., Sept. 22, 1910) gives the dimensions of Table 20 for U and 
cup leathers. 

Some constructors prefer the leather-ring form of packing to the 
cupped leather. This construction with dimensions is shown in 
Fig. 14. It has been used successfully for pressures of 6000 lbs. per 
sq. in. Cylinder packing is turned to fit the cyHnder and ram 
packing is bored to fit the ram, other dimensions being left rough. 
Multiple leathers are stitched or cemented together. The pressure 
water must be given free access to the spaces behind and below the 
ring and, contrary to what might be expected, this construction can 
only be used for pressure in one direction — double-acting cylinders 
requiring a ring for each direction of pressure. Obviously too, it 
will not make the double joint of Fig. 12. If used in this construction 

Table 20. — Dimensions of Hydraulic Cupped Leather Packings 



ff 



-D-- 



' Y i^=Clearance In 

Cavity 



Ji 



End U-packing 


U-wall, or neck 
packing 


Cup 
packing 


Stock 


Diam. of 


n 





TJ 


r 


D 





H 


K 


T) 


n 


Diam. 


Thick. 


ram 


























4" 


2r 


4" 


i" 


.3_// 


4" 


sl" 


i" 


A" 


4" 


1" 


6" 


A" 


6" 


4i" 


6" 


i" 


A" 


6" 


vi" 


I A" 


A" 


6" 


li" 


8i 


A" 


8" 


6J" 


8" 


li" 


^" 


8" 


9V' 


ji" 


A" 


8" 


li" 


11" 


A 


10" 


8J" 


10" 


U" 


!." 


10" 


Hi" 


if 


i" 


10" 


i-F' 


13" 


i" 


12" 


10^' 


12" 


li" 


i" 


12" 


I3i" 


li" 


i" 


12" 


il" 


15" 


\" 


13" 


Hi" 


13" 


i|" 


i" 


13" 


I4i" 


li" 


i" 


13" 


i|" 


I6i" 


\" 


14" 


I2i" 


14" 


i§" 


I" 


14" 


I5i" 


I J" 


i " 


14" 


If" 


I7i" 


J" 


18" 


16J" 


18" 


I J" 


i" 


18" 


I9i" 


ij" 


A" 


18" 


2" 


22" 


i" 


21" 


I9J" 


21" 


15" 


\" 


21" 


22I" 


ij" 


_S_II 


21" 


2" 


2S" 


i" 


23" 


2li" 


23" 


rV 


1'/ 


23" 


24§" 


il" 


A" 


23" 


2" 


27" 


i" 


24" 


22 1" 


24" 


i|" 


\" 


24" 


2Si" 


If" 


A" 


24" 


2" 


28" 


i" 



HYDRAULICS AND HYDRAULIC MACHINERY 



243 



a joint must be made under the gland flange and for this the Rapiefi 
joint, Fig. IS, is well tested and very suitable (see RapiefE joint). 

The action of the leather-ring packing is undoubtedly the same as 
that of the cupped form, that is, the pressure behind and below the 
leather forces it toward the joint, as Fig. lo shows it to do with cupped 
packing. This.makes it suitable for use as a flange packing as shown 
in Fig. i6, in which it is not pinched in place but is free to be forced 
into the joint after the manner of the Rapieff construction. 

Leather packings are usually made of hard, close-grained, oak- 
tanned leather, though some prefer Vim leather. They should be 
shaved to uniform thickness and soaked in warm water to make 
them pliable. Concensus of opinion favors the flesh side for the 
wearing side. After drying, they should be soaked for a half hour 
in warm tallow or paraffin Fig. 17 {Mechanical World, 1905) shows 
the form of the mold for cup, Fig. 18 for hat, and Figs. 19 and 20 for 
U leathers. In making U leathers, after the first operation. Fig. 19, 
the leather should stand an hour or more when the center block is 
inserted, the middle part is forced partly back and the whole allowed 
to dry. The mold should be forced together with a vice or press, 
the use of a central bolt being inadvisable as the leather draws away 
from the hole and sometimes tears. 




Fig. 21. — Dimensions of high-pressure hydraulic stuflSng boxes. 

The proportions of stuffing boxes for high-pressure hydraulic work 
may be determined from Fig. 21 by Professor Jenkins {Amer. 
Mach., Sept. 22, 1910). The diameter of the bolts should be figured 
for a stress at the root of the thread of from 10,000 to 15,000 lbs. 
per sq. in., remembering that the studs need carry no considerable 
stress due to screwing up and that the load due to the pressure is 
only that for the ring area of the cavity. 

There is a diversity of practice regarding the shape of the bottom 
of the stuffing box and the end of the gland, some claiming that the 
taper shown by the dotted lines leads to an undue pressure against 
the ram if leakage around and outside the packing is to be prevented, 
while others claim that the flat construction requires such pressure 
of the gland as to cause the packing to harden and score the ram. 
The construction shown in the small view of Fig. 2 1 would seem to 
be the most logical. 

Friction of Hydraulic Cup Leather Packing 

The friction of hydraulic cup leather packing formed the subject of a 
thorough and painstaking investigation with specially constructed 
apparatus by John Hick of Bolton, England, and published in a 
pamphlet by E. & F. N. Spon in 1867. Experiments were made on 
|-, 4-, and 8-in. rams under a great variety of pressures ranging 
between 200 and 6000 lbs. per sq. in., and showed the following 
general results: 



The total friction increases with the pressure. 

At constant pressure per sq. in. the total friction increases in 
direct proportion with the diameter of the ram, that is to say: 

At constant pressure per sq. in. the percentage lost by friction 
varies inversely with the diameter. 

The depth of the leather does not affect the friction of the ram. 

The experiments resulted in the following formula: 

Total friction, lbs. = CXram diam., ins.Xpres. per sq. in., lbs. 

The value of C for new, hard, badly lubricated leathers was .0471 
and for well-lubricated leathers in good condition .03 14. 

Mr. Hick concludes the account of his experiments with Table 2 1 
of the frictional resistance as a percentage of the total pressure on 
rams. These losses seem very small but they are well authenticated. 




Fig. 22. — High-pressure hydraulic stop valve. 

Table 21. — The Frictional Resistance of Cup Leather 
Hydraulic Packing 



Diameter, inches 


Friction, per cent. 


Diameter, inches 


Friction, per cent. 


2 


2.0 


12 


0.33 


3 


1-33 


13 


0.30 


4 


1 .0 


14 


0.28 


S 


0.80 


IS 


0. 26 


6 


0.66 


16 


0.2s 


7 


0.57 


17 


0.23 


8 . 


0.50 


18 


0.22 


9 


0.44 


19 


0.21 


10 


0.40 


20 


0.20 


II 


0.38 







Friction of Hydraulic Stuffing Boxes 

The friction of hydraulic stuffing boxes using braided hemp packing, 
compressed rather hard with fair lubrication and plunger condition, 
formed the subject of experiments made at the Pencoyd Iron Works, 
and reported by Walter Ferris (Amer. Mach., Feb. 3, 1898). The 
apparatus used was a hydraulic intensifier having rams of 17^, 141, 
and 8 ins. diameter used in various combinations. Initial pressures 



244 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



of 285, 335, 350 and 475 and terminal pressures of 750, 1450 and 1510 
lbs. per sq. in. were used. 

Unlike the cup leather, the stuffing box is subject to the outside 
pressure of the gland bolts and with the gland screwed down hard 
enough to hold high pressures, an increased percentage loss was 
naturally found when lighter pressures were used. With the packing 
compressed only enough to prevent leakage, Mr. Ferris finds the 
formula: 

Total friction, lbs. = .2Xram diam., ins.Xpres. per sq. in., lbs. 
to fairly represent the most efficient performance to be expected 
from machines having a single ram. 

For the percentage of loss Mr. Ferris deduces from this: 



Total friction, lbs. 



•25s 



Total pressure, lbs. diam. ram, ins. 

showing the percentage of loss to vary inversely with the diameter 
and, since low pressures involve large diameters, he advocates low 
pressures, with intensifiers where necessarJ^ 

For intensifiers, which have two rams, he deduces the formula: 



p2 = p\ 



.2D 



. 2d 



in which />i = initial pressure, lbs. per sq. in., 

/12 = intensified pressure, lbs. per sq. in., 
A =area of large ram, sq. ins., 

a = area of small ram, sq. ins., 
Z) = diameter of large ram, ins., 

(i = diameter of small ram, ins. 

Table 22 compares the results obtained with varying initial pres- 
sures under the same adjustment of the stuffing box with the results 
to be expected from this formula and brings out the progressive 
effect of an undue tightening of the gland bolts. 



Table 22. — Ferris's Experimental Results Compared with 
His Formula for the Friction of Hydraulic Intensifiers 
Fitted with Hemp Packed Stuffing Boxes 



Initial pressure, lbs. per sq. in 

Intensified pressure by experiment, lbs. per sq. in. 
Intensified pressure by formula, lbs. per sq. in. . . . 
experiment 



Ratio 



formula 



475 


350 


335 


I4S0 


iSio 


1450 


1433 


1643 


1572 


1. 01 


.92 


• 92 



28s 
750 
860 

.87 



High-pressure Hydraulic Valves and Fittings 

For high-pressure flange joints, see Index. 

A valve jor high-pressure (1500 lbs.) hydraulic service is shown in 
Fig. 22, by Jas. Clark {Amer. Mach., Aug. 10, 1911). 

The body 5 is a steel casting. The inlet and outlet are 2 ins. in 
diameter. The bushings C are 3I ins. inside diameter and were 
made from drawn brass tube | in. thick. They are provided with 
16 inlet ports d and 16 outlet ports e, each f Xi in. The cup seats, 
distance pieces, and screw are brass, while the stem F is soft steel, 
brass not having the requisite strength. The clampin'g parts and 
handwheel are cast-iron. 

The joints in the center are made tight by the use of gaskets, but 
it was not considered safe to use gaskets for the end joints; hence, 
the U cups were used instead. The bushings were made a slip-fit 
in the body. It is also necessary to have vent holes at the points 
g and h, so that in case a cup should leak, the balance would not then 
be destroyed. 

In service these valves have proven absolutely tight. 

A quick return hydraulic valve, intended to reduce the time required 
for the return stroke of hydraulic machinery, is shown in Fig. 23 
{Amer. Mach., Apr. &, 1897). The valve is used successfully in alarge 
steel works under pressures as high as 4000 lbs. per sq. in. The 
construction involves a main valve operated by the water pressure, 




III 1" Standard Pipe 
i— -r-J- Connection 



Section A-B-C 



-J"— Connection 

Fig. 23. — Quick return hydraulic valve. 



HYDRAULICS AND HYDRAULIC MACHINERY 



245 



which latter is controlled by the usual small hand-operated valve, 
which becomes, with this arrangement, a supplementary valve. 

The valve is attached at any convenient point, preferably directly 
to the hydraulic cylinder at a. The high-pressure water enters 
from below at h. At the top, at orifice c, is a connection to a small 
hand valve, which controls the action of the piston d, which forms 
the head of and actuates the main valve. The water is discharged 
from the main cylinder through the valve and the opening e, the valve, 
as shown, being in position for discharging the main cylinder. 

In operation, when the water above the piston d is discharged 
by a small hand valve, the water at h forces the valve up at once, 
closing the exhaust and introducing the high-pressure water into the 




Fig. 24.- Leather Rings in Valve Seat 




Fig. 25. . Leather Ring in Valve 

Figs. 24 and 25. — Leather sealed valves for high-pressure hydraulic 

service. 

main cylinder. A reverse operation forces the piston d down, shut- 
ting off the water from the main cylinder, and simultaneously opening 
the exhaust via the outlet e. It will be observed that the valve is 
metal-seated, no live packing being necessary except that shown in 
the small piston d. 

A leather-seated valve for high-pressure hydraulic pump service, 
as used at the Pencoyd Iron Works under 500 lbs. pressure, is shown 
in Figs. 24 and 2s{Amer. Mach., July 14, 1898). 

A groove about J in. square, but slightly dovetailed at the bottom, 
is cut in the face, in which is inserted the ring of leather, this ring, 
as inserted, standing slightly above the surrounding surface, but, 
of course, soon becoming flattened under the pressure until this 
projection above the surface is scarcely appreciable. Fig. 24 shows 
the construction as applied to the seat, while Fig. 25 shows it applied 
to a valve. 

This construction is practically a complete preventative of the 
tendency of high-pressure water to cut and score the valve seats. 



Valves for high-pressure hydraulic service as used on the pumping 
engines of the Pope Tube Mills (Riedler system) are shown in Fig. 26 
(Amer. Mach., Oct. 27, 1898). The illustration shows both suction and 
discharge valves which are identical. The engine operates at 60 
r.p.m. and under 1500 lbs. per sq. in. pressure with entire smoothness, 
indicator cords under those conditions being almost entirely free 
from oscillations. The special feature of the Riedler system of valves 
is that while they are closed positively by mechanism they are opened 
by fluid pressure — air or water as the case may be. The valve seat 
will be seen to be inserted and to be held down in place by the conical 
ended pins a, while it is packed against leakage by the cupped leather 




Fig. 26. — Poppet valve for high-pressure hydraulic service, Riedler 

system. 

b. By the aid of six webs it carries a central seat c, the opening being 
thus annular. From the seat c rises the guiding stem d, on which 
the valve slides as it opens, the stop e limiting its movement. The 
valve proper/ is a ring and carries above it a second ring g, having 
four cross webs h cast in one with it, the sleeve i, also in one with h, 
surrounding the stem d. The sleeve i has a threaded cap j screwed 
down to a defined position, and serving on its lower side as a stop 
for the flange k, which is pressed upward against^ by a strong spring I. 
The rock shaft m is the valve-closing shaft. Two disks, one of which 
is seen at n, stand one each side of the valve stem d, their simul- 
taneous action being secured by the connecting tie 0, while each 
disk carries a pin p, which, when moving downward, acts on the flange 



246 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



k and closes the valve through the medium of the spring I. At 
first sight there would appear to be still a spring action here, but in 
point of fact the spring is only a safeguard. It is of a strength 
sufiicient to cause the flange k and the sleeve i and with them the 
valve, to move as one piece. The action is strictly mechanical, and 
the valve moves with the mechanism as though the spring were not 
there, hut shovild a particle of foreign substance lodge under the valve, 
and so prevent its seating, or should the valve be set wrong, during 
the trial adjustments, the spring would give way and prevent damage. 
Without the provision of the spring, disaster might follow on slight 
causes. 

Another novel feature of this valve is seen in the leather ring q, 
placed between the valve / and the ring g. The tendency of high- 
pressure water to take advantage of any slight leak through a valve 




Cup Leather 
Oup Leather \ 

Ground Joint 45 



Fig. 27. 

Rapid Coupler for Hydraulic L.<^)) 
Connection 




Standard Pipe 
Thread 



Fig. 28. 
Hydraulic Hand Union 




Fig. 31- 



Fig. 32. 



Figs. 27 to 32. — Rapid and flexible hydraulic pipe connections. 

and cut the valve and seat into channels is well known, and this 
leather ring is provided to stop it. The valve is not leather packed 
in the usual sense, in that it has a metal to metal Joint, being in fact 
a perfect valve from usual standards without the leather ring, which 
simply serves to seal the joint and prevent the water from finding 
its way through any incipient leaks that may be present, and thus 
prevent grooving and channeling the valve and seat. 

Flexible hydraulic fittings, designed originally for a testing room 
but useful in other locations, are shown in Figs. 27-32 by F. S. 
Bunker {Amer. Mach., June 22, 1911). The rapid-connection coup- 
ling. Fig. 27, has its upper half connected into the stop valve on the 
main-line tap while the lower half is fitted to the pipe A of any of the 
flexible or semi-flexible joints. Figs. 29, 30 and 31. When connecting 
Fig. 27, it is only necessary to push the central pipe into the main 



housing and give it a quarter turn which seats the cross pin in the 
hook H. Fig. 32 is a swivel joint which, used in pipes A and B of 
Figs. 29, 30 and 31, makes them all universal. Fig. 28 shows a 
union joint.- 

A swivel pipe joint for hydraulic pipe work, compiled from actual 
experience, is supplied by U. Peters {Amer. Mach., June 11, 1903) 
and shown in Fig. 33 and the accompanying table of dimensions. 

An air-chamber charging device used by the Nordberg Mfg. Co. in 
connection with their large pumping engines is shown in Fig. 34 
(^Arner. Mach., Mar. 26, 1903). 

The main globe casting, which is of 13 ins. interior diameter, is 
located near the water end of the engine and is connected at a with 
one end of one of the pump cylinders and at b with the air space in the 
air chamber. The connection at a being with a pump cylinder, the in- 
terior of the globe is subject to alternate pressure and suction. Dur- 
ing the suction stroke the globe is filled with air by the valve d, and 
during the pressure stroke this air is expelled through the valve e 
and the pipe b to the air chamber. The valve c is introduced to 
control the ingress and egress of the water. On the one hand air 
must not enter so freely as to more than fill the globe and then escape 
to the pump cylinder, and on the other the globe must be so nearly 
full of water that the rise during the pressure stroke will expel the 
air. With free egress of the water it might easily escape in such 
volume that, opposed by the compressed air above it, it would not 
rise to a point where the air would be expelled. On the contrary, 
there is no danger of having too much water in the globe, as the return 
of the water from pipe b is prevented by the valve e, any drop of the 
water level insuring the entering of a fresh supply of air. To insure 
working conditions it is essential then that the water shall enter the 
globe more freely than it escapes from it. Valve c is therefore an 
obstruction valve only — that is, it has a hole through it for the water 
to escape. During the pressure stroke this valve opens freely and 
the water enters and expels the air, but during the suction stroke 
the water can only escape through the hole and the flow outward 
being more restricted than that inward, it is made certain that the 
volume of water escaping from the globe will not be in excess of that 
which will again completely fill it and expel the air on the succeeding 
pressure stroke. 

An air-chamber charging device for high-pressure work by Walter 
Ferris is shown in Fig. 35 {Amer. Mach., Nov. 2, i8gg). It consists of a 
vertical barrel of pipe, connected at the bottom with one water cylin- 
der of the main pressure pump, and ending at the top in a branch 
carrying two check valves, one inward and one outward opening. 
The connection to the main pressure pump is preferably— though 
not necessarily — double; a large pipe with a check valve opening 
toward the main pump, and a small pipe without a check valve. 
This is to partially equalize the amount of water passing from the 
main pump to the air pump under (say) 500 lbs. pressure per sq. in. 
with the amount returning to the main pump during its suction 
stroke, at a pressure of 5 or 10 lbs. Hence, during the suction 
stroke the water from the air pump comes into the main pump through 
both the ij-in. and 5-in. pipes, but it is expelled during the delivery 
stroke through the |-in. pipe only, as the check valve in the 1 5-in. 
pipe closes. It is absolutely necessary to the successful working 
of this pump that it should have no appreciable clearance spaces. 
During the delivery stroke of the main pump the water must rise 
into the air pump and fill it up to the upper check valve F2, expelling 
every particle of air. Some water goes through the check valve, 
too, but that does no harm. Then, during the suction stroke the 
valve V2 closes, the water is drawn back to the main pump, followed 
by air which enters through the check valve Vi, and which is expelled 
during the next delivery stroke. The branch carrying the two check 
valves Vi and V2 is inclined, in order to prevent the trapping of air 
under the caps of the valves or in other recesses. For the same 
reason the nipple a is tapped into the cap b at the highest point. 
In proportioning the pipes c and d it is also necessary to be sure that 
more water will leave the main pump during each delivery stroke 



HYDRAULICS AND HYDRAULIC MACHINERY 



247 




Nominal „ 






Nomina 




Size % 


l" 


l'/4 


Size 


34' 


A % 


34 


1 


Q 


1?4 


L 1%, 


8>l6 


934 


r 


H 


M 2% 


23/4 


3 


X 


H 


N 3 


3'/4 


334 


K 


234 


3i>i6 


334 


3>M6 


G 


iM 


D iH 


4^6 


4 1/2 


H 


IH 


I \H 


2H 


23/8 


B 


25/8 


C 3>lo 


354 


4 


U 


% 


F 1 


IH 


13/8 


V 


1 


P 2i>lo 


234 


3 


w 


34. 



Fig. 33. — Dimensions of swivel pipe joints for high-pressure hydraulic service. 



than will come back during the next suction stroke. Otherwise 
the barrel of the air pump will soon be pumped dry and fail to work. 
Long suciion pipes should he fitted with air chambers. It is not the 
pressure but the energy of the moving water that causes water 
hammer and, except for the increased diameter of the pipe and the 



■Air Deltvery, 



V2 - 1 Valve 




Fig. 34. — Air chamber charging device. 

resulting reduced velocity of the water, air chambers are about as 
necessary on suction as on pressure lines. F. M. Wheeler has pointed 
out {Trans. A.S.M. E., Vol. 14) that suction chambers are frequently 
wrongly connected. The air chamber should be so placed that when- 
ever the column of water is stopped or checked by the action of the 




Fig. 35. — Air chamber charging device for high pressures. 

pump it can flow on past the pump suction chamber or valves to the 
air chamber, so that its energy can be expended directly on the con- 
fined air. The chamber should not be so placed that the water 



248 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



passes under or past a right angle opening into it. Mr. Wheeler 
cites Fig. 36 as a bad and Figs. 37, 38 and 39 as good arrangements. 
The siphon, although the simplest of mechanical appliances, is 
subject to many vagaries and frequently refuses to operate. Lei- 
cester Allen {Amer. Mach., Sept. 21, 1893) after much experimental 




Fig. 36. 



Fic5. 37. 



r\ 



fkS" 




Suction 




Figs. 



Fig. 38. Fig. 39. 

36 to 39. — Correct and incorrect arrangements of suction 
air chambers. 



work, explains many of these actions and the means of overcoming 
them as follows: 

The siphon shown in Fig. 41 is clearly inoperative. The discharge 
from the free end h is so much greater than the inflow at a that the 
liquid in the branch ch runs out, admitting air and stopping the action 
almost immediately, unless the siphon be so small that the leg cb, 
through its capilarity, holds the liquid column from breaking up. 
In general, through the effect of capilarity, small siphons will often 
work even though large ones constructed in the same way will not. 

In Fig. 42 is shown a reverse construction in which the supply end 
is made much larger than the discharge end, the latter being still 
free as before. Here the supply always being more than suiScient 
for satisfying the capacity of the discharge end, the siphon, if operat- 
ing upon a liquid entirely free from absorbed gas and non-volatile, 




Fig. 41. Fig. 42. Fig. 43. 

Figs. 41 to 43. — Operative and inoperative siphons. 

will be continuously operative so long as the liquid in A is maintained 
at a level that will keep the end a submerged. 

Fig. 43 illustrates how the inoperative form of siphon, shown in 
Fig. 41, may be made operative by submerging the discharge end 
in the liquid of the receiving tank B. With a non-volatile liquid 
containing no absorbed gas, the discharge of the siphon, thus con- 
structed and arranged, would be continuous, so long as both ends 
are kept submerged. 

A practical conclusion from what has been said is, that siphons 
will be more certain in their action when the supply end is larger 
than the discharge end, and when the discharge end is submerged. 
A regulating valve or cock, at or near the discharge end of the siphon, 
may be used to adjust the discharge and regulate it into proper 
relation with the supply. When this appliance is used, the pipe 
may be made of equal size throughout; and, in such case, the dis- 



Table 23. — Percentage of the Total Amount of Water Supplied to Hydraulic Rams which is Delivered 

TO Various Elevations 
Elevation of discharge above delivery valve at ram 



Working head 



15 ft. I 18 ft. I 21 ft. I 24 ft. I 27 ft. I 30 ft. I 35 ft. I 40 ft. I 45 ft. I 50 ft. I 60 ft. 



70 ft. I 80 ft. I 90 ft. I 100 ft. 



2 ft. 

3 ft. 

4 ft. 

5 ft. 

6 ft. 

7 ft. 

8 ft. 

9 ft. 

10 ft. 

11 ft. 

12 ft. 

13 ft. 

14 ft. 

15 ft. 

16 ft. 

17 ft. 

18 ft. 

19 ft. 

20 ft. 

21 ft. 

22 ft. 

23 ft. 

24 ft. 



.0724 

■ 1327 
.i960 
. 2614 
.3282 

.3960 
.4647 

■ 5341 
.6040 
.6745 

•74S3 
.8166 
.8881 
.9600 



.0533 

. 1020 

.1535 
.2068 
.2614 

• 3170 

• 3733 
.4303 
.4877 

• 5459 

.6040 
.6627 
.7217 
.7809 
.8404 

.9001 
.9600 



.0402 
.0807 

• 1234 

• 1686 
.2146 

. 2614 

■ 3090 

• 3572 

• 4058 

■ 4549 

• 5043 

• 5540 
.6040 
.6543 
.7048 

■ 7555 
.8064 

• 8574 
.9086 
.9600 



• 0307 
.0651 

. 1020 
.1404 
. 1800 

.2203 
. 2614 
.3030 
.3450 
■3874 

.4302 
•4732 
.5166 
■ 5601 
.6040 

.6480 
.6921 
.7364 
.7808 
.8254 

.8701 
.9150 
.9600 



■0235 
.0530 
.0854 
.1189 

.1535 



. 2614 
. 2984 
.3357 

• 3733 

■ 4112 

• 4494 
.4877 

■ 5263 

• 5650 
.6040 

■ 6430 

■ 6823 
.7217 

• 7612 
.8007 
.8404 



.0181 

• 0441 

• 0724 

• 1020 
•1327 

• 1640 
. i960 
.2285 
. 2614 
•2947 

.3282 
.3620 
.3960 
■4303 
.4647 

• 4993 

• 5341 

• 5690 

• 6040 

• 6392 

• 674s 
.7098 
■7453 



.0112 
.0326 
.0560 
.0807 
.1063 

•1327 

• 1595 
.1868 

• 2145 

• 2425 

.2708 
.2994 
.3282 

• 3572 

• 3863 

■ 4157 
■4451 

■ 4746 
.5042 

■ 5340 

■ 5640 

• 5940 
.6241 



.0063 
.0243 
.0441 
.0652 
.0870 

. 1096 

• 1327 
.1561 
. 1800 
. 2041 

■ 228s 

■ 2532 
.2780 

• 3030 
.3282 

• 3535 

• 3790 
.4046 

■ 4303 
.4561 

.4820 
.5080 

• 5341 



.0027 
.0181 
.0348 

• 0533 
.0724 

.0920 

. II2I 
•1327 
•1535 

• 1746 

. i960 
.2177 
■2395 
.2614 
.2835 

.3058 
.3282 
■3507 

■ 3733 
.3960 

.4188 

■ 4417 
.4657 



.0132 
.0281 
.0441 
.0608 

.0782 
.0960 
. II42 

• 1327 

• 1514 

.1704 
.1896 
.2090 
.2285 
.2482 

.2680 
.2880 
■ 3081 
.3282 
.3486 

.3688 
.3892 
.4097 



.0063 
.0180 
.0307 
■0441 

.0580 
.0724 
.0870 

. 1020 
. II72 

■ 1327 
.1483 
. 1640 
. 1800 
. i960 

.2123 
.2286 
.2449 
.2614 
. 2780 

.2947 
•3II4 

■ 3282 



.0017 
.0112 
.0217 
•0325 

.0441 
.0560 
.0682 
.0807 
■0934 

.1063 
•II94 
.1327 
. 1460 

■ 1595 

•I73I 
.1868 
.2006 
■2145 
.2286 

■2425 

■ 2567 
.2708 



.0063 
.0150 
.0243 

.0340 
.0441 
•0545 
.0651 
. 0760 

.0870 
.0983 
. 1096 
. I2II 
.1327 

.1444 
.1561 
.1680 
. 1800 
. 1920 

.2041 
■ 2163 
.2085 



.0027 
.0099 
.0180 

.0264 
.0351 
.0441 
•0533 
.0627 

■0723 
.0821 
.0920 
. 1020 
. II2I 

. 1223 
• 1327 
.1430 
.1535 
. 1640 

.1746 
.1853 
. 1960 



.0063 
• OI32 

.0205 

.0281 
.0360 
.0441 
.0524 

.0608 
.0694 
.0782 
.0870 
.0960 

.1050 
.1142 
.1262 
■ 1327 
.1420 

• I5I4 
. 1609 
.1704 



I 



HYDRAXJLICS AND HYDRAULIC MACHINERY 



249 



37,800 


630 








W!? 


/ W 


iLuJA^l_y\^. 


'h/ 


J. 


<i/ 


■k^ 


y^/ 


■k. 


V 


v/ 


n 


—34,200- 
—32,400 
—30,600- 
-28,800- 
—27,000- 
—25,200- 
■^23,400- 
-g-21,600- 
5-19,800- 
a-18,000- 
-g-16,200 
S-14,400 

a 


—670- 
-640- 








6 


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9 


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/ 




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/- 


^.-v 


-480- 
—450- 














/ 


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f 


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y 


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y 










3-390- 

|-360- 
a-330- 
-1-300 
-fl-270 
3-240 

CO 










1 




/ 


/ 


/ 

t 


/ 


/ 


/ 


/. 


/ 


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< 


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y. 


/ 


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^/ 


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y 


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/ 


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^/ 


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^ 


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'/ 


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y 


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// 






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^ 
^ 


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(/, 


// 


y. 


^ 


^ 


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—10,800 
— 9,000 

7 "00 


—180 

—150 

inn 




1 










'//. 


^ 


% 


^ 




r' 






































In 










^ 


^ 


"x 


/ 








































; 


III 










^ 


t> 












































5,400 

3.600 


on 


/ 


'III 






% 


^ 
















































ffO 


11/ 


m 
























































EfficieDcj 
of Pump 


Water 
Hp. 






1 








i 


^ 








( 








8 


9 


10 


11 


12 


13 


14 1 


90 !« 


B.Hp. 




1 




2 




3 




4 


6 


6 




7 




8 




9 


10 


11 


12 




13 


1 

14 


1 

15 




80 5^ 







1 




2 


3 




4 






6 




7 


8 




9 


10 


11 


■ 


2' 


13 


' 


14 


15 


16 


— 1 — 
17 




70 J« 







1 


2 




3 


4 


' 


5 


6 


7 




s 


9 


10 


11 


15 


' 


13 


14 


15 


16 


17 




18 


19 


'2'0 1 


60 sS 







1 


2 


3 


4 




5 


6 


7 


8 


9'l'o' 


11 


12 


13 


14 15 16 17 


18 


19 


20 21 


22 


23 




50 i{ 




b 1' 2 




' 




5 6 ' 


'' J 


S 


10 11 12 13 14 15 16 l|7 18 19 20 21 22 23 24 25 26 27 28 1 


40 5S 




01 


2 


3 4 5 6 


7 


8 


9 I'o 1112 13141516 I7I 18 1192021 22 123 24 25 26 2712829 30 31 32 '33 34 35 1 






J_ 








' 


, 1 ,| , 


' n r 


1 


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L 


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1 ■ 11 -(-II r -I -| 



15, to 113 Ft. Head 



37,800 

36-.000 

34;200 

32;400 

30;600- 

28i800- 

27^000 

25,-200 

-S-23,-400- 
-W-21,600- 
-§,-19,-800- 
-g-18,000- 
-|-16;200- 

-^— 14;400 

^12i600 


630 
—600- 






























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y 




























31^3^ 


:^M3S^2S^ 


ft 


^^^^m£ 


I^^SMS^^^^ 


\..y\ 


tgs- 




—540- 
-510- 
—480- 
—450- 
-420- 
1-390- 
i-360- 
-5J-330- 

■g-3eo- 

|2-270 
O-240- 
—210- 


























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r? onn 


—120- 








1 


1 


1 


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^^ 










































































5 400 








^^ 














































































3,600 


60 




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^ 




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*' 
















































































Efficiency 
of Pump 


Water 
Hp. 





L 2 


3 ' 


I 5 


3 7 8 9 1 


li 12 13 14 15 ] 


6 17 18 19 20 21 22 23 24 25 26 27 28 2'9 30 3i 32 33 34 35 36 37 38 39 46 4i 42 43 44 45 4'6 1 


90 5i 


B.Hp. 




2' 


4 




? 




8 


10 


12 


14 




1,6 


18 


20 


22 




2,4 


26 


2,? 


30 


32 


34 


36 


38 


40 


42 




44 




46 


48 


50 


52 


80 5S 


•• 




2 




4 


6- 


8 


10 


12 


14 


16 18 


20 


22 


24 


26 '28 


30 


32 


34 


36 


88 


40 


42 


L 


4 


46 


48 


50 


52 


54 


56 




58 


70 5S 


•■ 




2 


4 




6 


8 


'I'o I12 


14 


16 


18 


20 22 


24 26 


28 


30 32 


34 36 


38 


40 42 


44 46 


48 


50 52 


54 


56 58 


60 62 


64 


66" 1 


60 sS 


" 




2 


4 


6 


8 


'I'O i2 


14 


16 


is' 20 '22 


24 


26 


28 30 32 34 36 


38 


40 42 


S4 


46 


48 ' 50 52 54 


56 


58 60 62 64 


66 


62 70 72 74 


76 




5055 


•• 




2' 4 ' 6 'S ip 12 14 16' 18 20 J2 2^4 26 28' 30 32 34 3(5 38 40 ' 42 '4!l 46 48 50 52 54 56 58 60 62' 64 66 68 70 72 74 76' 78 ' 80 82 84 86 88' SO 92 | 


40 58 


■■ 


2 4 


6 


8101214 


161820 22 24 


26 28 30 32 34 


36 38 4042 44'46 485052 54|56 5860 6264166 68 70 72 74 


7678 8082 84|8688 90 92 94'96 9|8 ° S = [§ § g S 


^% 




















r 1 ' 




' r 1 ' 




























' i' 1 1 ' 







100. to 300 Ft. Head 
From the line for gallons per minute or hour trace to the right to the line for the head and down to the line for the efficiency of 

the pump where read brake horse-power. 

Fig. 40. — Po-wer and capacity of water pumps. 



charge end may be left free without cessation of flow, when the liquid 
to be siphoned is non-volatile and free from absorbed gas. The 
supply end may even be smaller than the discharge end when the 
regulating valve is used. 

When the supply end of a siphon is only a little lower than the 
highest point of the bend, and when, also, the level of discharge is 



very much lower than the level of supply, if the discharge end be 
left free, and the cahber of the tube be of moderate size, the flow 
through it will be so free, and the supply will be so copious in pro- 
portion to the discharge, that, if the water flowing through the pipe 
be pure, the siphon may act satisfactorily for a long period. If, on 
the contrary, the difference of level of the supply and discharge be 



250 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



quite small, and the caliber of the tube be large, the longer leg, if the 
discharge end be left free and unregulated by a valve, will be very- 
apt to run out and render the siphon inoperative. It is scarcely 
necessary to add that any obstruction to inflow, such as the accumu- 
lation of silt in the supply end, or in any depressed portion of the 
tube, such as would form a trap wherein deposits of obstructive 
substances may collect, would produce a similar result. In such cases, 
a siphon, when refilled, may act for a time; but it slowly becomes 
inoperative provided the discharge end is not submerged, because 
the longer leg more or less gradually empties itself. This may be 
guarded against by a strainer placed over the supply end, but in 
many cases, the floating impurities in the liquid to be siphoned are 
so fine, that any strainer sufficiently fine to intercept them might 
itself become a sufficient obstruction to render the siphon inoperative. 
Siphons which are to operate in water containing free or unab- 
sorbed gas, should have a supply tank of sufficient size to permit 
the gas to rise and escape at the surface and the pipe should draw 
from the bottom of the tank. Dissolved air, which is always present 
in water, has a tendency to separate under the reduced pressure 
in the bend. Under active action, it is conimonly carried along 
with the water and does no harm but, if the action be stopped by 



closing one end only, the air will collect at the bend and eventually 
make the siphon inoperative. On the other hand, if both ends be 
stopped, the separation of the air is prevented. Hence it is clear 
that siphons which are to be used intermittently should be provided 
with means for stopping both ends when the action is suspended. 
The discharge of a siphon may be calculated from the formula: 



V = \/2g{h'-h) 

in which F= velocity, ft. per sec, 

A = difference of level between bend and surface of the 

supply tank, ft., 
h' = difference of level between bend and surface of the 

receiving tank, ft., 
g = acceleration of gravity = 32.2. 

The performance of hydraulic rams is subject to conditions which 
are usually unknown; for example, the adjustment of the delivery 
valve. Under these circumstances no universal figures for the per- 
formance can be given, but Table 23 {Amer. Engr., March 18, 1886) 
shows what may reasonably be expected. Should the delivery pipe 
be long, the friction head should be added to the static head. 



r 



PIPE AND PIPE JOINTS 



Table i. — Dimensions or Commercial Drawn Pipe 
From the National Tube Co.'s Book of Standards 



Standard 



Extra Strong 



The permissible variation in weight is 5 per cent, above and s per cent, below. 
Taper of threads is J in. diameter per ft. length for all sizes. 





Diameters, 
ins. 


a 


Weight per 
foot, lbs. 


J3 
u 


Couplings 


Size, 

ins. 


"3 

s 

1 


"a 
g 

a 


a 

IS 


a 

c 
•3 


Threads 

and 
couplings 


w 
H 


S3 

B.B 
a 

a 






i 


.405 


.269 


.068- 


.244 


.245 


27 


.562 


i 


.029 


i 


.540 


.364 


.088 


.424 


.42s 


18 


.685 


I 


.043 


i 


.675 


.493 


.091 


.567 


.568 


18 


.848 


li 


.070 


i 


.840 


.622 


. 109 


.850 


.852 


14 


1.024 


If 


.116 


i 


1. 050 


.824 


.113 


1. 130 


1. 134 


14 


1. 281 


If 


.209 


I 


I.3IS 


1.049 


.133 


1.678 


1.684 


Hi 


1.576 


li 


.343 


ri 


1.660 


1.380 


.140 


2.272 


2.281 


Hi 


1.950 


2i 


.535 


li 


1.900 


1. 610 


.145 


2.717 


2.731 


Hi 


2.218 


2| 


.743 


2 


2.375 


2.067 


.154 


3.652 


3.678 


Hi 


2.760 


2f 


1.208 


2^ 


2.875 


2.469 


.203 


S.793 


5. 819 


8 


3.276 


2f 


1.720 


3 


3-500 


3.068 


.216 


7.575 


7.616 


8 


3-948 


3i 


2.498 


3^ 


4.000 


3.548 


.226 


9.109 


9.202 


8 


4-S9I 


3l 


4.241 


4 


4-500 


4.026 


.237 


10.790 


10.889 


8 


5-091 


3i 


4.741 


4i 


5.000 


4-506 


.247 


12.538 


12.642 


8 


5-591 


3i 


5.241 


S 


5-563 


5 -047 


.258 


14.617 


14.810 


8 


6.296 


4i 


8.091 


6 


6.625 


6.06s 


.280 


18.974 


19.185 


8 


7.358 


4J 


9.554 


7 


7.62s 


7.023 


.301 


23.544 


23.769 


8 


8.358 


4i 


10.932 


8 


8.625 


8.071 


.277 


24.696 


25.000 


8 


9.358 


4f 


13. 90s 


8 


8,625 


7.981 


.322 


28.554 


28.809 


8 


9.358 


4f 


13.905 


9 


9.625 


8.941 


.342 


33.907 


34.188 


8 


10.358 


5i 


17.236 


10 


10.750 


10. 192 


.279 


31.201 


32.000 


8 


II. 721 


6J 


29.877 


10 


10.750 


10. 136 


.307 


34.240 


35.000 


8 


II. 721 


6i 


29.877 


10 


10.750 


10.020 


.365 


40.483 


41-132 


8 


II. 721 


6i 


29.877 


II 


11.750 


11.000 


.375 


4S.SS7 


46.247 


8 


12.721 


6J 


32.550 


12 


12.750 


12.090 


.330 


43.773 


45.000 


8 


13.958 


6i 


43.098 


12 


12.750 


12.000 


• 375 


49.562 


50.706 


8 


13.958 


6i 


43 . 098 


13 


14.000 


13-250 


.375 


54.568 


55.824 


8 


15.208 


6J 


47.152 


14 


15.000 


14-250 


.375 


58.573 


60.375 


8 


16.446 


6J 


59.493 


IS 


16.000 


15-250 


.375 


62.379 


64.500 


8 


17.446 


6i 


63.294 



Bursting and Collapsing Strength of Pipe 

Extended experiments on the bursting strength of pipe for the 
National Tube Co. by Prof. R. T. Stewart (Trans. A. S. M. E., 
Vol. 34) led the professor to advise the use of Barlow's formula for 
all ordinary calculations, with the proviso that the fiber stress be 
obtained as explained below. Barlow's formula is as follows: 

in which 5 = fiber stress, lbs. per sq. in., 

P = internal pressure, lbs. per sq. in., 
.R = inside radius of pipe, ins., 
r = thickness of pipe wall, ins. 



The permissible variation in weight is s per cent, above and s per cent, 
below. 



Size, 


Diameters, ins. 


Thickness, 
ins. 


Weight per 

ft. plain ends, 

lbs. 


ins. 


External 


Internal 


i 


.40s 


.215 


.095 


.314 


i 


.540 


.302 


.119 


.535 


1 


.675 


.423 


.126 


.738 


i 


.840 


.546 


.147 


1.087 


1 


1.050 


.742 


.154 


1.473 


I 


1.31S 


.957 


.179 


2. 171 


li 


1.660 


1.278 


.191 


2.996 


li 


1.900 


1.500 


.200 


3.631 


2 


2.375 


1.939 


.218 


5.022 


2i 


2.875 


2.323 


.276 


7.661 


3 


3-500 


2.900 


.300 


10.252 


3i 


4-000 


3.364 


.318 


12.505 


4 


4-500 


3.826 


.337 


14-983 


4i 


5-000 


4.290 


.355 


I7-61I 


5 


5.563 


4.813 


.375 


20-778 


6 


6.62s 


5. 761 


.432 


28.573 


7 


7.62s 


6.62s 


.500 


38.048 


8 


8.625 


7.62s 


.500 


43.388 


9 


9.62s 


8.625 


.500 


48.728 


10 


10.750 


9-. 750 


.500 


54-735 


II 


II. 750 


10.750 


.500 


60.07s 


12 


12.750. 


11-750 


.500 


65-415 


13 


14.000 


13.000 


.500 


72.091 


14 


15.000 


14.000 


.500 


77.431 


IS 


16. 000 


15 .000 


.500 


82.771 



Double Extra Strong 
The permissible variation in weight is lO per cent, above and lo per cent, 
below. 



Size, 


Diameters, ins. 


Thickness, 


Weight per 
ft. plain ends, 








ms. 


External 


Internal 


ins. 


lbs. 


i 


.840 


.252 


.294 


I. 714 


1 


1.050 


.434 


.308 


2.440 


I 


1. 315 


.599 


;3S8 


3.659 


li 


1 .660 


.896 


.382 


S.214 


li 


1.900 


1. 100 


.400 


6.408 


2 


2.375 


1.503 


.436 


9.029 


2i 


2.875 


I. 771 


.552 


13 -69s 


3 


3-500 


2.300 


.600 


18-583 


ii 


4.000 


2.728 


.636 


22-850 


4 


4-500 


3.152 


.674 


27-541 


4i 


5 -poo 


3-580 


.710 


32-530 


S 


5-563 


4-063 


.7S0 


38.552 


6 


6.625 


4.897 


.864 


53.160 


7 


7.62s 


5.875 


.875 


63.079 


8 


8.62s 


6.875 


.875 


72.424 



251 



Professor Stewart's recommendation regarding the fiber stress is 
that it be determined from the formulas: 

{Continued on page 2S4,first column) 



252 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Dimensions and Weight, Lbs. per Foot, of Large Sizes of 

Hydraulic Pipe 

From the National Tube Company 

The permissible variation in weight is s per cent, above and s per cent, 
below 



Dimensions and Weight of Lead Pipe 



Size ins. 


Thickness ins. 


=4 


H 


% 


I 


I. D. 


O. D. 


Weight 


Weight 


Weight 


Weight 


9 


9H 


60.08 


71.09 


81.77 


92. 12 


10 


loH 


67 -59 


80 . 10 


92.28 


104.13 


II 


11% 


74.26 


88.11 


lOI .63 


114 .81 


12 


12V4. 


80.94 


96.12 


110.97 


125. 49 



Dimensions and Weight of Standard Welded Square Pipe 
From the National Tube Company 

The permissible variation in weight is s per cent, above and 5 per cent, below 



Size, 


ins. 


Thickness, 
ins. 


Weight per foot 


External 


Internal 


plain ends, lbs. 


H 


.607 


.134 


1 .46 


I 


.800 


.100 


1.25 


I 


.750 


.125 


1. 55 


I 


.624 


.188 


2 .11 


iH 


1 .000 


.125 


1.97 


iH 


.982 


.134 


2.0s 


iM 


.938 


.I56_ 


2 .29 


lU 


.874 


.188 


2 .48 


iM 


.750 


.250 


3.28 


iH 


1 .250 


.125 


2.33 


iH 


1 .220 


.140 


2.55 


iH 


1. 188 


.156 


2.78 


iH 


1 .124 


.188 


3.05 


iH 


1 .000 


.250 


4.00 


iiMe 


1.407 


.140 


2.76 


iiMe 


I.37S 


.156 


3 .00 


I'Me 


1. 311 


.188 


3.75 


iiHs 


1. 187 


.250 


4.60 


2 


I.7S0 


.125 


3.10 


2 


1.732 


.134 


3.18 


2 


I .710 


■ 14s 


3-52 


2 


1 .624 


• .188 


4-39 


2 


I .500 


.250 


5.40 


2H 


2 . 124 


.188 


5 60 


3 


2 .600 


.200 


7 .06 



Dimensions and Weight of Seamless Brass and Copper Pipe 

OF Iron Pipe Sizes 

From the Bridgeport Brass Company 

Brass and copper pipe may be obtained in an almost endless variety of 
diameters and thicknesses. Pipe of the following sizes has the same system of 
threading as iron pipe. 



Size 




Btandar 


d iron pipe sizes 


Extra heavy iron pipe sizes 


same as 
iron pipe. 


O.D. 


I.D. 


Thick- 
ness 


Weight per 
ft., lbs. 


Thick- 
ness 


Weight per ft., 
lbs. 


ms. 


Brass 


Copper 


Brass 


Copper 


H 


.405 


.281 


.0620 


.246 


.2573 


. 100 


.35178 


.36902 


H 


.540 


.375 


.0825 


.437 


.4584 


.123 


.59168 


.62068 


H 


.675 


■494 


.0905 


.612 


.6394 


.127 


.80288 


.84224 


}ri 


.840 


.62s 


.1075 


.911 


.9514 


.149 


I .1867 


1.2449 


H 


1 .050 


.822 


.1140 


1.24 


1 .291 


.157 


I. 6173 


I .6966 


I 


1. 315 


I .062 


.1265 


1.74 


1. 818 


.182 


2.3783 


2.4949 


iH 


1 .660 


1.368 


. 1460 


2.56 


2.673. 


.194 


3.2798 


3 • 4406 


iH 


1 .900 


I .600 


.1500 


3.04 


3.178 


.203 


3.9694 


4 . 1 640 


2 


2.375 


2 .062 


■ I 565 


4 .02 


4.203 


.221 


5.4874 


5.7564 


2^4 


2.875 


2.500 


.1875 


5.83 


6.098 


.280 


8.7937 


8.7937 


3 


3.500 


3 062 


.2190 


8.32 


8.694 


.304 


II .167 


II. 715 


3H 


4.000 


3.500 


.2500 


10.85 


11.35 


.321 


13.624 


14.292 


4 


4.500 


4.000 


.2500 


12.30 


12.68 


.421 


16.359 


17 161 


aH 


5 .000 


4.500 


.2500 


13.74 


14-37 








5 


5.563 


5 062 


.2505 


15.40 


16. II 








6 


6.625 


6.125 


.2500 


18.45 


19.29 








7 


7.62s 


7 .062 


.281S 


23.92 


25 .02 









Caliber, 


Outside 

diam., 

ins. 


Weight 
per ft. 


Caliber, 
ins. 


Outside 

diam., 

ins. 


Weight 
per ft. 




Lbs. 


Oz. 


Lbs. Oz. 


H 


.48 




7 


I 


1 .20 


I 


8 




.53 




10 




1.23 


2 






.56 




12 




1.27 


2 


8 




.60 


I 






1. 35 


3 


4 




.66 


I 


4 




1.42 


4 






.72 


I 


8 




1.48 


4 


12 




.74 


I 


12 




1. 59 


6 






.76 


2 
























.80 


2 


8 


iH 


1.44 


2 


8 
















H 


.64 




9- 




I. S3 


3 






.68 




12 




1.59 


3 


12 




.72 


I 






1.67 


4 


12 




.76 


I 


4 




1-74 


5 


12 




.80 


I 


8 




1.83 


6 


12 




.84 


I 


12 






















.88 


2 




l'/2 


1.74 


3 






.96 


2 


8 




1.78 


3 


8 




1 .04 


3 






1.83 
1.87 


4 
5 


4 












5« 


.77 




12 




1.98 


6 


8 




.81 


I 






2.04 


7 


8 




.88 


I 


8 




2.07 


8 






.96 


2 






2 . 12 


8 


8 




.98 


2 


4 






















I .02 


2 


8 


iH 


2 .00 


4 






I .06 


2 


12 




2 .06 


5 






I .08 


3 






2. 12 


6 






I. 14 


3 


8 




2. IS 
2.18 


6 

7 


8 












H 


.88 


I 






2.31 


8 


8 




.94 
1 .00 
1.03 


I 
I 

2 


4 




2.50 


10 








2 


2.18 


3 






1.07 


2 


4 




2.24 


4 






1 .10 


2 


8 




2.27 


4 


12 




1. 14 


3 






2.31 


5 






1. 21 


3 


8 




2.34 


6 






1.27 


4 






2.38 


7 






1.34 


4 


4 




2.48 


8 
8 


















H 


1 .00 


I 






2.51 


9 






1 . 12 


I 


4 




2.71 


II 


12 



Dimensions and Weight of Block Tin Pipe 



Inside 


Outside 


Weight 


Inside 


Outside 


Weight 


diam., 


diam.. 


per ft.. 


diam.. 


diam., 


per ft.. 


ms. 


ms. 


oz. 


ms. 


ms. 


oz. 


He 


H 


M2 


Me 


Me Scant 


4 


H 


Vie 


% 




1%2 


6 


3/1 6 


5i6 


2H 




=/« 


8 
















Vi 


i%2 Scant 


AH. 








Vi 


H 


3 




% 


5 




i?fe 


4 




% Full 


SH 




Me 


5 




2 Hz 


6 




Me Full 


6 




2^2 Full 


7 




],i Scant 


7 




•He 


8 




Vi Full 


8 




2%2 


10 

12 


Me 


Ma Full 




4 










li Scant 


SVi 


H 


2^2 


8 




'%2 


7'/2 




Hi2 


9 




Me 


8 




'Me 


10 

14 


H 


^i 




4 










Vi Full 


4M2 


% 


% Full 


8 




»%2 


5 




2%2 


10 




Me Scant 


6 










Me Full 


7 










1%2 


8 










% Scant 


9 










% Full 


10 










2^2 


12 









PIPE AND PIPE JOINTS 



253 



Dimensions and Weight of Standard Large 0. D. Pipe 
From the National Tube Company 

The permissible variation in weight is S per cent, above and S per cent, below 



Size 
O.D. 



14 
IS 
16 
17 

18 
20 



24 
26 
28 
30 



Thickness and weight per ft., lbs. 



Vi 



36-713 
39-383 
42.053 
44-723 

47-393 



s/, 



45-082 
49.020 
52-357 
55-695 

59-032 
65-708 
6g.o^S 
72.383 



54-568 
58.573 
62.579 
66.584 

70-589 
78.599 
82 .604 
86.609 

94-619 
102 ,629 



Vie 



63.371 
68 . 044 
72.716 
77.389 

82.061 
91.407 
96.079 
100.752 

110.097 
119-442 
128.787 
138.132 



72 ,091 
77.431 
82.771 
88. Ill 

93.451 
104. 131 
109.471 
114. 811 

125.491 
136.172 
146.852 
157.532 



80.726 
86.734 
92.742 
98 . 749 

104-757 
116 .772 
122 .780 
128.787 

140.802 
152.818 
164.833 
176.848 



5i 



89.279 

95.954 

102 .629 
109.304 

llS-979 
129-330 
136-005 
142 .680 

156.030 
169.380 
182.730 
196.081 



iMs 



97-748 
105.091 
112.433 
119.776 

127. 118 
141 .804 
149.146 
156.489 

171. 174 
185.859 

200.545 
215.230 



106.134 
I 14 -144 
122. 154 
130.164 

138.174 
154.194 
162 .204 
170.215 

186.23s 
202.255 
218.275 
234.296 



% 



122.654 
132 .000 
141-345 
150.690 

160.035 

178.725 



I>S 



.842 
.522 
.202 
.882 



181 .562 
202 .923 



154 -69s 
166.710 
178.725 
190.740 

202 .756 
226.786 



Dimensions and Weight of Shelby Cold Drawn Seamless Steel Tubing 

Unless otherwise ordered, this pipe is supplied in mill lengths of 5 ft. and over 



Thick- 


Equiv- 
alent 

in 
deci- 
mal 

of 
inch 


Weight, lbs. per ft. 


ness 
B.W.G. 


Outside diameter, ins. 


& 
frac- 
tions 


Vi H 


M 


li 


I 


iH 


i!4 


i?i 


lii 


iH 


2 


2H 


2],i 


2% 


3 


3M 


3 1/2 


3H 


4 


4H 


Aii 


aH 


5 


sH 


5\i 


20 

I8 
i6 
14 
13 

12 
II 
10 

H2 


.035 

.049 
.065 
.083 
-095 

.109 
.120 
.134 
.156 
.188 

.218 
.250 
.312 
.375 
.500 

.62s 

.750 

-875 

1 .000 


.17 
-24 
• 30 

■37 

-41 

-45 
■49 


.22 
■30 

■ 39 

.48 

■54 

.60 
.65 
• 70 


-27 

• 37 
-47 
-59 
.66 

-75 
.81 
.88 
.99 
1. 13 


-31 
■ 43 
.56 

.70 
-79 

.89 

-97 

1 .06 

I .20 
1.38 

1-53 


-36 
.50 
.65 

.81 
.92 

1 .04 
1. 13 
1.24 
1. 41 
1.63 

1.82 
2 .00 


-41 
-56 
.74 
.92 
1 .04 

1. 18 
1.29 
1.42 
1. 61 
1.88 

2.12 
2.33 


-45 

-63 

.82 

1.03 

1. 17 

1-33 
1-45 
1 .60 
1.82 
2.13 

2.41 
2.67 
3-13 
3.50 


• 50 

.69 

.91 

1. 14 

1.30 

1.47 
1. 61 
1-77 
2.03 
2.38 

2.70 
3-00 

3-54 
4.00 


-55 
.76 

I .00 

1.25 
1.42 

1.62 

1.77 
1. 95 
2.24 
2.63 

2.99 
3-33 
3-96 
4-50 
5-33 


1. 17 
1.48 
1.68 

1. 91 
2 .09 
2.31 
2.66 
3.13 

3.57 
4.00 
4.79 
5.50 
6.67 


1.34 
1.70 
1.93 

2 .20 
2.41 
2,67 
3-07 
3-63 

4.16 
4-67 
5-63 
6.50 
8.00 

9-17 


1-52 
1 .92 
2.19 

2.49 
2.73 
3 03 

3-49 
4.13 

4-74 
5-33 
6.46 
7-50 
9-34 

10.84 


1 .69 
2.14 
2.44 

2.78 
3.0s 

3-39 
3-91 
4.63 

S-32 
6.00 
7-29 
8.50 
10.67 

12.50 


1.86 
2.36 
2.69 

3.07 
3-37 
3-74 
4-32 
S.13 

5-91 
6.67 
8.13 
9-51 
12 .00 

14.17 


2.95 

3.37 
3-69 
4.10 

4-74 
5. 63 

6-49 

7 33 

8.96 

10.50 

13.34 

15-84 
18.00 
19-84 
21-34 


320 

3.66 
4.01 
4-45 
5. 16 
6.13 

7-07 
8.00 
9.79 
11.50 
14.67 

17.50 
20.00 
22.17 
24.01 


3-45 

3^94 
4-33 
4.82 
5-57 
6.63 

7.66 

8.67 

10.63 

12.50 

16.00 

19-17 

22 ,00 
24-51 
26.67 


4-65 
S.I8 

5-99 

7-13 

8.24 
9-34 
II .46 
13-SO 
17-34 

20.84 
24.01 
26.84 
29-34 


4-97 
5-53 
6.41 
7.63 

8.82 
10.00 
12 .29 
14.50 
18.67 

22.50 
26.00 
29-17 
32,01 


5-89 
6.82 
8.13 

9-41 
10.67 
13-13 
15-50 
20,00 

24.17 
28.01 
31-51 
3467 


6.25 

7.24 
8.63 

9.99 

11.34 
13-96 
16.50 
21.34 

25-84 
30.01 
33-84 
37 -34 


6,61 
7-66 
9-I3 

10.58 
12,00 
14-79 
17.51 
22 .67 

27-51 
32.01 
36.18 
40.01 


6.96 
8.07 
9.63 

II . 16 

12 .67 
15-63 
18.51 
24,00 

29.17 

34-01 
38.51 
42.68 


7.31 

8.49 

10.13 

11.74 
13.34 
16.46 
19.51 
25.34 

30.84 
36,01 
40.84 
45 .34 


7.67 
8.91 
10.63 

12 .32 


H 








14.00 


Me 










17 -30 


H 














20.51 


H 














26,67 


H 


















32 .51 


H 






















38.01 


H 






























43-17 


I 






























48,01 











































Dimensions and Weight of Standard Welded Rectangular 

Pipe 
From the National Tube Company 

The permissible variation in weight is 5 per cent, above and 5 per cent, below 



Size, 


ins. 


Thickness, 
ins. 


Weight per foot 


External 


Internal 


plain ends, lbs. 


iHXi 


.970X .720 


.140 


1 .67 


iKXi 


.874X .624 


.188 


2.05 


iHXiK 


1 .256X1.006 


.122 


2.05 


iK2Xi!4 


1.210X .960 


.145 


2.24 


iJ^XiH 


1.188X ,938 


.156 


2.40 


iJ^XiH 


1.124X ,874 


.188 


2.8s 


iHXiH 


i.oooX .750 


.250 


3.67 


2 XiU 


i,732X .982 


.134 


2.53 


2 XiH 


1 .710X1 ,210 


.145 


3 00 


2 Xi^ 


1 ,624X1 ,124 


.188 


3-61 


2 XiH 


1 .500X 1 ,000 


.250 


4-65 


2I/2XIH 


2 .210X 1 ,210 


.145 


3-52 


2VzXiii 


2,124X1.124 


.188 


4-39 


zHXiH 


2 , 000 X 1 . 000 


.250 


5-40 


3 X2 


2 ,624X1 .624 


.188 


5 -60 


3 X2 


2 , 605 X 1 . 600 


.200 


6.00 



254 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 2. — British Standard Pipe Threads 

Threads of standard Whitworth form and straight. The joint is 
made on the incomplete taper threads made by the die mouth. 



Nominal 
inside 

diameter 
pipe, 
ins. 


Approximate 

outside 

diameter 

pipe, 

ins. 


Gage diam- 
eter top of 
thread, 
ins. 


Depth of 
thread 


Core 

diameter 


Number 

of 
threads 
per inch 


i 


a 


.383 


.0230 


■337 


28 


i 


a 


.518 


• 0335 


.451 


19 


f 


a 


.6s6 


• 0335 


.589 


19 


J 


u 


.825 


.0455 


.734 


14 


i 


a 


.902 


• 0455 


.811 




1 


lA 


1 .041 


• 0455 


• 950 




l 


lA 


1. 189 


• 0455 


1 .098 




I 


iH 


1.309 


.0580 


1 .193 




li 


Hi 


1.650 


.0580 


1.534 




li 


Iff 


1.882 


.0580 


1.766 




li 


2A 


2 . 116 


.0580 


2.000 




2 


2| 


2.347 


.0580 


2.23 




2i 


2| 


2.S87 


.0580 


2.471 




2^ 


3 


2.960 


.0580 


2,844 




2j 


3i 


3.210 


.0580 


3.094 




3 


3i 


3.460 


.0580 


3.344 




3i 


3l 


3.700 


.0580 


3.584 




3i 


4 


3.9S0 


.0580 


3.834 




3i 


4i 


4.200 


.0580 


4.084 




4 


4i 


4-450 


.0580 


4.334 




4i 


S 


4.9S0 


.0580 


4 834 




5 


Si 


S.4S0 


.0580 


5.334 




Si 


6 


S.9S0 


.0580 


5^834 




6 


6J 


6.450 


.0580 


6.334 


II 


7 


n 


7-450 


.0640 


7.322 


10 


8 


H- 


8.450 


.0640 


8.322 


10 


9 


9i 


9-450 


.0640 


9.322 


10 


10 


lOi 


10.450 


.0640 


10.322 


10 


II 


III 


11.450 


.0800 


II .290 


8 


12 


I2| 


12.450 


.0800 


12. 290 


8 


13 


I3i 


13.680 


.0800 


13-520 


8 


14 


I4f 


14.680 


.0800 


14-520 


8 


IS 


isl 


15.680 


.0800 


15-520 


8 


16 


i6| 


16.680 


.0800 


16. 520 


8 


17 


I7l 


17.680 


.0800 


17-520 


8 


18 


l8| 


18.680 


.0800 


18,520 


8 



40,000 

S = for butt- welded steel pipe 



50,000 

n 
60,000 

n 
28,000 

n 



for lap-welded steel pipe 
for seamless steel tubes 
for wrought-iron pipe 



in which 5 = working or safe fiber stress, lbs. per sq. in., 
n = factor of safety based on ultimate strength. 

Some of the results of Professor Stewart's tests are given in Table 5. 

The strength of tubes against collapsing pressure formed the subject 
of exhaustive tests by Prop. R. T. Stewart {Trans. A. S. M. E., 
Vol. 27). Over 500 tubes, provided by the National Tube Co., were 
tested, the diameters ranging between 3 and 10 ins., outside, and of 
all commercial thicknesses obtainable, the material being Bessemer 
steel, lap welded. The first result of the tests was to show that the 
collapsing pressure decreases as the length of the tube increases up 
to a length equal to about 6 diameters, beyond which there is no 
further material decrease in the collapsing pressure with increase of 
length. Beyond that length the collapsing pressure is given by the 
formulas: 



Table 3. — Length of Commercial Drawn Pipe for i Sq. Ft. 
OF Surface 

From the National Tube Co 's Book of Standards 



Size 
ins. 



I 

li 
I5 

2 

2j 
3 
35 

4 
41 
5 
6 



13 

14 



H 



.405 
.540 
.675 
.840 

1.050 
1. 315 
1.660 
1.900 

2.375 
2.875 
3-500 
4.000 

4-500 
S.ooo 
5.563 
6.625 

7.625 
8.625 
8.62s 
9.625 

10.750 
10.750 
10.750 
11.750 

12.750 
12.750 
14.000 
15.000 



Standard weight 
pipe 



16.000 .375 



,068 
.088 
.091 
.109 

.113 

• 133 

. 140 
-145 

• 154 
.203 
.216 
.226 

.237 

• 247 
.258 
.280 

.301 

• 277 
.322 

• 342 

.279 
.307 
.365 
.375 

.330 
-375 
-375 
-375 



Length of 

pipe in ft. 

per sq. ft. of 



W 



9.431 
7-073 
S.658 
4-547 

3-637 
2 .904 
2.301 
2.010 

1.608 

1.328 

1. 091 

.954 



.763 
.686 
.576 

.500 
.442 
.422 
.396 

.355 
.355 

• 355 
.325 

• 299 

• 299 

• 272 

• 254 

.238 



Extra strong pipe 



Length of 

pipe in ft. 

per sq. ft. of 



W 



14.199 
10.493 

7.747 
6. 141 

4-635 
3.641 
2.767 
2.372 

1.847 
1.547 
1.24s 
1.076 

.948 
.847 
.756 
.629 

.543 
.473 
.478 
.427 

.374 
.376 
.381 
■ 347 

.31S 
.318 
.288 
.268 



.095 
• 119 
. 126 

.147 

.154 
.179 
.191 
-200 

.21 
.276 
.300 
.318 

.337 

-355 
-375 
-432 

-500 
-500 

.500 

.500 



500 



.500 
.500 



500 



9-431 
7.073 
5.658 
4.547 

3.637 
2.904 
2.301 
2.010 

1.608 

1.328 

1.091 

.954 



■ 763 
.686 

■ 576 

.500 

-442 



-396 



-355 



.325 



-254 



-238 



Double extra 
strong pipe 



17- 766 
12.648 
9.030 
6.995 

5-147 
3-991 
2-988 
2.546 

1.969 
1.644 
I-317 
I-I3S 

.998 
.890 
■ 793 
.663 

.576 
.500 



.442 



.355 



.293 
.272 



-294 

.308 
.358 
-3 
.400 

.436 
• 552 
.600 
.636 

.674 
.710 
-750 



.875 
-87s 



Length of 

pipe in ft. 

per sq. ft. of 



4-547 
3-637 

2 -904 
2,301 
2 ,010 

1,608 
1.328 
1 .091 

■ 954 



• 763 
.686 
.576 

.500 

• 442 



15-157 

8.801 
6.376 
4.263 
3.472 

2.541 
2.156 
1.660 
1.400 

1.211 

1.066 

■ 940 

.780 

.650 
.555 



^^=5 



0,210,000 I -5 I 



and P = 86, 670-5— 1386 



{a) 



in which P = collapsing pressure, lbs. per sq. in., 
(f = outside diameter, ins., 
i = thickness, ins. 

Formula (a) is to be used when the ratio -5 is less than .023 and formula 

(6) when this ratio exceeds that figure. 

The dimensions of Bessemer steel lap-welded tubes of a greater 
length than six diameters against collapsing pressure may also 
be determined from Fig. i, by Professor Stewart {Trans. 
A.S.M. E., Vol. 27). 

In order to condense the size of the chart, the curve is broken 
into two parts, XX and YY; YY being the upper portion of XX 
transferred to the left and then dropped down, the break in the curve 
corresponding to a collapsing pressure of 2080 lbs. and a thickness 
divided by diameter of .040. The scales for the portion XX are 
at the lower and right-hand margins, while those for the portion 
YY are at the upper and left-hand margins. 

The use of the chart is best shown by an example: Find the 
probable collapsing pressure of a tube having an external diameter 
equal to 6 ins. and a thickness of wall equal to .203 in. 



PIPE AND PIPE JOINTS 



255 



Table 4. — Test Pressure of Commercial Drawn Pipe 
From the National Tube Co.'s Book of Standards 

Standard Extra Strong 



Size, 
ins. 



Il 



J4 
IS 



Weight per 
foot com- 
plete, lbs. 



.24s 
.42s 
.568 
.852 

I -134 
1.684 
2.281 
2.731 

3.678 
5. 819 
7.616 
9.202 

10.889 

12 .642 
14.810 

1-9. i8s 

23.769 
23.000 
28.809 
34-188 

32.000 

as. 000 

41.132 
46.247 

45.000 
SO. 706 
SS. 824 

60.37s 
64.500 



Test pressure, 
lbs. 



Butt 



700 
700 
700 
700 

700 
700 
700 
700 

700 
800 
800 



Lap 



1000 
1000 

1000 
1000 
1000 
1000 

1000 
1000 
1000 
1000 

1000 
800 

icoo 
900 

600 
800 
900 
800 

600 
800 
700 

700 
600 



Size 

ins. 



It 



2i 

3 
3/ 

4 
4-1 
S 
6 



9 
10 



13 

14 



IS 



Weight per 
foot plain 
ends, lbs. 



■ 314 

.S3S 

.738 

1.087 

1.473 
2. 171 
2 .996 
3.631 

S.022 
7.661 

10. 2S2 
12. SOS 

14.983 
17. 611 
20.778 
28.573 

38.048 

43-388 
48.728 
S4-73S 

60.07s 
6S.41S 
72 .091 
77.431 

82.771 



Test pressure, 
lbs. 



Butt Lap 



700 
700 
700 
700 

700 

700 

1500 

1500 

1500 

ISOO 

1500 



2S00 

2000 
2000 
2000 

2000 
1800 
1800 
1800 

ISOO 
ISOO 
ISOO 
1200 

1 1 00 
1 1 00 
1000 
1000 



In addition to the above test, on 
sizes i in. to I in. inclusive, the 
pipe is jarred with a hammer 
while under pressure. 

Double Extra Strong 



Size, 
ins. 



li 

i^ 

2 

2^ 

3 

3'i 

4 

4^ 

S 

6 

7 



Weight per 
foot plain 
ends, lbs. 



1.714 

2.440 

3 659 

S.214 

6.408 

9.029 

13.695 

18.583 

22.850 

27-541 

32.530 

38-552 

53-160 

63.079 

72.424 



Test pressure, 
lbs. 



Butt 



700 
700 
700 
2200 
2200 
2200 
2200 



Lap 



3000 
3000 
3000 
3000 
2500 
2500 
2000 
2000 
2000 
2000 
2000 



Dividing the outside diameter by the thickness of wall we get 
i 
2 equal .0338. Since this value is less than .04 we look for it 

on the scale at the lower margin of the chart and then trace upward 
until the line XX is reached; ^then trace to the right and read from 
the scale of probable collapsing pressures 1540 lbs. per sq. in. This 
is the probable collapsing pressure for a length of 20 ft., but is also 
substantially correct for any length greater than about six diameters, 
or 3 ft. for a 6-in. tube, between transverse joints tending to hold 
the tube to a circular form. 

A second chart by Professor Stewart, Fig. 2, shows the relation 
of the probable collapsing pressure to the plain-end weight, while 
the preceding chart shows its relation to the thickness of wall. 
This chart should be used in calculations relating to collapsing pres- 



Table 5. — Bursting Test Pressures of Commercial Drawn 
Pipe. From the National Tube Co.'s Book of Standards 

The column marked "See note above" gives the number burst by failure of 
material not at weld. 

C — Clavarino conditions. 
B — Birnie conditions. 







w 


'ft 
"0 

^^ 


"3 . 
ES 

-t-> - 


<u 
n c^ 

'.S in' 
-p .— . 

tu'rt 

tl 
(U 

> 

< 


Bursting pressures, 
lbs. per sq. in. 


a 



■,5 
c 


w 

•0 
a 

W 


> 



a 
a; 

C 

(B 
OJ 


» i-i 
d) 

is'" 

II 

rt s, 3 

fc.&a 
> 

< 




Size 
ins 


3 
J 

'S 


a 

3 

B 


0) 
u 

> 

< 


Class of 
material 




\ i 


10 


-405 


.066 


11,840 


i7,32o[l4,266 


C 


I 


44,oiiJStandard pipe 




X 


10 


.540 


.085 


8,830 


14,680 


12,206 


C 


I 


38,645 


Standard pipe 




i 


10 


.675 


.088 


S.8S0 


13,030 


10,330 


c 


I 


39,272 


Standard pipe 




i 


10 


.840] . lOI 


11,380 


16,310 


14,038 


c 





58,163 


Standard pipe 


'O 


1 


10 


1.050 


.109 


7,150 


9,150 


8,020 


c 





38,657 


Standard pipe 


Ti 


I 


10 


1. 315 


.131 


4,500 


8,800 


6,990 


c 





35,085 


Standard pipe 


OJ 

^ 

-i 


li 


10 


1 .660 


■ 139 


4,400 


7,300 


5, 808 


c 





34,603 


Standard pipe 


li 


15 


1.660 


.140 


5. 500 


11,900 


7,700 


c 


I 


45,215 


Redrawn 


? ^ 


li 


10 


1.900 


• 143 


3.000 


6,100 


4,960 


c 





33,031 


Standard pipe 


^ 


2 


II 


2.37s 


.149 


3.830 


6,060 


4,951 


c 





40,485 


Standard pipe 


0) 


2| 


10 


2.875 


.198 


4.310 


5,740 


5,134 


c 





37,351 


Standard pipe 


ai 


3 


10 


3-500 


.204 


4.650 


6,370 


S,398 


c 





46,234 


Standard pipe 




li 


10 


1 .660 


.180 


7,910 


14,280 


10,514 


c 





48,922 


Extra strong 




2 


10 


2.375 


-213 


7,250 


8,940 


8,238 


c 





45,935 


Extra strong 




2 


10 


2.375 


.220 


6,160 


8,920 


7,661 


c 





41.347 


Extra strong 




2 


10 


2.375 


-445 


8,500 


18,314 


14,992 


c 





40,023 


XX strong 














General average 


41,686 






2 


10 


2.375 


.155 


4,890 


7,940 


6,645 


c 


I 


50,962 


Standard pipe 




2 


10 


2.375 


.182 


4,860 


10,060 


7,361 


c 





47,889 


Standard pipe 




3 


10 


3-500 


.210 


3,830 


8,200 


6,368 


c 


7 


53,560 


Standard pipe 




4 


10 


4.500 


.232 


4,810 


5, 680 


S,249 


c 


I 


51,462 


Standard pipe 


•T3 


S 


10 


S-S63 


.258 


3.410 


5,260 


4,538 


c 


I 


48,882 


Standard pipe 


0) 


6 


5 


6.625 


.275 


2,450 


5,210 


4,088 


c 





49,286 


Standard pipe 


6 


s 


6.625 


-275 


3,170 


4,760 


3,666 


B 





44,I06 


Standard pipe 


10 


5 


I0.7S0 


-349 


3,560 


4,730 


4.290 


c 


I 


66,080 


Standard pipe 


^ 


10 


s 


10.750 


-347 


2,770 


3,940 


3,396 


B 


2 


52,692 


Standard pipe 





2 


10 


2.375 


.218 


2,500 


9,870 


7,909 


c 





43,254 


Extra strong 


m 


2 


10 


2 .000 


. 108 


S,ioo 


6,560 


6,062 


c 


7 


SS,607 


Boiler tubes 




3 


10 


3.000 


. 112 


3,220 


4,860 


3,967 


c 


I 


S2,9S7 


Boiler tubes 




4 


s 


4.000 


.135 


3,640 


4,070 


3,840 


c 


2 


56,978 


Boiler tubes 




4 


5 


4.000 


.136 


3,720 


4,040 


3,914 


B 


I 


57,440 


Boiler tubes 














General average 


52.225 




B 


2 


10 


2.000 


.098 


5,420 


6,590 


6,052 


C 


10 


61,530 


Boiler tubes 




3 


10 


3.000 


. 112 


3,940 


4,730 


4,272 


c 


10 


57,075 


Bioler tubes 


^ OJ ' 


4 


6 


4. 000 


.134 


4,160 


4,440 


4,318 


c 


6 


64,450 


Boiler tubes 


(D 


4 


4 


4.000 


•134 


4,250 


4,440 


4,328 


B 


4 


64,488 


Boiler tubes 


m 












General averag 


e 


61,886 




-)j 


i\ 


10 


I .660 


.136 


2,880 


6,290 


5,283 


C 


3 


32,126 


Standard pipe 


5 (u 


li 


10 


1.660 


.136 


3,640 


S,68o 


4,891 


C 


I 


29,817 


Standard pipe 


.2 i 


2 


10 


2.375 


.156 


2,930 


4,250 


3,687 


C 


2 


28,051 


Standard pipe 


s 


li 


10 


I .660 


.188 


2,770 


7,330 


S.895 


C 


I 


26,678 


Extra strong 


« 












General averag 


e 


29.168 






2 


10 


2-375 


.152 


2,400 


3,940 3,213 c 


I 


25,122 


Standard pipe 


2 


10 


2-37S 


.207 


5,530 


J7,I20 6,349 C 


8 


36,461 


Extra strong 


^ 


L 










General averag 


e 


30,792 





sure when the plain-end weight is either given or required, while 
the preceding chart should be used when the thickness of wall is 
given or required. 

Example. — Find the probable collapsing pressure of a 6| 
(7 0. D.) in. casing whose plain-end weight is 17 lbs. per ft. 

Dividing the plain-end weight in lbs. per ft. by the square of the 
It) 
outside diameter in in., we get -r^' equal .347. Finding this value 

on the scale at the lower margin of Fig. 2 we trace vertically until 
the line XX is reached, then horizontally toward the right and read 
1525 lbs. per sq. in. as the probable collapsing pressure required. 

While this value is for a 20-ft. length of tube, as in the preceding 
chart, it may be used without substantial error for any length greater 
than about six diameters, or in this case 35 ft., between joints 
tending to hold the tube to a cireular form. 



256 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Scale for Y 
Thickness -=- Outside Diameter 



.040 








.050 








.060 








.070 








.08C 


suoo 


































/ 








































""/ 


f 








































/ 








































/ 












1500 






























/ 






































/ 








































( 


/ 








































/ 








































/ 


' 
















^*4000 
























/ 
















/ 






















/ 
















X 


/ 






















/ 
















/ 




S 




















/ 
















t 


/ 




v.% 


















1 


f 
















/ 






le for 
sure, L 


















/ 
















/ 






















/ 


















/ 








S m 
W 2 
















/ 
















/ 










1 














/ 
















/ 


/ 




















/ 


/ 
















/ 












Isooo 












/ 
















/ 


f 




















/ 


















/ 
























/ 
















/ 
























/ 


















/ 
























/ 
















/ 
























/ 
















1 


/ 


















2500 




Y/ 


/ 
















/ 
























/ 
















/ 
























/ 














y 


y 


























/ 














y 
































X 


-^ 


-' 










L- 























200O 



IBOO 



0) 3 

1000 tg 2 



500 



.010 



.030 



.020 
Scale for X 
Thickness-^ Outside Diameter 

Fig. I. — Plotted in terms of thickness. 



.040 



6000 



4500 



4000 



§3500 



a cs 



wph 



"3 3000 



2500 



Scale for Y 

Plain End Weight -f Diameter ' 
.50 .60 .70 



.80 



































/ 








































/ 








































Y 


/ 








































/ 










































V 








































/ 








































/ 


Y 








































/ 








































A 










































/ 


















2ni¥i 






















/ 


















/ 
























/ 


















/ 






















/ 


















/ 
























/ 
















^/ 


/ 






















/ 






- 












/ 






















1 


/ 
















/ 








1500 
















/ 


















/ 






















/ 


t 
















/ 
























/ 


















/ 






















/ 


















/ 
























/ 
















J 


f 












1000 










/ 


















/ 
























/ 
















/ 
























/ 


















/ 






















/ 


/ 
















/ 
























/ 
















/ 


/ 






















/ 


V 














/ 


/ 
























/ 














y 


/' 
























/ 












^ 


y 




























/ 








-- 



































^ Si 
o . 



.10 



.30 



.20 

Scale for X 
Plain End Weigh t-p Diameter ' 

Fig. 2. — Plotted in terms of weight. 



Figs, i and 2.— The strength of Bessemer steel tubes against collapsing pressure. 



Professor Stewart's paper contains the following observations: 
The apparent fiber stress under which the different tubes failed varied 
from about 7000 lbs. for the relatively thinnest to 35,000 lbs. per sq. 
in. for the relatively thickest walls. Since the average yield point 
of the material was 37,000 and the tensile strength 58,000 lbs. per sq. 
in., it would appear that the strength of a tube subjected to a fluid 
collapsing pressure is not dependent alone upon either the elastic 
limit or ultimate strength of the material constituting it. 

The experiments show that the element of greatest weakness in 
a cpmmercial lap-welded tube is its departure from roundness, even 
when this departure is comparatively small, as was the case with 
the tubes tested. The thinnest portion of wall, while in itself an 
element of weakness, is wholly subordinate to out-of-roundness in 
its influence upon the collapsing strength of commercial lap-welded 
tubes. 

The weld is not an element of weakness for tubes subjected to 
external fluid pressure. 

The factor of safety, he concludes, should be determined from the 
following considerations: 

For the most favorable practical conditions, namely, when the 
tube is subjected only to stress due to fluid pressure and only the 
most trivial loss could result from its failure, a factor of safety of 
three would appear sufficient. 

When only a moderate amount of loss could result from failure 
use a factor of four. 

When considerable damage to property and loss of life might re- 
sult from a failure of the tube, then use a factor of safety of six. 

When the conditions of service are such as to cause the tube to 



become less capable of resisting collapsing pressure, such as the thin- 
ning of wall due to corrosion, the weakening of the material due to 
over-heating, the creating of internal stress in the wall of the 
tube due to unequal heating, vibration, etc., the above factors of 
safety should be increased in proportion to the severity of these 
actions. 

Additional experiments on the strength of hibes against collapsing 
pressure were made by Profs. A. P. Carman and M. L. Carr 
{University of Illinois Bulletin, 1906). These experiments relate 
more especially to drawn brass and to cold-drawn seamless steel 
tubes. They confirm Professor Stewart's conclusion that the in- 
fluence of the length on the strength ceases at about 6 diameters. 

The dimensions of drawn brass and of cold-drawn seamless steel 
tubes of a greater length than 6 diameters against collapsing pres- 
sure may be determined from Figs. 3 and 4, which are from the 
published record of these experiments. The charts are to be used 
in the same manner as Professor Stewart's charts for lap-welded 
Bessemer steel tubes. 

The bursting strength of cast-iron elbows and tees formed the subject 
of a series of tests at the Case School of Applied Science by S. M. 
Chandler (Amer.Mach., March 8, 1906) and the results are given in 
Table 6. Three samples of each size were tested and the individual 
and average results are given. 

Equation of Pipes 

For the equation of pipes, that is, finding the number of small pipes 
having the same frictional resistance as one large one, the most 



PIPE AND PIPE JOINTS 



257 



4500 -- 










~" 














































































































4000 




































































/ 
























/ 
























' 




3500 - 






































- 
























/ 








"T 
















/ 
































3000 














/ 






































~^ 








/' 














































/ 


/ 






















/ 
























/ 






















/ 
























/ 








































2000 






/' 






















J 
























T 
























/ 
























--i — 
























/ 
























/ 














































1\ 
























Z-. 
























/ 
























r 
























y 
























7 
























7 






















600 


y' 
























y'^ 
























^"^ : 






















^^^ 
























'-" 










_. 


._ 


_ 








__ 



.01 



.02 .03 .04 .05 

Thickness -^ Outside Diameter 



.06 



.07 



Fig. 3. — ^The strength of drawn brass tubes against collapsing pressure. 



4500 ~ 










: 7 




t 




~i 


4000 


t 








: 1 




t 




-I 




1 




r 


1 


: 7 




T 




1 


3000 


t 




/ 




t 




u 




I 


2500 


/ 




t 




J 




t 




l 


2000 


r — 




z 




I 




r 




~ z 




I 




^ 


/. 




j>. . 












7 




7 




y 




/ 




500 '^ 





















.01 



.06 



PI 



.02 .03 .04 .05 

Thickness ~ Outside Diameter 

Fig. 4. — The strength of cold drawn steel tubes against collapsing 
pressure. 



Table 6. — Bursting Strength in Lbs. per Sq. In. of Standard 
Screwed Gray Iron Elbows and Tees 



Size 


Elbows 


Aver- 
age 


Size 


Tees 


Aver- 
age 


2i 


3500 


3200 


3400 


3400 


li 


3400 


3300 


3300 


3333 


3 


2400 


2600 


2100 


2500 


li 


3400 


3200 


2800 


3300 


3i 


2100 


1700 


2400 


2250 


2 


2S00 


2800 


2500 


2600 


4 


2800 


2500 


2500 


2600 


2k 


2400 


2100 


2500 


2450 


4i 


2000 


2600 


2600 


2600 


3 


1400 


1900 


1800 


1850 


5 


2600 


2500 


2500 


2533 


Zh 


1200 


1500 


1800 


1650 


6 


2600 


2200 


2300 


2367 


4 


1800 


2100 


1700 


1867 


7 


1800 


2100 


1900 


I9SO 


4J 


1100 


1400 


1400 


1400 


8 


1700 


1600 


1700 


1667 


S 


1700 


1300 


ISOO 


1600 


9 


1800 


1800 


1900 


1833 


6 


1400 


1500 


IIOO 


1450 


10 


1800 


1700 


1600 


1700 


7 


1400 


1400 


ISOO 


1433 


12 


IIOO 


1200 


900 


IISO 


8 


1200 


1400 


1390 


1350 












9 


1300 


1400 


1200 


1300 












10 


IIOO 


1300 


1200 


1200 












1 12 


IIOO 


1000 


IIOO 


1067 



accurate formula, according to Prof. G. F. Gebhardt {Power, 
June, 1907) is: 

diWd + 2,.(> 
in which d = diameter of larger pipe, . 
<ii = diameter of smaller pipe, 
n = number of small pipes equivalent to one large one. 

Table 7 has been calculated from this formula. The table gives 
the equation of standard drawn pipes, and of pipes of which the 
17 



nominal and actual diameters are the same. Instructions for use 
are given above the table. 

The equation of extra strong and double extra strong pipes is given 
in Tables 8 and g by H. D. Nitchie, an engineer of the Watson Still- 
man Company {Power, Aug. 3, 1909). Instructions for use are given 
above the tables. 

Cast-iron, Riveted and Copper Pipe 

The thickness of cast-iron pipe, in the smaller diameters, is de- 
termined chiefly by the foundry consideration of the least thick- 
ness which it is desirable to cast. A critical examination of existing 
formulas and prevailing practice was made by P. H. Baerman in a 
paper read before the Engineers' Club of Philadelphia in 1882. The 
resulting formula for the least thickness was: 

Thickness, ins. = .3 in.+.oisXdiameter, ins. 

For water pipe this gives an excess of strength for heads up to 
300 ft. and diameters up to 10 ins. For cases beyond those condi- 
tions Mr. Baerman gives the formula: 

Thickness, ins. = .0001 5 X head, ft. X diameter, ins. 

The ultimate strength of cast-iron is taken at 18,000 lbs. per sq in. 
and the factor of safety at 125. For any given case the thickness 
should be calculated from both formulas and the greatest resulting 
thickness be used. 

The dimensions of the American Water Works Association standard 
cast-iron pipe are given in Tables 10 and 11, of the Abendroth and 
Root spiral rivited pipe in Table 12, of Abendroth and Root flanged 
fittings in Table 17 and of the Pelton Water Wheel Co.'s riveted 
hydraulic pipe in Table 21. 

The thickness of copper pipe to withstand internal pressure accord- 



258 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 7. — The Equation or Equalization of Pipes 

Follow the line for one size of pipe to the column for the other; the figures at the intersection give the number of small pipes equiva- 
lent to one large one. Use the upper right-hand portion of the table for standard drawn pipe and the lower left-hand portion for pipe of 
which the nominal and actual diameters are the same. 

Standard Drawn Pipes 



Dia. 1 


i 1 


i 


I 


iM 2 1 2M 3 


4 1 


5 


6 7 1 8 


9 10 1 II 


12 1 13 


14 15 1 16 


17 


Dia. 


i 




2.27 


4.88 


IS. 8 


31.7 


52.9 


96.9 


20s 


377 


620 


918 


1.292 


1,767 


2,488 


3,014 


3.786 


4.904 


5.927 


7.321 8,535 


9.717 


i 


i 


2.60 




2.0s 


6.97 


14.0 


23.3 


42. S 


90.4 


166 


273 


40s 


569 


779 


1,096 


1,328 


1,668 


2,161 


2,615 


3,226 3,761 


4.282 


i 


I 


7.5s 


2.90 




3.45 


6.82 


II. 4 


20.9 


44.1 


81. 1 


133 


198 


278 


380 


536 


649 


815 


1.070 


1.263 


1,576 1,837 


2,092 


I 


li 


24.2 


9.30 


3.20 




1.26 


3.34 


6.13 


13.0 


23.8 


39.2 


58.1 


81.7 


112 


157 


190 


239 


310 


375 


463 


539 


614 


li 


2 


54.8 


21.0 


7.2s 


2.26 




1.67 


3.06 


6.47 


II. 9 


19.6 


29.0 


40.8 


55.8 


78.5 


95.1 


119 


ISS 


187 


231 


269 


307 


2 


2j 


102 


39.4 


13.6 


4-23 


1.87 




1.83 


3.87 


7.12 


II. 7 


17.4 


24.4 


33.4 


47.0 


S6.9 


71. S 


92.6 


112 


138 


161 


184 


2i 


3 


170 


65-4 


22.6 


7.03 


3. II 


1.66 




2.12 


3.89 


6.39 


9.48 


13.3 


20.9 


23.7 


31.2 


39.1 


50.6 


61. 1 


75. S 


88.0 


100 


3 


4 


376 


144 


49.8 


IS.S 


6.87 


3.67 


2.21 




1.84 


3.02 


4.48 


6.30 


8.61 


12.1 


14.7 


18. s 


23.9 


28.9 


35.7 


41.6 


47.4 


4 


5 


686 


263 


90.9 


28.3 


12. 5 


6.70 


4-03 


1.83 




1.65 


2.44 


3-43 


4.69 


6.60 


8.00 


10. 


13.0 


15.7 


19.4 


22.6 


25. 8 


5 


6 


1,116 


429 


148 


46.0 


20.4 


10.9 


6.56 


2.97 


1.63 




1.48 


2.09 


2.85 


4.02 


4.86 


6. II 


7.91 


9.56 


II. 8 


13.8 


IS. 6 


6 


7 


1,707 


656 


226 


70. S 


31.2 


16.6 


10. 


4.54 


2.49 


I. SI 




1. 41 


1.93 


2.71 


3.28 


4.12 


5.34 


6.45 


7.97 


9.31 


10.6 


7 


8 


2,435 


936 


322 


lOI 


44. 5 


23.8 


14-3 


6.48 


3-54 


2.18 


1.43 




1.35 


1.93 


2.33 


2.92 


3.79 


4-57 


S.67 


6.60 


7.52 


8 


9 


3.335 


1,281 


440 


137 


60.8 


32.5 


19.5 


8.85 


4-8s 


2.98 


1.95 


1.37 




1.41 


1. 71 


2.14 


2.77 


3-35 


4.14 


4.83 


S.50 


9 


10 


4.393 


1,688 


582 


181 


80.4 


42.9 


25.8 


II. 7 


6.40 


3.93 


2.57 


1.80 


1.32 




1. 21 


1.52 


1.97 


2.38 


2.94 


3.43 


3-91 


10 


II 


5,642 


2,168 


747 


233 


103 


55-1 


33-1 


iS-o 


8.22 


S.05 


3-31 


2.32 


1.70 


1.28 




1.26 


1.63 


1.88 


2.43 


2.83 


3.22 


II 


12 


7,087 


2,723 


938 


293 


129 


69.2 


41.6 


18.8 


10.3 


6.34 


4-15 


2.91 


2.13 


1. 61 


1.26 




1.30 


1.57 


1.93 


2.26 


2.58 


12 


13 


8,657 


3,326 


1,146 


358 


158 


84.5 


50.7 


23.0 


12.6 


7.75 


5.07 


3.56 


2.60 


1.98 


I. S3 


1.22 




1. 21 


1.49 


1.74 


1.98 


13 


14 


10,600 


4.070 


1,403 


438 


193 


103 


62.2 


28.2 


15.4 


9.48 


6.21 


4-35 


3.18 


2.41 


1.88 


I. so 


1.22 




1.24 


1.44 


1.64 


14 


15 


12,824 


4.927 


1,698 


530 


234 


125 


75-3 


34.1 


18.7 


II. 5 


7.52 


S.27 


3.85 


2.92 


2.27 


1. 81 


1.48 


1.21 




1. 17 


I 35 


IS 


16 


14.978 


S.758 


1,984 


619 


274 


146 


88.0 


39.9 


21.8 


13.4 


8.78 


6. IS 


4.51 


3-41 


2.66 


2.12 


1.73 


1.42 


1. 18 




1. 14 


16 


17 


17.537 


6,738 


2,322 


724 


320 


171 


103 


46.6 


25.6 


15.7 


10.3 


7.20 


S.27 


3.99 


3. II 


2.47 


2.03 


1.66 


1.37 


1. 17 






18 


20,327 


7.810 


2,691 


840 


371 


198 


119 


54-1 


29.6 


18.2 


II. 9 


8.35 


6. II 


4-63 


3-6o 


2.87 


2.35 


1.92 


1.59 


1.36 


1. 16 




20 


20,676 


10,249 


3.532 


1,102 


487 


260 


157 


70.9 


38.9 


23-9 


IS. 6 


10.9 


8.02 


6.07 


4.73 


3.76 


3.08 


2.52 


2.08 


1.78 


I. 52 




^A 


42,624 


16,376 


5.644 


1,761 


778 


416 


250 


113 


62.1 


38.2 


25.0 


17.5 


12.8 


9.70 


7.55 


6.01 


4.92 


4.02 


3.32 


2.84 


2.43 




30 


7S.4:;3 


28,990 


9.990 


3.117 


1,378 


736 


443 


201 


IIO 


67.6 


44-2 


31.0 


22.7 


17.2 


13.4 


10.7 


8.72 


7.14 


S.88 


5. 03 


4-30 




36 


120,100 


46,143 


IS.902 


4,961 


2,193 


1,172 


• 705 


319 


I7S 


108 


70.4 


49.3 


36.1 


27.3 


21.3 


16.9 


13.9 


II. 3 


9-37 


8.01 


6.85 




42 


177.724 


68,282 


23.531 


7.341 


3.245 


1.734 


1,044 


473 


259 


159 


104 


73 


53-4 


40.5 


31.5 


25.1 


20.5 


16.8 


13-0 


II. 9 


10. 1 




48 


249.351 


9S,8i8 


33.020 


10,301 


4.554 


2,434 


1,46s 


663 


363 


223 


146 


102 


75.0 


S6.8 


44.2 


35-2 


28.8 


23. S 


19.4 


16.6 


14.2 




Dia. 


i 


f 


I 


li 


2 


2j 


3 


4 


5 


6 


7 


8 


9 


10 


II 


12 


13 


14 


15 


16 


17 





Actual Internal Diameters 



ing to the rules of the U. S. Board of Supervising Inspectors of Steam- 
boats may be determined from the formula: 

pd 

<= -~ h .0625 

6000 -^ 

in which < = thickness, ins., 

^ = working pressure, lbs. per sq. in., 

d = inside diam. of pipe, ins. 
Tke bursting strength of lead pipe under cold water pressure may 
be calculated for a tensile strength of 1740 lbs. per sq. in., the usual 
factor of safety being five. Lead looses its strength rapidly under 
moderate increase of temperature — even to that of boiling water — 
and for hot water service, using the same nominal value for tensile 
strength, the factor of safety should be doubled. 

Standard Pipe Flanges and Fittings 

Tables 13 to 16 give the dimensions of standard pipe flanges and 
fittings according to the report of the A. S. M. E. committee as 
revised in 1914. The following explanatory notes apply: 

(a) Standard and extra heavy reducing elbows carry same dimen- 
sions center to face as regular elbows of largest straight size. 

(J) Standard and extra heavy tees, crosses and laterals, reducing on 
run only, carry same dimensions face to face as largest straight size. 

(c) If flanged fittings for lower working pressure than 125 lbs. are 
made, they shall conform in all dimensions except thickness of shell, 
to this standard and shall have the guaranteed working pressure cast 
on each fitting. Flanges for these fittings must be of standard 
dimensions. 

(d) Where long radius fittings are specified, it has reference only 



to elbows which are made in two center to face dimensions and to be 
known as elbows and long radius elbows, the latter being used only 
when so specified. 

(e) All standard weight fittings must be guaranteed for i2S-lb. 
working pressure and extra heavy fittings for 250-lb. working pres- 
sure and each fitting must have some mark cast on it indicating the 
maker and guaranteed working steam pressure. 

(/) All extra heavy fittings and flanges to have a raised surface 
of J^6 in. high inside of bolt holes for gaskets. 

Standard weight fittings and flanges to be plain faced. 

Bolt holes to be J^ in. larger in diameter than bolts. 

Bolt holes to straddle center line. 

(g) Size of all fittings scheduled indicates inside diameter of ports. 

{h) The face to face dimension of reducers, either straight or 
eccentric, for all pressures, shall be the same face to face as given in 
table of dimensions. 

(i) Square head bolts with hexagonal nuts are recommended. 

For bolts, 1% in. diameter and larger, studs with a nut on each 
end are satisfactory. 

Hexagonal nuts for pipe sizes i in. to 46 ins., on 125-lb. standard 
and I in. to 16 ins. on 2So-lb. standard can be conveniently pulled 
up with open wrenches of minimum design of heads. Hexagonal 
nuts for pipe sizes 48 ins. to 100 ins. on 125-lb. and 18 ins. to 48 ins. 
on 2so-lb. standards, can be conveniently pulled up with box or 
socket wrenches. 

( J ) Twin elbows, whether straight or reducing, carry same dimen- 
sions center to face and face to face as regular straight size ells and 
tees. 



PIPE AND PIPE JOINTS 



259 



Table 8. — The Equation of Equalization of Extra Strong Pipe 

Follow the line for one size to the column for the other and at the intersection find the 
number of the smaller pipes equivalent to one of the larger 



Pipe 


i 


f 


i 


i 


I 


li 


a 


2 


2^ 


3 


3i 


4 


4i 


5 


6 


Pipe 


size 


in. 


in. 


in. 


in. 


in. 


ins. 


ms. 


ins. 


Ins. 


ins. 


ins. 


ins. 


ins. 


ins. 


ms. 


size 


Int. 
area 


.068 


.139 


.231 


• 452 


.71 


1. 271 


1.753 


2.935 


4.209 


6.569 


8.8s6 


11.45 


14.18 


18.193 


25-97 


Int. 
area 


Ins. 
































Ins. 


i 


I 


2.05 


3-4 


6.7 


10.4 


18.8 


25.8 


43. 


62. 


96. s 


130. 


168. s 


208. 


267. 


382. 


i 


1 




I 


1.66 


3.26 


S-i 


9.2 


12.6 


21 . 1 


30.3 


47-3 


63.8 


82. s 


102. 


131. 


186. 


1 


i 






I 


1.96 


3-08 


S-5 


7.6 


12.7 


18.2 


28.4 


38.4 


47.5 


61. s 


78.8 


112 . 


i 


i 








I 


1-57 


2.82 


3-9 


6.5 


9.3 


14-5 


19.6 


25-3 


31.3 


40. 2 


57.4 


J 


I 










I 


1.79 


2.47 


4.14 


5. 93 


9.2s 


12.5 


16. 1 


20 


25.6 


36. 5 


I 


li 












I 


1.38 


2.3 


3-31 


5.16 


6.96 


8.4 


II . I 


14.3 


20.4 


u 


li 














I 


1.67 


2,4 


3-74 


5-05 


6.54 


8.1 


10.4 


14.8 


li 


2 
















I 


1-43 


2.24 


3.02 


3-91 


4.84 


6.2 


8.9 


2 


2i 


















I 


1.56 


2. 1 


2.72 


3.36 


4-31 


6. IS 


2i 


3 




















I 


1.35 


1-75 


2.16 


2.77 


3-95 


3 


3h 






















I 


1.29 


1.6 


2.05 


2.92 


3i 


4 
























I 


1.24 


1-59 


2.26 


4 


4^ 


























I 


1.28 


1.83 


4i 


S 




























I 


1.37 


5 


6 






























I 


6 




iin. 1 


fin. 


im. 


iin. 


I in. 1 1 i ins| i i ins 1 2 ins. 1 2 J ins 


3 ins. 


3i ins 


4 ins.U^ ins 


5 ins. 


6 in. 





Side outlet elbows and side outlet tees, 
■whether straight or reducing sizes, carry 
same dimensions center to face and face 
to face as regular tees having same reduc- 
tions. 

(k) Bull head tees or tees increasing on 
outlet, will have same center to face and 
face to face dimensions as a straight fit- 
ting of the size of the outlet. 

(/) Tees and crosses 16 in. and down, 
reducing on the outlet, use the same dimen- 
sions as straight sizes of the larger port. 

Size 18 in. and up, reducing on the outlet 
are made in two lengths depending on the 
size of the outlet as given in the table of 
dimensions. Laterals 16 ins. and down, 
reducing on the branch, use the same 
dimensions as straight sizes of the larger 
port. 

(m) Sizes 18 ins. and up, reducing on 
the branch, are made in two lengths de- 
pending on the size of the branch as given 
in the table of dimensions. 

The dimensions of reducing flanged fit- 
tings are always regulated by the reduc- 
tions of the outlet or branch. Fittings 
reducing on the run only, the long body 
pattern will always be used. 

Y's are special and are made to suit 
conditions. 

Double sweep tees are not made reduc- 
ing on the run. 

(n) Steel flanges, fittings and valves are 
recommended for superheated steam. 

Pipe Joints 

The common gasket joint is a constant 
source of trouble and a poor thing at best. 
When used, its security will be greatly 
increased if the gasket does not extend 
beyond the bolts. Still greater security 
may be obtained by facing the flanges 
slightly concave, with which construction 
the gasket must be entirely within the bolts. 
The types of joint shown below apply to 
most conditions and should be used in 
preference to the common construction. 

{Continued on page 262, first column) 

Table io. — American Water Works Association Standard Cast-iron Pipe for Fire Lines and other High-pressure Service 

Adopted May 12, 1908 
The weights are per length to lay 12 ft., including standard sockets; proportionate allowance to be made for any variation. 



Table q. — The Equation or Equalization of Double Extra Strong Pipe 



Pipe 


f 


\ 


I 


I 


li 


li 


2 


2i 


3 


3h 


4 


4^ 


5 


6 


Pipe 


size 


in. 


m. 


m. 


m. 


ms. 


ms. 


ms. 


ms. 


ms. 


ms. 


ms. 


ms. 


ms. 


ms. 


size 


Int. 
area 


.042 


.047 


.139 


.271 


.615 


.93 


1.744 


2.419 


4.097 


5-794 


7.724 


10 


12.96 


18.66 


Int. 
area 


Ins. 






























Ins 


1 


I 


1 . 12 


3.32 


6.4s 


14.6 


22. 1 


41-5 


57.5 


95.6 


137 


184, 


236 


308 


444 


1 


J 




I 


2.96 


5. 77 


13 -I 


19.7 


37.2 


51. 5 


87 


123 


164 


213 


276 


398 


i 


3 






I 


1.95 


4.43 


6.7 


12.5 


17.. 4 


29.4 


41 .6 


55.5 


72 


93-5 


134 


i 


I 








I 


2.27 


3-42 


6.4s 


8.95 


15. 1 


21.4 


28. 5 


37 


47.9 


69 


I 


li 










I 


I.SI 


2.84 


3.94 


6.65 


9-4 


12. 5 


16.3 


21 . I 


30.4 


li 


li 












I 


1.88 


2.6 


4-4 


6.21 


8.3 


10.8 


14 


20 


ih 


2 














I 


1-39 


2.34 


3 31 


4.42 


5.74 


7.45 


10.7 


2 


2h 
















I 


1 .69 


2.39 


3-19 


4-1 


5-33 


7.72 


2i 


3 


















I 


I. 41 

I 


1.88 
1-33 


2.44 
1-73 


3.16 
2.24 


4-55 

3-22 


3 

3i 


3-5 
































4 






















I 


1.3 


1.68 


2.42 


4 


4-i 
























I 


1.3 


1.87 


4* 


5 
6 


























I 


1.44 

I 


5 
6 




f in.l 


\ in. 


fin. 


I in. 


1} inslii insU ins.U^ insU ins. |3i insU ins.|4^ insis ins.|6 ins. 





Nominal 
inside 


Class E, soo-ft. head, 
217 lbs. pressure 


Class F, 600-ft. head, 
260 lbs. pressure 


Class G, 
340 


700-ft. head, 
lbs. pressure 


Class H, 800-ft.head, 
347 lbs. pressure 


Nominal 
inside 


diameter. 


Thickness, 
ins. 


Weight per 
Foot 1 Length 


Thickness, 
ins. 


Weight per 


Thickness, 
ins. 


Weight per 


Thickness, 
ins. 


Weight per 


diameter. 


ms. 


Foot 1 Length 


Foot 1 Length 


Foot 1 Length 


ins. 


6 

8 

10 

12 

14 
16 
18 
20 

24 


.58 
.66 

^74 
.82 

.90 

.98 

1.07 

IIS 

1. 31 

1-55 
1.80 


41-7 

61.7 

86.3 

113. 8 

145.0 
179.6 
220.4 
263 .0 

359.6 
521.7 
725.0 


500 

740 

1035 

136s 

1740 
2155 
264s 
3155 

431S 
6260 
8700 


.6t 
.71 

.80 

.89 

• 99 
1 .08 
1. 17 
1.27 

1-45 
1-73 

2 . 02 


43.3 

65.7 

92. 1 

122. 1 

157-5 
195-4 
238.4 
286.3 

392.9 
585.4 
820.0 


520 

790 

iios 

1465 

1890 
234s 
2860 
3435 

4715 
702s 
9840 


.6s 
.75 
.86 
.97 

1.07 
1. 18 
1.28 
1.39 


47-1 

70.8 

100.9 

135-4 

174-2 
219.2 
267 . 1 
320.8 


56s 

850 

1210 

162s 

2690 
2620 
3205 
3850 


.69 

.80 

.92 

1.04 

1.16 
1.27 
1.39 
1-51 


49-6 

75-0 

106.7 

143.8 

186.7 
232. 5 
286.7 
344-6 


595 

900 

1280 

1725 

2240 
2790 
3440 
4135 


6 

8 

10 

12 

14 
16 
18 
20 

24 


30 
















36 














36 



260 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table ii. — American Water Works Association Standard Cast-iron Pipe 

Adopted May 12, 1908 
The weights are per length to lay 12 ft., including standard sockets; proportionate allowance to be made for any variation. 



Nominal 


Class A, ioo-£t. head. 


Class B, 200-ft. head. 


Class C, 300-ft. head, | 


Class D, 400-ft. head. 


Nominal 


inside 


43 lbs. pressure 


86 lbs. pressure 


130 


lbs. pressure 


173 


bs- pressure 


inside 


diameter, 


Thickness, 
ins. 


Weight per 


Thickness.l Weight per 


Thickness, 
ins. 


Weight per 


Thickness, 
ins 


Weight per 


diameter. 


ms. 


Foot 1 Length 


ins. 


Foot Length | 


Foot 1 Length 


Foot 1 Length 


ms. 


4 


• 42 


20,0 


240 


.45 


21.7 


260 


.48 


23.3 


280 


.52 


25.0 


300 


4 


6 


• 44 


30.8 


370 


.48 


33.3 


400 


-SI 


35.8 


430 


.55 


38-3 


460 


6 


8 


.46 


42.9 


515 


■ 51 


47-5 


570 


.56 


52.1 


625 


.60 


55.8 


670 


8 


10 


.50 


57.1 


68s 


.57 


63-8 


76s 


.62 


70-8 


850 


.68 


76-7 


920 


10 


12 


• 54 


72. s 


870 


.62 


82.1 


985 


.68 


91.7 


1,100 


.75 


100. 


1,200 


■ 12 


14 


• 57 


89-6 


1,075 


.66 


102.5 


1,230 


• 74 


116. 7 


1,400 


.82 


129. 2 


1,550 


14 


16 


-60 


108.3 


1.300 


.70 


125.0 


1.500 


.80 


143-8 


1,725 


.89 


158.3 


1,900 


16 


18 


.64 


129. 2 


1,550 


.75 


150.0 


1,800 


.87 


175-0 


2,100 


.96 


191. 7 


2,300 


18 


20 


-67 


ISO-O 


1,800 


.80 


175-0 


2,100 


.92 


208.3 


2,500 


1.03 


229. 2 


2,750 


20 


24 


.76 


204.2 


2,450 


.89 


233-3 


2,800 


1.04 


279.2 


3,350 


1. 16 


306.7 


3,680 


■ 24 


30 


.88 


291-7 


3,500 


1.03 


333-3 


4,000 


1.20 


400.0 


4,800 


1.37 


450.0 


5,400 


30 


36 


-99 


391-7 


4,700 


I-I5 


454-2 


5,450 


1.36 


545-8 


6,550 


1.58 


625.0 


7,500 


36 


42 


1. 10 


S12-5 


6,150 


1.28 


591-7 


7,100 


1.54 


716.7 


8,600 


1.78 


825.0 


9,900 


42 


48 


1 .26 


666-7 


8,000 


1.42 


750-0 


9,000 


1. 71 


908.3 


10,900 


1 .96 


1050.0 


12,600 


48 


54 


I-3S 


800.0 


9,600 


1. 55 


933-3 


11,200 


1 .90 


I14I-7 


13,700 


2.23 


1341-7 


16,100 


54 


60 


1.39 


916.7 


11,000 


1.67 


I 104 -2 


13,250 


2.00 


134I-7 


16,100 


2.38 


1583-3 


19,000 


60 


72 


1 .62 


1283.4 
1633-4 


15,400 
19,600 


1 .95 


1545- 8 


18,550 


2-39 


1904.2 


22,850 








72 


84 


1.72 


2.22 


2104. 2 


25.250 












84 













Table 12. — Dimensions, Strength and Weight of Abendroth & Root Black Spiral Riveted Pipe 







Approximate 


Plain end 


WithA. &R. 


With Root 






Approximate 


Plain end 


With A. & R. 


With Root 


Diam- 


Thickness, 


bursting 




flanges, bolts 


bolted joint 


Diam- 


Thickness, 


bursting 




flanges, bolts 


bolted joint 


ter. 


B. W. 
gage 


pressure, 
lbs. per sq. 


pipe 


and gaskets 


complete 
Weight per 


ter, 
ins. 


B. W. 
gage 


pressure, 
lbs. per sq. 


pipe 


and gaskets 


complete 


ins. 


Weight per 


Weight per 


Weight per 


Weight per 


Weight per 






in. 


100 ft. 


100 ft. 


100 ft. 






in. 


100 ft. 


100 ft. 


100 ft. 


3 


22 


1060 


115 


139 


153 


13 


16 


570 


1 106 


1274 


1346 




20 


1325 


147 


171 


185 




14 


730 


1420 


1588 


1660 




18 


i860 


205 


229 


243 




12 
10 


950 
1165 


1866 

2294 


2034 
2462 


2106 
2534 


4 


20 


1000 


195 


227 


247 
















18 


1390 


273 


30s 


325 


14 


16 


530 


1 199 


1399 


1465 




16 


1845 


360 


392 


412 




14 
12 


675 
890 


1539 
2022 


1739 
2222 


1805 
2288 


5 


20 
18 


795 

I 100 


242 
340 


282 
380 


304 
402 




10 


1090 


2486 


2686 


2752 




16 


1480 


448 


488 


510 


15 


14 
12 


630 
825 


1649 
2167 


1889 
2407 


1973 
2491 


6 


18 
16 


930 

1220 


385 
508 


433 
556 


475 
598 




10 


1015 


2664 


2904 


2988 




14 


1580 


653 


701 


743 


16 


14 


590 


1771 


2051 


2149 




12 


2060 


858 


906 


948 




12 
10 


770 
950 


2327 
2861 


2607 
3141 


2705 
3239 


7 


18 


790 


446 


Sio 


540 
















16 


1060 


588 


652 


682 


18 


14 


525 


1974 


2334 


2394 




14 


1340 


755 


819 


849 




12 


690 


2593 


2953 


3013 




12 


1780 ■ 


992 


1056 


1086 




10 


850 


3188 


3548 


3608 


8 


18 


690 


507 


587 


604 


20 


14 


470 


2180 


2556 


2608 




16 


945 


669 


749 


766 




12 


620 


2863 


3239 


3291 




14 


I180 


860 


940 


957 




10 


760 


3521 


3897 


3949 




12 


1540 


1130 


I2I0 


1227 


























22 


14 


430 


2390 


2830 


2830 


9 


16 


820 


753 


873 


863 




12 


s6s 


3140 


3580 


3580 




14 


1040 


967 


1087 


1077 




10 


695 


3860 


4300 


4300 




12 


1380 


1271 


1391 


1381 


























24 


14 


395 


2604 


3108 


3084 


10 


16 


740 


835 


963 


1025 




12 


515 


3421 


3925 


3901 




14 


945 


1071 


1199 


1261 




10 


635 


4216 


4720 


4696 




12 


1024 


1408 


1536 


1598 


























26 


12 


475 


3558 


4718 


4090 


II 


16 

14 


670 
860 


916 
I176 


1060 
1320 


II22 
1382 




10 


580 


4380 


5540 


4912 




12 


II20 


1546 


1690 


1752 


28 


12 
10 


440 
545 


3894 
4720 


5274 
6100 


4478 
5304 


12 


16 


615 


1003 


1 163 


I215 
















14 


790 


1287 


1447 


1499 


30 


12 


410 


411S 


5531 


475S 




12 


1025 


1692 


1852 


1904 




10 


510 


5063 


6479 


5703 




10 


1265 


2080 


2240 


2292 















PIPE AND PIPE JOINTS 



261 



Table 13. — American Standard Pipe Flanges for 125 Lbs. Working Pressure 
Notes. — Bolt holes should straddle center lines. Flanges should be plain faced. 

Square head bolts with hexagonal nuts are recommended. For bolts 1% ins. diameter and larger stud, with a nut, at each end is satisfactory. 
Hexagonal nuts for pipe_^ sizes I in. to 46 ins. can be conveniently pulled up with open wrenches of minimum design of heads. Hexagonal nuts for pipe sizes 
48 ins. to 100 ins. can be conveniently pulled up with box or socket wrenches. 
Rules approximately followed in compiling these data: 

Bolt circle = i. 10D-I-3 

Flange thickness = .03 15D+ 1.25 (for sizes 26 ins. to 100 ins.) 

D = inside diameter of pipe 
Flanges to be spot bored for nuts for sizes 32 ins. to 100 ins. inclusive. 





























Bolt Spacing 




Minimum thick- 






















and Clearance, Ins. 


Diame- 
ter of 


ness 


ins. 


Stress 
on pipe, 
lbs. per 
sq. in. 


Diame- 
ter of 
flange, 
ins. 


Thick- 
ness of 
flange, 
ins. 


Width 

of 

flange 

face, ins. 


Diame- 
ter of 

bolt 
circle, 

ins. 


No. of 
bolts 


Diame- 
ter of 
bolts, 
ins. 


Effect- 
ive area 
of bolts, 

sq. ins. 


Stress 
on bolt 

metal, 
lbs. per 

sq. in. 


Diame- 
ter of 
bolt 

holes, 
ins. 


/ \ 1^ 


\ \\ 


As cal- 
culated 


To 

nearest 
fraction 


/ \ 


' 


.>J_ 


pipe, ins. 


U A — * 

(On Chord) 




A 


B 


c 


I 


•43 


Me 


143 


4 


Me 


iH 


3 


4 


Me 


■ 093 


264 


?i6 


2 . 12 


.91 


1.2 I 


iH 


■ 44 


Me 


178 


4H 


y2 


m 


3% 


4 


Me 


■ 093 


412 


Ma 


2.38 


■ 91 


1.47 


iH 


■ 4S 


Me 


214 


5 


?ie 


m 


3li 


4 


H 


. 126 


438 


H 


2.73 


1. 00 


1.73 


2 


.46 


Me 


286 


6 


H 


2 


4% 


4 


% 


.202 


486 


% 


3.35 


I. 21 


2. 14 


2H 


.48 


Me 


357 


7 


'Me 


2H 


sH 


4 


% 


.202 


750 


% 


3.88 


1.2 I 


2.67 


3 


• SO 


Me 


428 


7H 


H 


2)4 


6 


4 


5^ 


.202 


1093 


Vi 


4^23 


1.2 I 


3.02 


3'/2 


•S2 


Me 


SOO 


8H 


iMe 


2H 


7 


4 


% 


.202 


1488 


% 


4.94 


1.2 I 


3.73 


4 


• S3 


H 


500 


9 


1^6 


2^ 


iH 


8 


% 


.202 


972 


% 


2.87 


I. 21 


1. 56 


4M 


•SS 


H 


S62 


9H 


IMe 


2% 


yH 


8 


H 


.302 


823 


% 


2.96 


1.44 


1-52 


S 


.56 


H 


62s 


10 


iMe 


2H 


8H 


8 


% 


• 302 


10 16 


H 


3-25 


1.44 


I. 81 


6 


.60 


Me 


667 


II 


I 


2H 


9H 


8 


H 


• 302 


1463 


H 


3.63 


1.44 


2. 19 


7 


• 63 


H 


700 


12!^ 


iHe 


2Vi 


loH 


8 


% 


.302 


199 I 


H 


4. II 


1.44 


2.67 


8 


.66 


H 


800 


13 W 


i« 


2% 


im 


8 


% 


.302 


2600 


H 


4.50 


1.44 


3.06 


9 


• 70 


iMe 


818 


IS 


iH 


3 


13 H 


12 


% 


.302 


2194 


H 


3^43 


1.44 


1.99 


10 


.73 


H 


833 


16 


iMe 


3 


I4M 


12 


H 


.420 


1948 


I 


3^69 


1.66 


2.03 


12 


.80 


'Me 


923 


19 


iH 


zH 


17 


12 


H 


.420 


2805 


I 


4.40 


1.66 


2.74 


14 


.86 


li 


1000 


21 


m 


3\i 


18% 


12 


I 


.550 


29 IS 


iH 


4.86 


1.88 


2.98 


IS 


.90 


'A 


1072 


22H 


m 


3% 


20 


16 


I 


.S50 


25 10 


iH 


3.90 


1.88 


2.02 


16 


• 93 


1 


1000 


23 H 


iMe 


3% 


2IH 


16 


I 


• 550 


2856 


iW 


4. 14 


1.88 


2.26 


18 


I. 00 


iMe 


1059 


25 


irm 


3yi 


22?4 


16 


iH 


.694 


2865 


iH 


4.44 


2.09 


2.35 


20 


1.07 


m 


nil 


27H 


I' Me 


3% 


25 


20 


iH 


.694 


2829 


iH 


391 


2.09 


1.82 


22 


1. 13 


iMe 


1 158 


29 K2 


l'?-t6 


3H 


27H 


20 


iH 


•893 


2660 


m 


4.26 


2.31 


1-95 


24 


1.20 


iM 


1200 


32 


1% 


4 


29H 


20 


iM 


.893 


3166 


iH 


4.62 


2.31 


2.31 


26 


1.27 


iMe 


1238 


34 M 


2 


4H 


3I?4 


24 


iH 


• 893 


3096 


m 


4. 14 


2.31 


1.83 


28 


1-33 


\H 


1273 


36 H 


2H6 


4H 


34 


28 


iW 


• 893 


3078 


m 


3.81 


2.31 


I. SO 


30 


1.40 


iMe 


1304 


38M 


2H 


4% 


36 


28 


m 


1057 


2985 


m 


4 03 


2.53 


I. SO 


32 


1.47 


IH 


1333 


41H 


2\\ 


4% 


38H 


28 


iVi 


1.294 


2775 


15^ 


431 


2.75 


1. 56 


34 


I.S4 


l^-ie 


1360 


43H 


2Me 


AH 


40H 


32 


M 


1.294 


2741 


1=4 


3-97 


2.75 


1.22 


36 


1.60 


m 


1385 


46 


2% 


5 


42% 


32 


l\'2. 


1.294 


3073 


m 


4- 19 


2.75 


1.44 


38 


1.67 


inu 


1407 


48% 


2H 


sH 


4SK4 


32 


1% 


I- 5 IS 


2924 


1% 


4-43 


2 .96 


1.47 


40 


1.73 


iH 


1428 


S0V4 


2H 


sH 


47M 


36 


1=4 


1^5 IS 


2880 


1% 


4. II 


2.96 


I- 15 


42 


1.82 


I'Me 


1448 


S3 


25i 


sH 


49 H 


36 


1=4 


IS IS 


3175 


1% 


431 


2.96 


1-35 


44 


1.87 


iH 


1467 


ssVi 


254 


s^A 


Si% 


40 


\% 


I. 5 IS 


3136 


1% 


4.06 


2.96 


I. 10 


46 


I 94 


liMe 


1484 


57H 


2IH6 


sH 


53% 


40 


m 


I.SIS 


3428 


1% 


4.22 


2.96 


1.26 


48 


2.00 


2 


isoo 


59!-^ 


2H 


sH 


56 


44 


iH 


1^5 IS 


3393 


1% 


3.98 


2.96 


1.02 


SO 


2.07 


2H6 


ISIS 


61H 


2% 


5% 


S8M 


44 


1% 


1.746 


3195 


Hi 


4. 14 


3. 19 


•95 


S2 


2.14 


2H 


IS30 


64 


2% 


6 


60J/2 


44 


1% 


1.746 


3456 


iZi 


4.30 


3- 19 


I. II 


S4 


2.20 


2?'ie 


IS43 


66 H 


3 


6H 


62% 


44 


1% 


1.746 


3726 


m 


4-45 


3. 19 


1.26 


56 


2.27 


2K 


ISSS 


68% 


3 


6H 


65 


48 


1% 


1.746 


3674 


m 


4.26 


3- 19 


1.07 


S8 


2^34 


2M6 


1567 


71 


3H 


6M2 


67M 


48 


1% 


1.746 


3941 


m 


4.40 


3^ 19 


1.2 I 


60 


2.41 


2Me 


1538 


73 


3H 


6H 


69H 


52 


1% 


1.746 


3892 


lU 


4. I?) 


3- 19 


I. 00 


62 


2.47 


2 1/2 


ISSO 


7S?4 


3M 


6% 


ym 


52 


m 


2.051 


3538 


2 


4-34 


3^41 


■ 93 


64 


2. 54 


2M6 


IS6l 


78 


3H 


7 


74 


52 


lA 


2.051 


3770 


2 


4.48 


3^41 


i.07 


66 


2.6l 


2H 


1572 


80 


3?^ 


7 


76 


52 


m 


2.05 I 


4010 


2 


4.60 


3.41 


I. 19 


68 


2.68 


21H6 


1582 


82)4 


3?^ 


iH 


78H 


56 


m 


2.05 I 


3952 


2 


4.38 


3^41 


• 97 


70 


2.74 


2H 


159 I 


84 Mi 


3M 


7H 


80K2 


56 


iH 


2.05 I 


4188 


2 


4^51 


3-41 


I. 10 


72 


2.81 


2l?-i6 


1600 


86 H 


zVi 


7H 


82 H 


60 


lA 


2.05 I 


4136 


2 


4 33 


3^41 


.92 


74 


2.88 


2% 


1609 


881.^ 


35/i 


7H 


84K2 


60 


^H 


2.051 


4368 


2 


4.44 


3.41 


1.03 


76 


2.94 


2IM6 


1617 


90?^ 


35i 


7H 


86 H 


60 


1% 


2.05 I 


4608 


2 


4^54 


3.41 


I. 13 


78 


3.01 


3 


1625 


93 


3?4 


iVi 


88% 


60 


2 


2.302 


4325 


2],i 


4.66 


3.63 


I 03 


80 


3.08 


3H6 


1633 


9SK 


3?^ 


7H 


91 


60 


2 


2.302 


4549 


2H 


4.78 


3.63 


1. 15 


82 


3^ IS 


3^^ 


1640 


97 H 


3li 


7% 


93M 


60 


2 


2.302 


4779 


2A 


4.90 


3^63 


1.27 


84 


321 


3Me 


1647 


99H 


Sli 


7^4 


95 H 


64 


2 


2.302 


4702 


2]^ 


4.68 


3^63 


I. OS 


86 


3.28 


3H 


1653 


102 


4 


8 


97% 


64 


2 


2.302 


4928 


2H 


4^79 


3^63 


I. 16 


88 


3.3s 


3M6 


1660 


104 M 


4 


8H 


100 


68 


2 


2.302 


4857 


2% 


4.60 


3.63 


•97 


90 


3-41 


3H 


1667 


106 M2 


4H 


8M 


I0214 


68 


2>4 


2.648 


4416 


2)4 


4.71 


3.83 


.88 


92 


3.48 


3H 


1643 


108H 


4l'i 


8?i 


\04yi 


68 


2^^ 


2.648 


461S 


2H 


4.81 


3^83 


• 98 


94 


3-55 


3Me 


1649 


II I 


AM 


81/2 


I06M 


68 


2 14 


2.648 


4817 


2)4 


4.89 


3.83 


1.06 


96 


3.62 


35^ 


I6SS 


113M 


4M 


854 


I08M2 


68 


2K 


3 023 


4401 


2H 


4.99 


4.06 


•93 


98 


3.68 


31 Me 


1661 


I IS 1/2 


4H 


8?4 


110% 


68 


2M 


3^023 


4587 


2% 


5 09 


4.06 


1.03 


100 


3. 75 


3?4 


1667 


117W 


4% 


8Ji 


113 


68 


2M 


3023 


4776 


2% 


5-20 


4.06 


I. 14 



262 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 14. — American Standard Extra Heavy Pipe Flanges for 250 Lbs. Working Pressure 
Notes. — Bolt holes should straddle center lines. Flanges should have Me-in. raised face for gaskets. 

Square head bolts with hexagonal nuts are recommended. For bolts l% ins. diameter and larger stud with a nut at each end is satisfactory. 
Hexagonal nuts for pipe sizes I in. to 16 ins. can be conveniently pulled up with open wrenches of minimum design of heads. Hexagonal nuts for pipe sizes 
18 ins. to 48 ins. can be conveniently pulled up with box or socket wrenches. 
Rules approximately followed in compling above data: 

Bolt circles = l.i7lD-|-3.75 

Flange thickness = .0546 Z34-I.37S (for sizes 10 ins. to 48 ins.) - 

I> = inside diameter of pipe 
Distance between inside edge of bolt holes and raised face to be )42 in. 
Thickness of flange given in table includes raised face. 
Flanges to be spot bored for nuts. 





Minimum thick- 






















Bolt Spacing 
and Clearance, Ins. 


Diame- 


ness, 


ins. 


Stress 
on pipe, 
lbs. per 

sq. in. 


Diame- 
ter of 
flange, 
ins. 


Thick- 
ness of 
flange, 
ins. 


Width 

of 

flange 

face, ins. 


Diame- 
ter of 

bolt 
circle, 

ins. 


No. of 
bolts 


Diame- 
ter of 
bolts, 
ins. 


Effect- 
ive area 
of bolts, 

sq. ins. 


Stress 
on bolt 
metal, 
lbs. per 
sq. in. 


Diame- 
ter of 

bolt 
holes, 

ins. 






ter of 


As cal- 
culated 


To 
nearest 
fraction 


4—^ 


\ / 1 \ 




pipe, ins. 


V. 


y 


-HV 






'(On Chord)' 




A 


B 


c 


I 


• 4S 


H 


250 


4^2 


'Me 


m 


3H 


4 


J2 


. 126 


389 


5^ 


2.29 


1.00 


1. 29 


iK 


■ 47 


Vi 


312 


5 


H 


ili 


3V4. 


4 


H 


.126 


609 


H 


2.63 


1. 00 


1.65 


i^^ 


• 49 


K2 


375 


6 


'Me 


2H 


4H 


4 


H 


.202 


547 


Vi 


3. 17 


1. 21 


1.96 


2 


• SI 


H 


500 


tVi 


li 


2M 


5 


4 


H 


.202 


972 


H 


3-53 


1. 21 


2.3^ 


. 21.^ 


.53 


ri6 


555 


IM 


I 


2!.!2 


Sl'i 


4 


% 


.302 


10 16 


li 


4^15 


1.44 


2.71 


3 


.56 


?i6 


667 


m 


1% 


zH 


6H 


8 


% 


.302 


731 


H 


2.53 


1.44 


1.09 


3H 


• 59 


yi6 


778 


9 


lYis, 


2% 


■?H 


8 


H 


.302 


99S 


■'A 


2.77 


1.44 


1.33 


4 


.61 


H 


800 


10 


m 


3 


7% 


8 


H 


.302 


1300 


li 


3.01 


1.44 


1.57 


4H 


.64 


H 


900 


I0l.'2 


iMe 


3 


8H 


8 


% 


.302 


1646 


% 


3.25 


1.44 


1. 81 


5 


.67 


iHe 


909 


II 


m 


3 


9H 


8 


Vi 


.302 


2032 


% 


3.53 


1.44 


2.09 


6 


.72 


H 


1000 


I2K2 


lj>i6 


3H 


1054 


12 


H 


.302 


I9S0 


% 


2.75 


1.44 


1. 31 


7 


.78 


me 


1077 


14 


m 


3 ^'2 


11% 


12 


% 


.420 


1909 


I 


3.07 


1.66 


J. 41 


8 


• 83 


^Vie 


1230 


IS 


1% 


3}'2 


13 


12 


H 


.420 


2493 


I 


3.36 


1.66 


1.70 


9 


.89 


H 


1285 


16H 


m 


m 


14 


12 


I 


• 550 


2410 


iH 


3.62 


1.88 


1.74 


10 


.94 


iMe 


1333 


I7}^2 


m 


3M 


I5M 


16 


I 


■550 


2231 


iH 


2.97 


1.88 


1.09 


12 


I. OS 


1 


1500 


20l.!2 


2 


AM 


nH 


16 


iM 


.694 


2546 


iH 


3.46 


2.09 


1.37 


14 


I. 16 


i}i 


1355 


23 


2\i 


4K2 


2054 


20 


iH 


.694 


2773 


-iH 


3-17 


2.09 


1.08 


IS 


1. 21 


13/16 


1579 


24'/2 


2yi6 


4% 


2 11/2 


20 


i\i 


■ 893 


2473 


m 


3.36 


2.31 


I. OS 


16 


1.27 


iH 


1600 


25K2 


2H 


4M 


22H 


20 


iH 


■ 893 


2814 


m 


3.52 


2.31 


I. 21 


18 


1.37 


1% 


1636 


28 


2?i 


S 


24% 


24 


iVi 


■893 


2968 


m 


3.23 


2.31 


•92 


20 


1.48 


iH 


1666 


3OK2 


2],i 


sM 


27 


24 


m 


I^057 


3096 


iH 


3.52 


2.53 


■ 99 


22 


1.59 


me 


1760 


33 


2% 


5M 


29H 


24 


iVi 


1.29s 


3058 


m 


3.81 


2.7S 


1.06 


24 


1.70 


m 


1846 


36 


2Yi 


5% 


32 


24 


1% 


I^SIS 


3 1 10 


m 


4. IS 


2.96 


1.22 


26 


1.81 


iWi-i 


1793 


38K 


21^6 


6\i 


34H 


28 


1% 


I^SIS 


3126 


1% 


3.86 


2.96 


.90 


28 


1. 91 


m 


1866 


40M 


21^6 


6% 


37 


28 


1% 


I. SIS 


3629 


1% 


4.14 


2.96 


1. 18 


30 


2.02 


2 


1875 


43 


3 


6I/2 


39M 


28 


m 


1.746 


3615 


1% 


4.38 


3- 19 


I. 19 


32 


2. 13 


2H 


1882 


45M 


3\i 


(>% 


4I}'2 


28 


1% 


2.051 


3501 


2 


4.64 


3-41 


1.26 


34 


2.24 


2H 


1889 


47H 


3H 


6% 


43}.^ 


28 


m 


2.051 


39S2 


2 


4.87 


3-41 


1.46 


36 


2. 35 


2% 


1894 


50 


3H 


7 


46 


32 


m 


2.051 


3877 


2 


4.50 


3-41 


1.09 


38 


2.46 


2^6 


1948 


52M 


3lU 


iVk 


48 


32 


iH 


2.051 


4320 


2 


4.70 


3-41 


1.29 


40 


2.56 


2M6 


1953 


S4K2 


39'!e 


iVi 


SOH 


36 


m 


2.051 


4255 


2 


4.38 


3.41 


.97 


42 


2.67 


21 He 


1953 


57 


3'H6 


nVi 


52M 


36 


m 


2.051 


4691 


2 


4-59 


3.41 


I. 18 


44 


2.78 


2l?'i6 


1955 


59H 


3?l 


1% 


55 


36 


2 


2.302 


4587 


2\i 


4.79 


3-63 


I. 16 


46 


2.89 


2% 


2000 


61K2 


3% 


7% 


S7H 


40 


2 


2.302 


4512 


2H 


4.49 


3.63 


.86 


48 


3 00 


3 


2000 


65 


4 


SH 


6034 


40 


2 


2.302 


4913 


2H 


4-76 


3.63 


1.13 



Professor Sweet's joint for the cylinder covers of steam engines 
is shown in the section on steam engines (See Cylinder Cover Joints). 
It has also been adopted by the Ball Engine Co. with entire success. 

The joint is metal to metal and without grinding, the surfaces 
being ordinary tooled surfaces. The only, and a necessary, precau- 
tion is to make the joint narrow — not over f in. wide. 

The narrow metal to metal joint is also entirely successful for 
high pressure air as will be shown later. 

The Rapieff joint used with invariable success for the numerous 
joints of the Zalinski dynamite gun and its air plant, where it 
regularly withstood pressures of 2000 lbs. per sq. in., is shown in 
Figs. 6-14. It is thus described by B. C. Batcheller, Chf. Engr. 
Amer. Pneumatic Service Co. {Amer. Mack., Apr. 23,1908). Just 
inside the bolt circle a groove of peculiar shape, abc Fig. 6, is 



turned in the face of each flange into which a ring of round rubber 
cord is laid and the flanges are bolted up metal to metal. The 
combined cross-sectional area of the grooves is made slightly less 
than the sectional area of the rubber cord, resulting in compression 
of the rubber, the surplus flowing into the narrow space d, which 
is about ^ in. wide, and opens into the interior of the pipe. 

The fluid pressure acts against the rubber, tending to force it out- 
ward and, putting the entire ring of rubber under static pressure, seals 
the joint at c. Thus the higher the pressure the tighter is the joint. 

The rubber gasket ring is shown in Fig. 7. It is made from 

rubber cord that can be bought by the yard and made into 

rings as required. A splice is shown at /, which is made by cutting 

the cord obliquely and joining the ends with rubber cement. The 

(Continued on Page 266 first column) 



PIPE AND PIPE JOINTS 



263 





c 






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p 


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U 


m 


a 








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n 


W 










o 






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G 


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« 


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3 








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Cll 






ffl 

H-1 


s 






4 


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■g 










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JS 


« 


<u 




.a 


o 


O 


3 
f) 




a 




O 

55 


ci! 

c 


.2 


J3 




H 


CO 

e 




lU 


a 


H 


^ 




3 


fe 




rt 


It) 


a 


w 

CM 






3 
C 


a 


^ 




C 


J3 


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w 


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M 


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3 








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bo 


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o 


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O 


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m 








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Q E H 




1^^ 


















fQ 


o 


Q tl] fe, o 










■^ 




















1 



PIPE AND PIPE JOINTS 



265 



EegularSO Elbow 
-A 



BeKular45 Elbow 




Special Return Bend 



Regular Reducer 



Table 17. — Center to Face Measurements of Abendroth & 

Root Flanged Fittings 

Spaces Filled from Center to Face 

Dimensions in inches 



Inside 


90° elbows. 


45° 
elbows 




1 




diameter. 


tees and 


Y-branches 1 


bends 


ms. 


crosses 
A 


A 








D 


A 1 B 


c 


A \ B 


3 


3l 


2f 


9 


2i 


9 


3f 


7i 


4 


4i 


2if 


11 


2| 


II 


4f 


85 


5 


Si 


3J 


12 


3 


12 


Si 


loi 


6 


6J 


3l 


I3i 


3i 


I3i 


6i 


I2i 


7 


7i 


4 


IS 


4i 


IS 


7i 


I4J 


8 


8i 


4ii 


17 


s 


17 


8i 


16} 


9 


9f 


Sit 


18J 


si 


i84 


9i 


I8J 


10 


lOi 


Si 


21 


si 


21 


lOi 


20} 


II 


II 


Sf 


22 1 


si 


22J 


II 


22 


12 


12} 


6i 


24 


6 


24 


12} 


24i 


13 


13 


si 


26 


6i 


26 


13 


26 


14 


I3J 


si 


27i 


6J 


274 


14 


28 


15 


IS 


si 


29J 


6i 


29i 


IS 


30 


16 


16 


6J- 


314 


7 


3li 


16 


32 


18 


18 


7t 


3S 


7J 


3S 


18 


36 


20 


20 


loj 


381 


8 


38i 


20 


40 


22 


22 


II 


41 


9 


41 


22 


44 


24 


24 


12 


44 


10 


44 


24 


48 


26 


26 
28 

30 














28 














30 























Length of Reducers 



Inside Diameter. 
Total length 



4 


S 


6 


7 


8 


9 


10 


II 


12 


13 


14 


16 18 


23 


23 


23 


22 


22 


22 


33I33I33I33I32 


32 32 




45 Ell 





45 T 



I- — e ^ 



Table 18. — Cast-iron Screwed Pipe Fittings for Pressures up to 100 Lbs. per Sq. In. 

Walworth Mfg. Co.'s Standard 
Dimensions in inches 



Pipe 
dimensions 


Body dimensions of fittings 


Pipe 
dimensions 


Body dimensions of fittings 


Nominal 


•0 


For all fittings 


Cen. 

to 

face 


90° 

ell 
rad. 


4S° 
ell, 

face 
to 

face 


45° Ys. 


Nominal 

inside dia. 

of pipe 


•0 

0) 

.s 


For all fittings 


Cen. 

to 

face 


90° 

ell 
rad. 


45° 
ell, 
face 
to 
face 


45° Ys. 


inside dia. 
of pipe 


In- 
side 
dia. 




Dia. 

of 

bead 




J3 TJ 

^0 


Face 

to 
face 


Cen. 

to 
face 


In- 
side 
dia. 




Dia. 

of 
bead 




J3 -6 


Face 

to 

face 


Cen. 

to 

face 






A 


B 


C 


D 


£ 


F 


G 


H 


J 


K 






A 


B C 


D 


E 


F 


G 


H 


J 


K 


} 


18 
18 
14 
14 
114 

Ilj 
Ilj 
III 

8 

8 


i 

i 

If 

If 
2 

2i 

2il 

3A 


.... 


I 

li 
lA 
If 

2A 

. 2j 
2i 
3f 
4i 

4i 




f 

A 

A 
f 

i 
I 
I 


3 

i 
lA 

lA 

ii 

iH 
2 

2| 
2| 

3A 


A 

if 

I 

li 
li 
lA 
ij 

2A 


A 
A 
ii 

lA 

lA 

If 

If 

li 






3i 

4 

4i 

S 

6 

7 

8 

9 
10 
12 


8 
8 
8 
8 
8 

8 
8 
8 
8 
8 


4A 

4S 

S 

SA 

6il 

7H 
8| 
9j 
loj 
1 = 1 




5i 
6 

6A 
7A 
81 

9l 
loj 
12J 
I3i 

isf 




lA 

li 

li 

li 

H 

li 
If 
ij 
li 
il 


3ii 
4 

4A 
4ii 

sA 
6A 
li 

H 

9A 


2i 

2lf 

3A 

3f 

4 

4l 
SA 

sH 

6A 
7! 


2A 

2i 

2A 
2A 
2H 

3i 

3A 

3l 

4A 

4i 


8f 

9i 

lOl 
Hi 
13i 

I4t 
16J 
'9, 
20i 

24 


61 

7J 
71 
8i 
10 

Hi 

13 

I4i 

16 
19 


i 

i 

I 

I 

li 

li 

2 

24 

3 


2A 
2| 
2| 
3i 

3f 

4l 
Si 
6i 
7t 


lA 

ij 

2 

2i 

3 

3i 

4 

S 

sS 



266 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



ring should have the same diameter as the grooves in the face of 
the flanges. Rubber cord | in. diameter is large enough for the 
largest joints, and it is not convenient to use cord much less than 
I in. in diameter. The rubber should be of good quality, and pref- 
erably what is known in the trade as "pure gum." When a joint 
is made in a horizontal pipe, the rubber ring can be held in the 
groove of one flange by rubber cement when the joint is put 
together. The surface be may have an angle of 60 deg. 

When alignment of the pipe sections is required it is readily ob- 
tained by the construction shown in Fig. 8. Fig. 9 shows a modified 
form and Fig. 11 an application to a cylinder head. The rubber 
rings, of which sections are shown, may be cut from flat sheets. In 
making the joint shown in Fig. 11, the ring should be stretched 
over the head to hold it in place, talc powder being used to prevent 
its sticking. 

Two or more joints at as many shoulders on the same piece may 
be made with this joint as shown in Fig. 13 in which a lantern A 
is bolted to an annular casting B. Joint C is like Fig. 6 and joint 
D like Fig. 11. 

This joint is used with complete success for low pressures in the 
Batcheller pneumatic postal tubes, in which the pressure seldom 
exceeds 5 lbs. per sq. in. A rectangular groove, Fig. 14, J in. wide 
by I in. depth, is turned in one flange and a tongue f in. wide by 
^ in. high on the face of the opposite flange. The outside diameter 
of the tongue fits the outside diameter of the groove. A rubber 
ring, as shown, | in. wide and | in. thick, is laid in the groove and 
the flanges are bolted together. The rubber ring is compressed to 
a thickness of -^ in., the surplus flowing into th'e space provided by 
making the tongue narrower than the groove. The tongue and 
groove of this form are easily machined and alignment of the sections 
is insured. 



This joint is also regularly used by the Nordberg Mfg. Co. for 
mine-pumping plants where the pressure is heavy and the service 
severe. 

Pipe joints and threaded unions for high^pressure air, according to 
H. V. Haight, Chf. Engr. Canadian IngersoU-Rand Company {Amer. 
Mach., Apr. 23, 1908) should have metal to metal joints with narrow 
faces and are preferably of the ball and socket type. Regarding the 
actual joint, the rule is the higher the pressure the narrower the joint. 
Fig. 15 shows details of a ball and socket joint used for pressures 
up to 1000 lbs. per sq. in. The radius at the end of the pipe is 
slightly less than in the socket, giving line contact. The thread 
not being subject to air pressure is made straight and the flange 
screwed on by hand. The ball and socket feature makes the 
joint tight even if the parts are not in perfect alignment. Extra 
heavy pipe was used in order to have sufficient thickness after 
threading. 

A high-pressure flange union is shown in Fig. 16. It permits a 
movement of 5 deg. in any direction, as indicated in the smaller 
illustration. The recesses a are for calking with lead should it be 
necessary, which it seldom is. Fig. 17 shows a type of fitting 
used in the United States Navy for torpedo service and for air pres- 
sures up to 3000 lbs. per sq. in. Note especially the knife-edge 
joints. 

Fig. 18 shows a fitting for connecting copper tubing. The tube 
is swelled outward and the end pinched between the nipple and 
swivel, the former being turned to an angle of 30 deg. with the center 
line. 

Flange joints for high-pressure hydraulic work are shown in Figs. 
19 and 20 and Table 19 by U. Peters (Amer. Mach., Apr. 18, 
1901). 

Fig. 19 shows a joint for bored or seamless drawn steel pipes 





Fig. 8. 



Fig. 7. 




Fig. 10, 



Fig, ii.' 



Fig. 12. 
Figs. 6 to 14. — The Rapieff pipe joint. 




Fig. 14. 



PIPE AND PIPE JOINTS 



267 




steel Spherical 
Washers 




Showing Contract 
between.Eipe and.EIbow 



EiG. IS-Details of an Elbow and Companion Flange of a Ball and Socket Joint 

~?i"w.I.Pipe 

1 Pipe Thread 






Wall o£ Tank- 

yiG. i8.-Tank Connection for ^' Inch 
Copper Tu.be 




W/////M,', 



ITnion Tor Copper Tube Connection 




i - ^-. 



Fig. 17.- Fittings used in U.S. Torpedo Service 

Figs. 15 to 18. — Pipe fittings for high-pressure air. 



Fig. 16.- Flange Union for 
1000 Pounds Air Pressure 



from 3 to 8 ins. and larger diameters with ring-grooved male and 
female ends for the copper gasket. The flanges and bolts are also of 
steel, and the accompanying table gives some dimensions for the 
various sizes. The table is made up from connections actually in 
use and approved. They stood a trial test of 7 gross tons per sq. 
in. without showing any sign of leakage, if properly connected. 

The pipes and flanges are threaded either by the United States 
standard of 8 threads, or by the Whitworth screw standard with 
6 threads per inch. 

For smaller pipes the connection shown in Fig. 20 is applied. Such 
pipes are generally called by the catalog names of extra heavy or 
double extra heavy steel or wrought-iron pipes. As given in the 
tables of the various makers, they are of difierent dimensions, 
for pressures from 500 to 7000 lbs. per sq. in. The flanges are 
usually of forged or cast steel and of different shapes, corresponding 
to the number of bolts from oval-like, triangular and square to 
round, and it would take too much space to tabulate all these 
dimensions. More difficult to determine than the size of the pipes 
for heavy pressure is the size of the flange bolts. A formula is there- 
fore here given: 

4000 to 5000 (Z)^— rf2) 



number of bolts 



- = safe tensile strength of bolt. 



The factor 5000 is used for higher pressures. The length of thread 
for cast-steel and wrought-iron flanges may be made: 

/=2.2S {D-£) 
and for cast-iron flanges: 

S= 2.50 {D-d) 

f\=f-\-is in. and e=i^ to j in. The thickness of the copper 
gasket is usually not over g in. 



For high-pressure superheated steam or hydraulic work S. D. Love- 
kin, Chief Engr. New York Shipbuilding Company {Amer. Mack., 
June 8, 1905), considers the joint shown in Fig. 21 superior to all 
others. The faces are serrated and for steam a plain gasket of 
annealed copper is placed between them. For hydraulic work dealing 
with pressures up to 6000 lbs. per sq. in. Mr. Lovekin uses lead 
gaskets. 

The Van Stone or Walmaco pipe joint for high-pressure (250 lbs.) 
steam is shown in Fig. 22. In making this Joint the flange is slipped 
on the pipe, the pipe is brought to a red heat, the end is rolled over 
against the smooth face of the flange by a special machine which 
insures perfect contact and, finally, the pipe is placed in a lathe and 
a light cut is taken from the face which is to make the joint. Rubber 
gaskets are used for pressures up to 125 lbs. and copper gaskets for 
higher pressures. In some cases the ends are ground together. 
The advantages of the joint are: The pipe is not weakened by 
threads; the joint is made between the ends of the pipe; the flanges' 
simply act as collars to hold the ends of the pipe in contact; the 
flanges swivel, thus greatly reducing the labor of erecting the 
work. 

Table 20 gives the dimensions of this joint as made by the Wal- 
worth Mfg. Co. 

Pipe Markings 

The standard pipe markings of the American Society of Mechanical 
Engineers {Trans. A. S. M. E., Vol. 33) are as follows: 

In the main engine rooms of plants which are well lighted, and 
where the functions of the exposed pipes are obvious, all pipes shall 
be painted to conform to the color scheme of the room; and if it is 



268 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 





Fig. ig. 
Table 19. — Dimensions of High-pressure Hydraulic Pipe Flanges. 



Dimensions in Inches 


For 6 Bolts 


For 8 Bolts 


d 


D 


b 


e 


t 


/ 


i^ 


c 


S 


F 


c 


5 


3 


iVz 


3.6 


U 


H 


1'4 


934 


7U 


IH 








3'4 


m 


3.9 


'4 


H 


2 


lOVa 


7% 


1^8 








35^ 


514 


4.2 


He 


'4 


2'/8 


ll?i 


834 


iH 








4 


6 


4.8 


'A. 


H 


2^6 


12% 


9H 


iVs 


liH 


834 


IH 


4H 


6% 


5.4 


% 


H 


2'/2 


14 


10^2 


l?4 


13 


10 


IH 


5 


IVz 


6.0 


H 


H 


2^4 


1534 


1134 


2 


1414 


11 


15/8 


6 


9 


7.2 


H 


'4 


3 


18 '4 


1334 


2M 


1634 


13 


1% 


7 


10 H 


8.4 


\ 


'4 


3!4 


20^4 


1534 


2H 


1834 


1434 


2 


8 


12 


9.6 


% 


'4 


3'/2 


23 "4 


1734 


234 


21!'4 


1534 


2H 



A=S+^(i 




Fig. 21. — Pipe joint for high-pressure superheated steam or hydraulic 

work. 



desirable to distinguish pipe systems, colors shall be used only on 
flanges and on valve fitting flanges. 

In all other parts of the plant, such as boiler house, basements, 
etc., all pipes (exclusive of valves, flanges, and fittings) except the 
fire system, shall be painted black, or some other single, plain, 
durable, inexpensive color. 

All fire lines (suction and discharge) including pipe lines, valve 
flanges and fittings, shall be painted red throughout. 

The edges of all flanges, fittings or valve flanges on pipe lines larger 
than 4 inches inside diameter, and the entire fittings, valves and 
flanges on lines 4 inches inside diameter and smaller, shall be painted 
the following distinguishing colors: 





Fig. 22. — The Van Stone pipe joint. 



Distinguishing Colors 



Steam division 



Water division 



TO BE Used on Valves, 
Fittings 



Flanges and 



High pressure — white. 
Exhaust steam — buff. 
Fresh water, low pressure — blue 
Fresh water, high pressure 

boiler feed lines — blue and 

white. 
Salt water piping — green. 



PIPE AND PIPE JOINTS 



269 



Table 20. — The Van Stone Pipe Joint for Pressures up to 250 Lbs. per Sq. In. 

Dimensions in inches 




Long Hub Flanges 
made from cast-iron or ferrosteel 



Ring Flanges 
made from malleable iron or cast steel 




Long Hub Flanges 
made from rolled steel 




Short Hub Flanges 
made from malleable iron, cast steel or rolled steel 



Size 


4 


4i 


5 


6 


7 


8 


9 


10 


12 


14 


15 


16 


18 


20 


22 


24 


D. Diameter of hub, cast-iron or ferrosteel 


6A 


6H 


71 


8J 


9! 


loi 


Hi 


I3i 


ISf 


i6f 


I7i 


I9i 


2lh 


23i 


26 


28i 


/. Diameter of hub, rolled steel . . 


5l 


6i 


6i 


7J 


9 


lOi 


Hi 


I2| 


I4f 


16A 


17A 


18A 


20f 


22i 


24i 


27 








10 


I0§ 


II 


I2i 


14 


IS 


16 


I7i 


20 


22| 


23i 


2S 


27 


29i 


3ii 


34 






B. Thickness of flange, cast-iron ferrosteel, cast steel, or 
malleable iron. 


li 


lA 


l| 


lA 


ij 


If 


li 


li 


2 


2\ 


2A 


2i 


2f 


2i 


2f 


2i 


H. Thickness of flange, rolled steel, with long hub 


II 


li 


ij 


ll 


lA 


li 


lA 


li 


If 


li 


lit 


li 


2 


2i 


2i 


2A 




6f 


1\ 


7i 


9 


10 


II 


I2i 


13I 


I Si 


17 


18 


19 


2I| 


23i 


2Si 


27i 






7l 


8i 


9i 


lof 


III 


13 


14 


I5i 


I7¥ 


20 


21 


22i 


24i 


26i 


28i 


3li 




Number of bolts 


8 


8 


8 


12 


12 


12 


12 


16 


16 


20 


20 


20 


24 


24 


28 


28 








i 


i 


i 


3 


i 


I 


i 


f 


J 


i 


I 


I 


I 


li 


li 


li 






3i 


35i 


A\ 


4i 


4A 


4f 


4ii 


4ii 


sA 


s\ 


5l 


6 


6i 


6i 


6i 


7i 




F. Height of flange, rolled steel, with long hub 


3l 


3\ 


3\ 


3i 


3! 


3J 


3l 


3i 


4 


4f 


4i 


4i 


S 


Si 


Si 


6i 






E. Height of flange, cast steel or malleable iron 


li 


iH 


li 


2 


2A 


2* 


2\ 


2i 


2A 


2M 


2M 


2i 


3A 


3i 


3A 


3l 



Oil division 
Pneumatic division 

Gas division 

Fuel oil division 
Refrigerating system 

Electric lines and feeders 



f Delivery and discharge — brass 
\ or bronze yeUow. 

All pipes — gray. 

City lighting service — alumi- 
num. 

Gas engine service — black, red 
flanges. 

All piping — black. 

White and green stripes alter- 
nately on flanges and fittings, 
body of pipe being black. 

Black and red stripes alter- 
nately on flanges and fittings, 
body of pipe being black. 



The standard pipe markings of the ships of the United States Navy 
are given in Fig. 23 {Amer. Mach., Nov. 26, 1908). 



Outside of machinery spaces all pipes, except pneumatic pipes, 
are painted white (the general color of neighboring work). The 
contents of each pipe are indicated by distinctive color bands placed 
upon the flanges, or at intervals between the flanges, or in both 
places, as shown. 

The valves also are painted distinctive colors, indicating the 
contents of the pipe. 

The direction of flow of the contents of the pipes is indicated by 
a narrow color band (red or black) painted around the center of 
the band that indicates the contents of the pipe. 

In general, the narrow black band indicates the flow toward the 
motive power of the system, or toward an auxiliary, and the red band 
indicates the flow away from the motive power or auxiliary. Ex- 
cept ventilation pipes in the coal bunkers, under the fire and engine 
rooms and store-room floors in double bottoms and wing passages, 
all pipes are painted the color of the compartment and retain their 
distinctive bands. 



270 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Salt Water Pipes 
Sea Suction Piping to Pumps for Flushing Systems Etc. 
Salt Water Delivery from Pumps for Flushing Mains Etc. 



Fresh Water Pipes 
Fresh Water Suction to Pumps 
Tresh Water Mains and Branches 



Ventilation Pipes 




Ventilation Piping Supplying Air tolpCompartments 
whether Natural or Artificial 



Ventilation Piping Exhausting Air from'"' Compartments 
whether Natural or Artificial 



Steam Pipes 



K3 £^ 



^^K3 



Steam Supply to Auxiliary Exhaust Steam from Auxiliary 

Drainage Pipes j. Color of Wall of Comp't 



Where Neighboring Worl: is White Where Neighboring Worit is Bed 



Fire Jlain 



Flanges Lagged and White 
Bed Bands between Flanges 



Magazine Flood 



Magazine Flood direct from Sea Fire 
Main when used as Magazine Flood 



Pneumatic Pipes 



Entirely Black, No Indications for Direction 



Hydraulic Pipes 



Supply 



Key 



Exhaust 



Fig. 23. — Standard pipe markings of the United States Navy. 



Table 21. — Dimensions, Strength and Weight of Pelton 
Water Wheel Co.'s Double Riveted Hydraulic Pipe 



ft 
'5 

g 

Q 


H 0) 

& g, 
a'H 

16 


1'" 

> 


ft 

•u a 

"^ a 
•S " 

•0 ^ 

t ^ 






"ia 


ft" 


ft 

'a 

"0 

u C 

4J 
g 

a 
Q 




0) 

C 

".5 

i.s 

> 


0) 

ft 

•S " 

K 


■4^ 

§ 

H 

•9 

ft 

M 

'S 


3 


18 


.05 


810 


2^25 


18 


12 


.109 


29S • 


25. 2S 


4 


18 


• OS 


607 


3.00 


18 


II 


.125 


337 


29.00 


4 


16 


.062 


760 


3^75 


18 


10 


.14 


378 


32.50 


S 


18 


-OS 


48s 


3^7S 


18 


8 


■ 171 


460 


40.00 


S 


16 


.062 


605 


4^50 


20 


16 


.062 


151 


16.00 


S 


14 


.078 


7S7 


5.75 


20 


14 


.078 


189 


19.75 


6 


18 


• OS 


40s 


4^25 


20 


12 


. 109 


26s 


27.50 


6 


16 


.062 


505 


5^25 


20 


II 


.125 


304 


31. SO 


6 


14 


.078 


630 


6.50 


20 


10 


.14 


340 


35.00 


7 


18 


• OS 


346 


4-75 


20 


8 


.171 


41S 


45.50 


7 


16 


.062 


433 


6.00 


22 


16 


.062 


138 


17.7s 


7 


14 


.078 


540 


7.50 


22 


14 


.078 


172 


22.00 


8 


16 


.062 


378 


7.00 


22 


12 


.109 


240 


30.50 


8 


14 


.078 


472 


8.75 


22 


II 


.125 


276 


34 SO 


8 


12 


. 109 


660 


12.00 


22 


10 


.14 


309 


39^00 


9 


16 


.062 


336 


7.50 


22 


8 


.171 


376 


50.00 


9 


14 


• 078 


420 


9.25 


24 


14 


.078 


158 


23^75 


9 


12 


• 109 


S87 


12^75 


24 


12 


.109 


220 


32.00 


10 


16 


.062 


307 


8.25 


24 


II 


• 125 


253 


37-50 


10 


14 


.078 


378 


10.25 


24 


10 


.14 


283 


42. 00 


10 


12 


■ • 109 


530 


14.25 


24 


8 


• 171 


346 


50.00 


10 


II 


• 125 


607 


16.25 


24 


6 


. 20 


40S 


59^00 


10 


10 


• 14 


680 


18.25 


26 


14 


.078 


14s 


25 -50 


II 


16 


.062 


275 


9.00 


26 


12 


.109 


203 


35-50 


II 


14 


• 078 


344 


11.00 


26 


II 


• 125 


233 


.39-50 


II 


12 


. 109 


480 


IS. 25 


26 


10 


•14 


261 


44-25 


II 


II 


.125 


553 


I7S0 


26 


8 


.171 


319 


54-00 


II 


10 


• 14 


617 


19 SO 


26 


6 


.20 


373 


64.00 


12 


16 


.062 


252 


10. 00 


28 


14 


.078 


135 


27-25 


12 


14 


.078 


316 


12.25 


28 


12 


. 109 


188 


38.00 


12 


12 


. 109 


442 


17^00 


28 


11 


.125 


216 


42.25 


12 


II 


.125 


S06 


19.50 


28 


10 


.14 


242 


47.50 


12 


10 


.14 


567 


21^75 


28 


8 


.171 


295 


58.00 


13 


16 


.062 


233 


10, 50 


28 


6 


.20 


346 


69.00 


13 


14 


.078 


291 


13^00 


30 


12 


.109 


176 


39. SO 


13 


12 


. 109 


407 


18. 00 


30 


II 


• I2S 


202 


45.00 


13 


II 


.125 


467 


20.50 


30 


10 


•14 


226 


50.50 


13 


10 


.14 


522 


23.00 


30 


8 


• I7I 


276 


61. 75 


14 


16 


.062 


216 


11.25 


30 


6 


.20 


323 


73-00 


14 


14 


.078 


271 


14.00 


30 


i 


• 25 


404 


90.00 


14 


12 


. 109 


378 


19. so 


36 


II 


.125 


168 


54-00 


14 


II 


.125 


433 


22 .25 


36 


10 


.14 


189 


60.50 


14 


10 


.14 


485 


25.00 


36 


A 


.187 


252 


81.00 


15 


16 


.062 


202 


II. 75 


36 


1 


.25 


337 


109.00 


15 


14 


.078 


252 


14-75 


36 


A 


• 312 


420 


135 -00 


IS 


12 


. 109 


352 


20.50 


40 


10 


.14 


170 


67 -SO 


15 


II 


.'125 


405 


23.25 


40 


A 


.187 


226 


90.00 


15 


10 


.14 


■4S3 


26 . 00 


40 


i 


.25 


303 


120.00 


16 


16 


.062 


190 


13 00 


40 


A 


.312 


378 


150.00 


16 


14 


.078 


237 


16.00 


40 


1 


.375 


4SS 


180. 00 


16 


12 


. 109 


332 


22.25 


42 


10 


.14 


162 


71 .00 


16 


II 


• 125 


379 


24.50 


42 


A 


.187 


216 


94-50 


16 


10 


• 14 


42 s 


28.50 


42 


i 


•25 


289 


126.00 


18 


16 


.062 


168 


14.75 


42 


A 


• 312 


360 


158.00 


18 


14 


.078 


210 


18. SO 


42 


i 


• 375 


435 


190.00 



MINOR MACHINE PARTS 



Tapers 

The most commonly used standard tapers are the Morse and 
the Brown & Sharpe. The former has, nominally, a taper of f in. 
per ft. measured on the diameter, as are all tapers, but having been 
established before the days of accurate measurements, the different 
numbers range between .6 and .63 in. per ft. The Brown & Sharpe 
taper is i in. per ft., except the No. 10, of which the taper is .5161 
in. per ft. 

The most desirable taper is the Jarno, which is .6 in. per ft. or .05 
in. per in., which latter figure brings out its desirable features most 
clearly. The number system, instead of being arbitrary as with 
others, indicates the dimensions, the number of any taper being 
equal to the diameter in tenths of an inch at the small end, the 
diameter in eighths of an inch at the large end, and the length in 
halves of an inch. Again, the length is equal to five times the small 
diameter or four times the large diameter, any one of the dimensions 
being the key to the others. The leading machine tool builders 
who use this taper are the Pratt & Whitney Co. and the Norton 




Table i. — The Brown & Sharpe Taper 
Dimensions in inches 



u 

\ 

6 
3 , 


P. a 

n M 
a 

S 


J3 
0. 

M 

s 


V 

"o 
.0 

"0 
ft 

a 


■d 
c 

V 

"d 
>- & 
t° 


>> 

c 

0) 


(0 






"I „ 

M 
J! 
M 
C 

<u 


•s ^ 

n cm 
H 13 


4^ 



.2 

ft 
Ui 




D 


p 


H 


K 


i 


R' 


r 


< 




I 


.20 


a 


lA 


a 


f 


.135 


A 


i 


.500 


2 


.25 


lA 


lA 


iH 


} 


.166 


i 


A 


.500 


3 


.312 


2 


2j 


iH 


i 


.197 


A 


A 


.500 


4 


■ 35 


li 


If 


lU 


a 


.228 


H 


A 


.500 


S 


.45 


If 


If 


iH 


J 


.260 


1 


i 


■ 500 


6 


.50 


2| 


2j 


2il 


i 


.291 


A 


A 


■ 500 


7 


.60 


3 


3J 


2H 


« 


.322 


M 


A 


.500 


8 


• 75 


3* 


3U 


3H 


I 


.353 


i 


H 


.500 


9 


.90 


4 


4-J 


3J 


li 


.385 


A 


i 


.500 


9 


.90 


4i 


4f 


4i 


li 


• 385 


A 


f 


.500 


10 


I . 0446 


S 


Si 


4!5 


lA 


■447 


f^ 


A 


.5161 


10 


I . 0446 


sH 


SH 


sH 


lA 


■ 447 


a 


A 


.5161 


II 


I.2S 


(>i 


6J 


6H 


lA 


.447 


a 


A 


.500 


12 


I. SO 


7i 


7i 


6H 


li 


.510 


i 


i 


.500 


13 


I.7S 


7i 


7J 


7A 


li 


.510 


i 


J 


• Soo 


14 


2 


8i 


81 


8A 


iH 


.572 


a 


A 


.500 


IS 


2.2s 


Si 


8J 


m 


iH 


.572 


a 


A 


.500 


16 


2. SO 


9i 


9i 


9 


ij 


.635 


H 


f 


.500 


17 


2.7S 


9i 


9i 












.500 


18 


3 


loi 


lOf 












.500 



Grinding Co. The Reed taper is the same as the Jarno, but without 
the convenient relationship of numbers, diameters and lengths 
The Sellers taper is f in. per ft. In this taper the customary driving 
tang of twist drills and boring bars is omitted and in its place the 
socket is provided with a key and the shank with a keyway to fit. 
Unlike the tang, this key has ample driving power and eliminates 




Fig. I. — Taper of steam-hammer piston-rod ends. 

the well known trouble due to the twisting off of the usual tang. 
There is nothing to prevent its use with other tapers. The sockets 
being fitted with keys, it is only necessary to mill the keyways in 
twist drill shanks and thereby get rid of a universal nuisance. 

The following tables give dimensions of these various tapers, all 
dimensions being in inches. 

Table 2. — The Jarno Taper 
Dimensions in inches 



L 



Taper per ft. 
Taper per in. 



.6 in. 
.05 in. 



Dia. of large end- 



Dia. of small end = 



Length of taper ■ 



No. of taper 
' 8 

No. of taper 

10 
No. of taper 



Number 


A 


B 


c 


2 


.250 


.20 


I 


3 


.375 


.30 


Ij 


4 


.500 


.40 


2 


S 


.625 


.SO 


2i 


6 


.750 


.60 


3 


7 


.875 


.70 


3i 


8 


1 .000 


.80 


4 


9 


I. 125 


.90 


4i 


10 


1.250 


1. 00 


S 


II 


1.375 


1. 10 


Si 


12 


1.500 


1.20 


6 


13 


1.62s 


1.30 


6J 


14 


1. 750 


1.40 


7 


IS 


1.87s 


I. SO 


7j 


16 


2.000 


1.60 


8 


17 


.2.125 


1.70 


8J 


18 


2.250 


1.80 


9 


19 


2.375 


1.90 


9J 


20 


2.500 


2.00 


10 



271 



272 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 





Table 5. — The Sellers Taper 
Dimensions in inches 



A 


B 


c 


D 


B 


F \ C \ H \ I 


J \K 


\L 


A/ 1 AT 1 1 P 1 Q 1 R 


i 


I 


lii 


i 


i 


A 


2A 


2i 


i 


a 


3 


i 


A 


A 


A 


A 


A 


i 


§ 


I 


lil 


i 


i 


A 


2A 


2| 


i 


i 


35 


A 


A 


A 


A 


A 


A 


i 


^ 


lA 


2f 


i 


A 


i 


2II 


3l 


i 


Is 


4i 


i 


i 


K 


A 


A 


i 


A 


1 


If 


3f6 


I 


f 


A 


3i^ 


4l 


i 


if 


Si 


1 


a 


ii 


A 


A 


A 


i 


i| 


2 


4i 


I 


1 


1 


4f 


5 -J 


i 


If 


6i 


A 


i| 


M 


a 


i 


A 


i 


If 


2| 


Si 


i-J 


1 


i 


6i 


7l 


i 


2 


8 


i 


If 


a 


a 


A 


A 


A 


2| 


3i 


7i 


If 


1 




9l 
















a 


i 























Table 3. — The Morse Taper 
Dimensions in inches 



u 
ID 

ft 
CS 




Diam. 

plug at 

small 

end 


Stand- 
ard 
plug 

depth 


Depth 
of 
hole 


End of 
spin, to 
key- 
way 


Length 
of 
key- 
way 


Width 

of 
key- 
way 


Length 

of 
tongue 


Diam. 

of 
tongue 


Thick- 
ness 
of 
tongue 


Rad. of 

mill 

for 

tongue 


Uad. of 
tongue 
a 


Shank 
depth 


Whole 
length 

of 
shank 


Taper 
per 
foot 


Diam. 
at end 

of 
socket 


Diara. 

of point 

of 

shank 


Taper 
per 
inch 





D 


P 


H 


K 


L 


W 


T 


d 


i 


R 


a 


5 


B 




A 






I 

2 

3 

4 
S 
6 


.369 

.572 

.778 

1.02 

1.475 

2. 116 


2j 

2A 

3A 

4A 
SA 

7i 


2A 
2f 

3i 

4f 
Si 
7t 


2A 

2\ 

3A 

35 

4il 

7 


f 

i 

lA 
li 
i\ 
If 


.213 

.26 

.322 

.478 

.63s 

.76 


A ■ 

f 

A 

4 

1 

1 


.33 

3 

¥s 
iM 
2 


if 

i 

A 

1 

a 

* 


A 
1 

A 
A 

4 


.05 
.06 
.08 
.10 
. 12 
• IS 


1 
2i 

3A 

A\ 
Si 
8 


2A 
3A 
3l 

4l 

6 

8A 


.600 
.602 
.602 
.623 
.630 
.626 


■ 475 

.7 

.938 
I. 231 
1.748 
2.494 


.356 
.556 
.759 
.997 
1.446 
2.077 


• OS 

.05016 

.05016 

.05191 

.0525 

.05216 



08 -2-V, 



IZ'fZ ^i 
ZZ'OZ — %T— ^' 





Table 6.- 



-The Standard Tool Co.'s Taper for Twist Drill 
Shanks 
Dimensions in inches 













Length 




Number of 
taper 


Diameter 


Diameter 


Total 


Depth of 


of tongue 


Thick- 


small end 


large end 


length of 


hole in 


to end 


ness of 


of shank 


of shank 


shank 


socket 


of socket 


tongue 












hole 






A 


B 


C 


D 


E 


F 


I 


.378 


.484 


2i 


If 


i 


i 


2 


.587 


.706 


2i 


III 


A 


i 


3 


.800 


.941 


2if 


2i 


i 


i 


4 


1.050 


1.244 


. 3i 


3 


A 


i 


5 


I-SIS 


1.757 


4l 


3f 


1 


■ I 


6 


2.169 


2.501 


6f 


5 


i 


li 


7 


2.81S 


3.283 


9 


7i 


I 


li 



Table 4. — ^The Reed Taper 



Number of 


Width of 


End of 
socket to 
keyway 


Length of 


Diameter 


Taper 


Taper 


taper 


keyway 


keyway 


of socket 


per foot 


per inch 




G 


H 


7 


K 






I 


.263 


If 


f 


M 


.600 


.0500 


2 


.388 


i| 


I 


lA 


.602 


.05016 


3 


.520 


2 


li 


lA 


.602 


.05016 


4 


.645 


2ii 


14 


lii 


.623 


.05191 


5 


1 .020 


3i 


2 


2A 


.630 


.0525 


6 


1.270 


4f 


2i 


2j 


.626 


.05216 


7 


1.520 


7 


3 




.62s 


.05208 



MINOR MACHINE PARTS 



273 



An accurate method of originating taper gages is by the use of two 
disks and a pair of straight edges, Figs. 2 and 3, the diameters and 
center distances of the disks being such that the angle between 
their two common tangents is that of the desired taper. The taper 
being usually given in ins. per in. or ins. per ft., the equivalent angle 
must first be determined. If the taper is measured perpendicular 
to the center line as in- Fig. 2, this angle for most cases in practice, 
may be taken directly from Table 7 and we have the relation: 



D — d = 2lz\n a. 

in which Z) = diam. of large disk, ins., 

cf = diam. of small disk, ins., 

^ = distance between centers of disks, ins., 

included angle 
a= 

2 

= angle of one edge with center line. 



{a) 




Fig. 2. — Measured perpendicu 
lar to center line. 

Figs. 2 and 3. — Originating tapers, 



Fig. 3. — Measured perpendicu- 
lar to one side. 



If the taper is not found in the table, divide the taper per inch 
by two, when it becomes the tangent of half the included angle. 
Find the angle corresponding to this tangent in a table of tangents 
and use the value so found in the above formula. 

Table 8 gives diameters of disks and center distances for the 

Table 7. — Tapers and Corresponding Angles 



Taper, 

ins. per 

ft. 



Included 
angle 



Deg. 



6H 
He 



?i6 

% 

1%2 

Vie, 
1^2 

1%2 



2H2 
nu 

2^2 

% 
25^2 

•Mo 

2Jfe 

29^2 



Min. 



4 

9 

18 

27 

36 

4S 

54 

3 



30 
38 
47 
56 
S 

14 
23 
32 
41 
SO 



8 

17 
26 
35 

44 

53 

2 



Angle with 
center line 



Deg. 



Min. 



2 

4}^ 

9 
13}^ 
18 

22\i, 
27 

36 

40 

43 
49 

53^ 
58 

2^. 

7 
16 

20j/i 
25 

30 

34 

38K2 

43 

52 
S6}^ 



Taper, 

ins. 

per ft. 



Included 
angle 



Deg. 



3 Hi 









2 

2M 

2y2 

2% 

3 

3H 

3'/2 

3M 

4 

4K 

4'/2 

4M 

s 

51/2 



SA 



13 

14 

IS 

16 
17 
18 
20 



23 

24 
25 

26 
28 



Min. 



Angle with 
center line 



Deg. 



28 
37 
46 

22 
58 

33 

9 

45 

20 
56 

32 

43 

54 

4 

15 

25 
36 
46 
55 



14 
23 
32 
40 
48 



13 

14 



Min. 



14 
iSM 
23 
41 

59 

16H 

52}4 
10 

28 

46 

21}^ 

57 
32 

m 

4214 
18 
53 
271^ 

2M. 
37 

46 
20 

54 

2814 
2 



Morse and Brown and Sharpe standard tapers. The disks may be 
spaced at the required distance by spanning over them with a 
micrometer, in which case the sum of their radii is to be added to 
the center distance, or a distance piece may be placed between them 
equal to the center distance less the sum of the radii. 

If the taper is measured perpendicular to one side. Fig. 3, the 
entire taper per inch is the tangent of the entire included angle 
which as before may be found in a table of tangents (Table 7 not 
applying to this case) and we have the relations: 



D—d = 2l sin Of 
and D—d = 2bta.na 



(&) 
(c) 



in which & = base distance between centers. Fig. 3, remaining nota- 
tion as before. 

Formula (6) is to be used when the disks are spaced by a microm- 
eter spanning over them and formula (c) when a distance piece is 
placed between them with its ends perpendicular to the horizontal 
line of Fig. 3. 

Table 8. — Diameters and Center Distances of Disks for 
Measuring Tapers 

Morse Standard Tapers 



No. of 


Diameters of 

disks, ins. 


Distance 

between 

centers 

of disks, 

ins. 


Taper 


taper 


Large 


Small 


per m. 


I 

2 
3 
4 
S 
6 


2% 


H 

2 


2.4990 
2.4990 
4.9980 
4.8281 
4.7746 
7.1619 


•OS 

.05016 
.05016 
.05191 

■0525 
.05216 


- 


Brown and Sharpe Standard Tapers 


I 
2 
3 
4 
5 

6 

7 
8 

9 
10 

II 
12 
13 
14 
IS. 

16 

17 
18 


H 
Vie 

H 
Vs 

2 
2M 

3 

sH 

3H 


H 
He 
H 
Vie 

H 

H 
% 
I 

iH 

2 
2M 

2H 

3 


I . 5009 
I . 5009 
I . 5009 
I . 5009 
I . 5009 

I • So°9 
3.0019 
3.0019 
3.0019 
2.9043 

6.0038 
6.0038 
6.0038 
6.0038 
12 .0077 

12.0077 
12.0077 
12 .0077 


■S 

■ s 
• s 

-5 

■5 

•S 
•S 
•S 
■5 
.5161 

■S 

-s 
-s 

■ s 

-5 

•5 
•5 

■5 



18 



Split-ring expanding mandrels will hold well, and at the same time 
release freely when the nut is loosened, if given a taper of 3 ins. per 
ft. measured on the diameter. 

The taper required in steam-hammer piston-rod ends and similar 
pieces in order to permit separation and yet hold the parts together 
without keys, as determined at the Crescent Steel Works and shown 
in Fig. I, is I in. per ft. measured on the diameter. The taper should 



274 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 9. — S. A. E. Standard Split Cotter Pins 
The applications of these sizes are given in Table ga. 





B. W. gage, upper line; 




Shank 
length, 
inches 


nominal diameter in inches, lower line 




He 


13 

%2 


12 

K4 


II 


10 

%4 


8 

1^4 


6 

IM4 


Me 


* 














Ke 


* 














M 


* 


* 












% 


* 


* 












M 


** 


* 












y% 




* 


A 


* 








I 




** 


* 


* 








^H 




** 


+ 


* 








^y^ 






** 


* 


A 






1% 






** 


* 


* 






iK 








* 


A 






1% 








A 


* 


* 




i^ 








** 


** 


* 




2 










** 


* 


* 


2K 










A 


** 


* 


2K 












** 


** 


2M 














** 


3 














** 


No. drill for 
















hole 


48 


36 


30 


28 


21 


II 


2 



* Short series. 



' Long series. A — Arbitrary sizes. 



Table 9a. — Applications of S. A. E. Standard Cotter Pins 



"3.S 


S. A. E 


. bolts 




u. s. s 


. bolts 




Yoke and rod ends 


'w 


Pin 


Pin length 


6 
J5 


Pin 


Pin length 





Pin 


Pin length 


d 


"S 

_n 


Dia. 


Short 


Long 




Dia. 


Short 


Long 





Dia. 


Short 


Long 


Q 


3/[6 


















Me 


Me 


M2 


48 


M 


Me 


M 


% 


48 


Me 


M2 


56 


48 


Me 


yi6 


H 


48 


Ms 


Me 


H 


H 


48 


Me 


5i 


?4 


48 


%2 


H 


^A 


36 


"■A 


%2 


% 


Vi 


36 


H2 


?4 


J6 


36 


%2 


H 


% 


36 


lU 


%2 


5« 


% 


36 


%2 


H 


1 


36 


%2 


H 


% 


36 


M 


%2 


H 


% 


36 


H2 


% 


i'/^ 


36 


%2 


% 


I 


36 


9l6 


}6 


% 


m 


28 


%4 


I 


iM 


30 










56 


5-6 


I 


IK 


28 


7/04 


iW 


iH 


30 










'He 










^64 


1I/6 


1% 


30 










% 


M 


m 


1% 


28 


H 


iM 


1 1/2 


28 










'4 


H 


iH 


m 


28 


%i 


m 


i?4 


21 










I 


}i 


i^i 


m 


28 


%i 


1% 


2 


21 










1% 


^Hi 


iH 


2 


II 


1M64 


i?4 


2M 


II 










iH 


^%i 


m 


2K 


XI 


nu 


2 


2H 


II 










iH 


mi 


2 


2H 


2 


mi 


2M 


2M 


2 










n^ 


Wm 


2K 


2?4 


2 


mi 


2M 


3 


2 











have a length of 3 diameters, 
within the head. 



The enlarged end prevents breakage 



Taper Pins and Their Correct Use 

The diameter of drills for Pratt & Whitney taper pins may be 
obtained from Table 11 by C. Talbot {Amer. Mach., Jan. 28, 1912) 
The drill sizes given are to be used when the diameter of the work 
and the length of the pin are the same. If the pin is to be cut off, 
use drill C or D according to the end cut off. 

Correctly used, taper pins are capable of far wider application 
than they have received. When used under alternating stresses 



they should be given a key draft — opposing pins taking the alternat- 
ing stresses. A pair of rods joined together as shown in Fig. 4 will, 
if subjected to alternating pull and thrust, invariably work loose at 
the pins while, if made as in Fig. 5, they will give no trouble. Again 
crank arms subject to alternating stresses and pinned to a shaft as in 
Fig.6 will work loose, while, if pinned as in Fig. 7, they will not. 

The essential feature is the key draft which may be obtained by 
filing out the holes as shown, or, in the case of Fig. 5, the holes may 
be reamed a little too deep for a seat without the key draft and then 
a thin shim — even a piece of paper of substantial thickness — may be 
placed within the coupling and between the rods. The amount of 
opening required is but slight, the only essential being that there is 
enough to insure that each pin pulls positively in one direction only. 
A suitable diameter for the pins is one-third the diameter of the male 
member — two or more pins being used if the stresses call for them. 

Table 10. — Total Taper from Taper per Foot 



h 

a-" 

^.2 










Tape 


r ins. per foot 










0) 


A 


Iz 


I 


i 


1 


i' 


.62 


i" 


i* 


I 


li 


s^ 


.0002 


.0002 


.0003 


.0007 


.0010 


.0013 


.0016 


• OOI6 


.0020 


.0026 


•0033 


A 


.0003 


.0005 


.0007 


.0013 


.0020 


.0026 


.0031 


• 0033 


• 0039 


.0052 


.0065 


i 


.0007 


.0010 


.0013 


.0026 


.0039 


.0052 


.0062 


.0065 


.0078 


.0104 


.0130 


A 


.0010 


.0015 


.0020 


.0039 


.0059 


.0078 


.0094 


.0098 


.0117 


.0156 


.0195 


i 


.0013 


.0020 


.0026 


.0052 


.0078 


.0104 


.0125 


.0130 


.0156 


.0208 


.0260 


A 


.0016 


.0024 


.0033 


.0065 


.0098 


• 0130 


.0136 


.0163 


.0195 


.0260 


.0326 


1 


.0020 


.0029 


.0039 


.0078 


.0117 


.0156 


.0187 


.0195 


.0234 


.0312 


.0391 


rs 


.0023 


.0034 


.0046 


.0091 


■ 0137 


.0182 


.0219 


.0228 


• 0273 


.0365 


• 0456 


i 


.0026 


•0039 


.0052 


.0104 


.0156 


.0208 


.0250 


.0260 


• 0312 


• 0417 


.0521 


A 


.0029 


.0044 


■0059 


.0117 


.0176 


• 0234 


.0281 


•0293 


.0352 


.0469 


.0586 


1 


.0033 


.0049 


.0065 


.0130 


■ 0195 


.0260 


.0312 


.0326 


.0391 


.0521 


.0651 


tt 


.0036 


.0054 


.0072 


.0143 


■ 0215 


• 0286 


.0344 


.0358 


• 0430 


■ 0573 


.0716 


1 


.0039 


■ 0059 


.0078 


.0156 


.0234 


.0312 


• 0375 


• 0391 


.0460 


• 0625 


.0781 


11 


.0042 


.0063 


.0085 


.0169 


■ 0254 


• 0339 


• 0406 


• 0423 


• 0508 


• 0677 


.0846 


i 


.0046 


.0068 


.0091 


.0182 


.0273 


• 0365 


■ 0437 


• 0456 


• 0547 


.0729 


.0911 


a 


.0049 


.0073 


.0098 


.0195 


.0293 


• 0391 


• 0469 


.0488 


• 0586 


• 0781 


• 0977 


I 


.0052 


.0078 


.0104 


.0208 


■ 0312 


• 0417 


• 050 


• 0521 


.0625 


• 0833 


.1042 


2 


.0104 


.0156 


.0208 


.0417 


.0625 


• 0833 


.100 


.1042 


• 125 


.1667 


• 2083 


3 


.0156 


■ 0234 


.0312 


.0625 


■0937 


• 1250 


• ISO 


• 1562 


• 1875 


.250 


• 312s 


4 


.0208 


.0312 


.0417 


■0833 


.125 


• 1667 


.200 


■ 2083 


• 250 


• 3333 


• 4167 


s 


.0260 


■ 0391 


.0521 


.1042 


.1562 


■ 2083 


■ 250 


.2604 


• 3125 


.4167 


.5208 


6 


.0312 


.0469 


.0625 


.125 


.1875 


• 250 


• 300 


•3125 


• 375 


.500 


.62s 


7 


• 0365 


.0547 


.0729 


■1458 


.2187 


• 2,917 


.350 


.3646 


•4375 


•5833 


.7292 


8 


.0417 


.0625 


.0833 


.1667 


.250 


• 3333 


.400 


• 4167 


.500 


.6667 


■ 8333 


9 


.0469 


.0703 


.0937 


■187s 


.2812 


•375 


.450 


.4687 


.5625 


• 750 


•937S 


10 


.0521 


.0781 


. 1042 


.2083 


• 3125 


• 4167 


• 500 


.5208 


.625 


• 8333 


I. 0417 


II 


• 0573 


.0859 


. II46 


.2292 


•3437 


•4583 


• 550 


■ 5729 


.6875 


.9167 


I. 1458 


12 


.0625 


.0937 


•125 


.250 


•375 


■ 500 


• 600 


.62s 


.750 


I^OOO 


1.250 


13 


.0677 


. 1016 


• 1354 


.2708 


.4062 


■ S4I7 


.650 


.6771 


.8125 


1.0833 


I ■ 3542 


14 


.0729 


.1094 


.1458 


.2917 


•4375 


•S833 


.700 


.7292 


.875 


I. 1667 


1.4583 


IS 


.0781 


. I172 


.1562 


.3125 


• 4687 


• 62s 


• 750 


.7812 


• 9375 


1.250 


1.5625 


16 


.0833 


• 125 


.1667 


• 3333 


• 500 


■ 6667 


.800 


.8333 


I. 000 


1^3333 


1.6667 


17 


.0885 


.1328 


.1771 


■ 3542 


■ 5312 


• 7083 


.850 


• 8854 


I •0625 


I. 4167 


1.7708 


18 


.0937 


.1406 


.1875 


.3750 


• 5625 


• 750 


• 900 


.9375 


1^125 


1.500 


I-87S 


19 


.0990 


.1484 


.1979 


■3958 


•5937 


• 7917 


• 950 


.9896 


1^1875 


1.5833 


1.9792 


20 


.1042 


. 1562 


.2083 


.4167 


• 62s 


•8333 


1. 000 


I. 0417 


1^250 


1.6667 


2.0833 


21 


.1094 


.1641 


.2187 


■ 4375 


• 6562 


• 875 


1.050 


1.0937 


1^3125 


1.750 


2.1875 


22 


.1146 


• I719 


.2292 


.4583 


• 6875 


• 9167 


1 .100 


1.1458 


I^37S 


1.8333 


2.2917 


23 


.1198 


.1797 


.2396 


■ 4792 


.7187 


.9583 


1. 150 


I. 1979 


1^4375 


I. 9167 


2.3958 


24 


.125 


.1875 


.250 


.500 


• 750 


1 .000 


1.200 


I. 250 


i^SOO 1 


2.000 


2.500 



1 Brown & Sharpe Taper (except No. 
2Jarno & Reed Tapers. 
'Nominally Morse Taper. 
1 Sellers Taper. 



10). 



Fig. 8 shows a cheap, neat and entirely successful eccentric-rod 
construction of which the author has made many. The rod proper 



MINOR MACHINE PARTS 



275 



Table ii. — Reamer Drills for Taper Pins 



->>,a'k 



Lshortest Length 
- -'- of Pin Only 




Length 


No. 





I 




2 




3 


4 




5 


A 


.156 


.172 


.193 


,219 


.250 


.289 




C 


No. 27 


No. 


21 


No. 


IS 


No. 5 


No. 


A 


No. I 


• 75° 


D 


No. 29 


No. 


24 


No. 


17 


No. 8 


No. 


I 


No. P 




C 


No. 28 


No. 


22 


No. 


16 


No. 6 


No. 


A 


No. / 




D 


No. 30 


No. 


26 


No. 


19 


No. II 


No. 


2 


No. G 


I. 250 


C 
D 


No. 28 
No. 31 


No. 
No. 


23 

28 


No. 
No. 


17 
20 


No. 7 
No. 12 


No. 
No. 


I 
3 


No. H 
No. F 


1. 500 


C 
D 


No. 29 
No. 32 


No. 
No. 


25 

29 


No. 
No. 


17 
22 


No. 8 
No. 14 


No. 
No. 


I 
3 


No. H 

No. i 


I.750 


C 
D 


No. 30 
No. 33 


No. 
No. 


24 
30 


No. 

No. 


18 

24 


No. 9 
No. 16 


No. 

No. 


I 
4 


No. H 
No. D 


2. ooo 


C 
D 




No. 
No. 


26 
31 


No. 
No. 


19 
26 


No. 10 
No. II 


No. 
No. 


2 
S 


No. G 
No. C 


2.250 


C 
D 








No. 
No. 


19 
28 


No. II 
No. J9 


No. 
No. 


2 
8 


No. G 
No. B 


2.500 


C 
D 












No. 12 
No. 20 


No. 
No. 


2 
10 


No. F 
No. I 


2.7SO 


C 
D 












No. 13 
No. 22 


No. 
No. 


3 
13 


No. J 
No. 2 


3.000 


C 
D 












No. 14 
No. 24 


No. 
No. 


3 

14 


No. i 
No. 2 



is made from round bar stock, without any forge work whatever. 
At the left-hand end, where it enters the boss on the eccentric strap, 



3- 



m 



3 



Fig. 4. 



Fig. s. 





Fig: 6. Fid. 7. 

Figs. 4 to 7. — Correct and incorrect use of taper pins. 

it is turnedJ;o the largest even size which the stock will hold up to. 
The shoulder a at the right-hand end, is likewise made as large as 




Fig. 8. — Eccentric rod construction. 

the bar will allow, and the taper between the two ends provides an 
appropriate shape. The eye is a simple cast affair in halves, held 



Length 


No. 


6 


7 


8 


9 


10 


A 


.341 


■ 409 


-492 


-591 


.706 


.750 


C 
D 


No. P 
No. 










1 .000 


C 
D 


No. P 
No. 


No. W 
No. V 








1.250 


C 
D 


No. 
No. N 


No. W 
No. V 








1 .500 


C 
D 


No. 
No. N 


No. W 
No. U 




A 




1.750 


C 
D 


No. 
No. M 


No. W 
No. U 


A 


A 




2.000 


C 
D 


No. N 
No. L 


No. V 
No. T 




A 

a 




2.250 


C 
D 


No. N 
No. Z. 


No. V 
No. T 


ii 
A 






2.500 


C 
D 


No. N 
No. K 


No. V 
No. 5 


1^ 


n 




2.750 


C 
D 


No. N 
No. J 


No. U 




6-1 


i 


3-000 


C 
D 


No. N 
No. / 


No. U 
No. R 




ii 




3.250 


C 
D 


No. M 
No. H 


No. U 
No. Q 


A 

a' 




i 


3-500 


C 
D 


No. M 
No. G 


No. S 


A 
M 


ii 


Ii 
1 


3-750 


D 


No. M 
No. F 


No. T 
No. P 


A 


i 




4.000 


C 
D 


No. L 
i 


No. T 
No. 


A 
If 


i 




4.250 


C 
D 






A 


a 




4-500 


C 

D 






A 
! 




ii 


4-750 


C 
D 








a 
a 


fi ■ 


5.000 


C 
D 










ii 

1! 


5-250 


C 
D 








ii 

a 


ii 

ii 


5-500 


C 
D 












5.750 


C 
D 










A 


6 .000 


C 
D 










1 
A 



together with studs and having a socket into which the end of the 
rod enters. It may be of polished brass though painted cast-iron is 




Fig. 9. — Knuckle joint construction. 

better. Wear may be taken up by filing the joint, or paper may be 
inserted in the joint before boring the hole. 



276 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Fig. 9 shows a knuckle joint put together in the same way which 
serves its purpose just as well as much more expensive constructions. 
Like the eccentric rod, the head is preferably of cast-iron. 

DovetaHs and T Slots 

The leading dimensions of dovetail slides and gibs may be deter- 
mined in accordance with Fig. lo, and the formulas which accompany 
it, by John Richards {A Manual of Machine Construction). 
Two methods of arranging the adjusting screws are shown, of which 
the one at A, with an angular point, is correct, and the one at B, 
with flat end, is wrong. The screw A exerts its power on the line m, 
pressing the surfaces together at n. The one at B exerts its force 
parallel to the faces at n, and by forcing the gib into the corner 
opens the joint at n, defeating the very purpose intended. 

Either of the methods shown in Fig. 1 1 is preferable to the set-screw 
plan. The one at a, consisting of a wedge the whole length of the 
joint, or two wedges, one at each end, can be appUed in nearly all 
cases and is reliable in every way. There is full contact of aU sur- 
faces, and the rigidity of the joint is not impaired by the gib. 

The one at e is also reliable, but not so rigid as the other and re- 
quires more width for the saddle, which is frequently objectionable. 

Fig. 12 shows a kind of joint, designed by Mr. Richards and em- 
ployed very successfully for the cross slides of engine lathes. Twist- 
ing strains are successfully resisted because of the width between 



tt = -T— or thicker 



4 

5 

C = 2A or wider 
A_ 
5 

4- ^°-2 



d 



e=- 



or thicker 



-4-^ 



[>= = 



Am^ 



Table 12. — Dovetail Slides and Gibs 
Dimensions in inches 



A I 



21 

2i 
2l 

3 

37 

4 

47 

S 



.018 
.036 

.072 
.144 
.216 

.288 
.360 
•433 
.505 
.577 

.649 
.721 
.794 
.866 
1. 010 

1. 154 
1.298 
1.442 
1.588 
1.732 

2.020 
2.308 
2.598 
2.885 






B 



B 



.022 
.044 
.087 
.175 
.262 

.350 
.437 
.525 
.612 
.700 

.787 

.875 

.962 

1.050 

1.225 
1.400 

I. 575 
1-750 
1-925 
2 . 100 

2-450 
2.800 
3-150 
3-501 



. 027 
.053 
.105 

.210 

.314 

.420 
.525 
.629 
.734 
.839 

.944 
1.049 

I -153 
1-259 
1.469 

1.677 
1.888 
2.097 
2.307 
2.517 

2.937 
3.356 
3-776 
4-I9S 



1 D 


D 


D 


.144 


-152 


-163 


.216 


.228 


.244 


.289 


.305 


.326 


.361 


.381 


• 407 


.433 


.457 


.489 


.577 


.610 


.652 


.721 


.762 


.815 


.866 


.915 


.979 


1 .010 


1 .067 


1. 142 


I-IS4 


1 . 220 


I-30S 



. 176 
.264 

.353 

.442 
-530 

.707 
.883 

i .060 

!.237 

[.414 



h — ^^ 




t^-2 



Fig. 10. 



Fig. 14. 



■2H- 5-J 



-n 



P'^^ 




Fig. II. Fig. 12. 

Figs. 10 to 12. — Dovetail slides. 

the fulcra at a and e, and the surfaces are well protected from chips 
and dirt. Some layers of thin paper are placed in the joint at c, 
so the screws can be set up haird and leave means of compensation. 

Exact dimensions of dovetail slides and gibs may be determined 
from Table 12 which shows the amount to be added or subtracted 
in dimensioning dovetail slides and their gibs, for the usual angles 
up to 60 deg. The column for 45-deg. dovetails is omitted, as A 
and B are alike for this angle. 

In the application of the table, assuming a base with even dimen- 
sions, as in Fig. 13, to obtain the dimensions x and y of the slide Fig. 
14, allowing for the gib ; in. thick, the perpendicular depth of the 
dovetail being f in., and the angle 60 deg., look under column A 
for f in. and find opposite 5 = .360 in., which subtracted from 2 ins. 
gives 1.640 ins., the dimension x. To find y, first get the dimension 
1.640 ins., then look under the column for 60-deg. gibs, and find D 
(for C = j in.) to be .289 in., which, added to 1.64, gives 1.929 ins. 
as the value of y. 

The measurement of dovetails is greatly facilitated by the use of 
Tables 13 and 14 by W. A. Colt {Amer. Mach., June 4, 1914), which 
also include the formulas for use in cases not included in the tables. 

Standard T slots for machine tools are greatly to be desired. 
Table 15 and the accompanying sections give the well-considered 
proportions of Wm. Sellers & Co., which, providing as they do 
for all conditions, deserve general adoption. 




Table 13. — The Measurement of Acute Angle Dovetails 



D 


30° 


35° 


40° 


45° 


1 50° 


55° 


1 60° 


Values of x 


M6 


.1478 


-1303 


.1171 


.1066 


.0982 


.0912 


■ 0853 


% 


-2957 


.2607 


.2342 


.2133 


-196s 


-1825 


.1707 


Me 


-4436 


.3910 


■ 3513 


.3200 


.2947 


-2738 


.2561 


U 


-5915 


■ S214 


.4684 


.4267 


-3930 


■3651 


-341S 


Mo 


-7393 


.6518 


.5855 


-5334 


-4913 


-4563 


.4268 


H 


.8872 


.7821 


.7026 


.6401 


-5895 


-5476 


-5122 


lU 


I. 0351 


-9125 


.8197 


.7468 


.6878 


-6389 


-5976 


M 


I. 1830 


I .0429 


-9368 


.8535 


.7861 


.7302 


.6830 


51 6 


1.3309 


I -1733 


I .0540 


.9602 


.8843 


.8214 


.7683 


H 


1.4787 


1-3036 


I .1711 


I .0669 


.9826 


.9127 


-8537 


iMe 


1.6266 


1.4340 


I .2882 


1.1736 


I .0809 


I .0040 


.9391 


M 


1-7745 


I ■ 5644 


I -4053 


I .2803 


I. 1792 


I -0953 


I -024s 


'Me 


1.9224 


1.6947 


1-5224 


1.3870 


1.2774 


I. 1866 


I. 1099 


Ji 


2.0702 


I-8251 


1-6395 


1-4937 


I. 3757 


1.2778 


1.1952 


'Mo 


2.2181 


1-9554 


1.7566 


I .6004 


1-4739 


1.3691 


I .2806 


I 


2 .3660 


2.0858 


1-8737 


1.7071 


1.5722 


I .4604 


1 .3660 



MINOR MACHINE PARTS 



277 




Table 14. — The JMeasueement of Obtuse Angle Dovetails 



D 


30° 


35° 


40° 


45° 


50° 


55° 


60° 


Values o£ X 


He 


.0396 


.0411 


.0426 


.0442 


.0458 


.0475 


.0492 


M 


.0792 


.0822 


.0852 


.0884 


.0916 


.0950 


.098s 


Me 


.1188 


• 1233 


.1278 


.1326 


• 1374 


.1425 


.1478 


M 


.1584 


.1644 


.1704 


.1768 


.1833 


. 1900 


.1971 


Me 


.1980 


.2055 


.2130 


.2210 


.2291 


.2375 


.2464 


H 


■ 2377 


.2466 


.2557 


.2652 


.2749 


.28SI 


.2957 


lU 


.2773 


.2877 


.2983 


.3094 


.3207 


.3326 


.3450 


H 


• 3169 


.3288 


.3409 


.3536 


.3666 


.3801 


.3943 


Me 


.3565 


.3699 


.3835 


.3978 


.4124 


.4276 


•443S 


^A 


• 3961 


.4110 


.4261 


.4420 


.4582 


.4751 


.4928 


iMe 


.4358 


.4521 


.4688 


.4862 


.5040 


.5227 


.5421 


M 


•4754 


.4932 


.5114 


.5304 


■ 5499 


.5702 


• 5914 


'Me 


• S150 


.5343 


.5540 


• 5746 


.5957 


.6177 


.6407 


^^ 


• SS46 


.5754 


.5966 


.6188 


.641S 


.6652 


.6900 


iMe 


• 5942 


.6165 


.6392 


.6630 


.6873 


.7127 


.7393 


I 


.6339 


.6576 


.6819 


.7072 


• 7332 


.7603 


.7886 



The proportions are based on the use of square-head bolts where 
the size of the head is ij times the diameter of the body plus \ in. 
They represent three conditions: First, both parts of the slot, that 




J_tJ 



wmmmm 



y, ° 



oh 






cU 



'--A 






im/^MmwM^' 



m////////////////m 

Table 15. — -Wm. Sellers & Co.'s Standard T Slots 
Dimensions in inches 



\a\b\c\v\-e\f\g 



II 



If 



A 


i 


\ 


I 


1 


lA 


A 


I 


i* 


i* 


1 


lA 


1 


lA 


ii 


lA 


i3. 


li 


\ 


If 


1 


lA 


1 


lA 


if 


lA 


\ 


lA 


I ' 


lil 


I 


If 


lA 


U 


I 


If 


i\ 


2 


li 


iK 


lA 


lii 


ij 


iH 


li 


2i 


li 


2 


lA 


ij 


li 


^\ 


If 


2i 


li 


2A 


lA 


2A 


If 


2 A 


U 


2j 


a 


2| 



(i + A in. 
d+Ain. 
d-\-\ in. 
<i + iin. 
i + i in. 

d + Jin. 
d+J in. 
(i + iin. 



i.Sd + i in. 
i.S<i + i in. 
l.Sd + Ain. 
I- 5(^-1- A in. 
I-Sfi-I- A in. 

l.S<i+Ain. 
1.5'i+Ain. 
i-Sfi + Ain. 



for the head and for the body of the bolt, are finished; second, both 
are rough, that is, unfinished cores; third, the slot for the head is 
cored and that for the body of the bolt finished. 

In the first two cases the two parts of the slot are, of necessity, 
concentric, and only a moderate amount of clearance is required. 
In the last case the two parts are almost certain to be out of parallel, 
and it is obvious that more side clearance must be allowed for the 
head. As much width as possible without danger of allowing the 
head to turn is therefore provided. 

The consequences of the prevailing confusion of T slots may be 
greatly mitigated by the use of adapters between fixtures and 
machine-tool platens as shown in Fig. 15. Instead of making the 
fixtures with integral tongues, they are made with slots 0/ standard 
•width. A pair of adapters for each machine tool having the dimen- 



sion a to suit the slot in the platen and the dimension h to suit the 
tandard fixture slot will enable any fixture to be used on any tool. 

Face plates of lathes, boring mills, etc., should never have 8 or 16 
slots, but 12, which permits the use of either 3 or 4 straps, the former 
insuring against distortion, while the latter is most convenient in 
adjustment. 




Fig. 15. — Adapter for miscellaneous T-slots and fixtures. 

Shaft Couplings 

The accompanying tables, 19 and 20, of flexible shaft couplings 
represent the practice of the General Electric Co. {Amer. Mach., Sept. 
29, 1910). The form shown in Table 19 uses flat leather links and is 
self-explanatory. That shown in Table 20 uses two endless belts 
placed side by side on the forms or arms of spiders. The belting 
used is a specially prepared leather which is designed to be used with 
a tension of 400 lbs. per sq. in. of cross-section, the rating being given 
both in kilowatts and horse-power, per revolution. The work per- 
formed by couplings of this kind is greater than would be expected 
of the same cross-section of belting, owing to the absence of slippage 
and the fact that the leather is firmly supported by the other parts 
of the coupling. 

Silent pawls of different constructions are shown in Figs. 20-23. 

Fig. 20 illustrates the principle of a device used with the brake 
mechanism on heavy cranes and hoists in steel mills. The driving 
member A is shown rotating in the direction R, thus driving the 
ratchet B by means of the pawl P. When A reverses and moves 
in the direction L, the pawl is raised until it meets the stop pin F, 
which motion is caused by the change in position of the links D 
and E. The locked linkage now causes the spring clamp G to move 
with it, while the resistance due to its spring tension keeps the pawl 
raised above the ratchet teeth as long as rotation in that direction 
is continued. When A again begins to move in the direction R, the 
pawl instantly drops, as clamp G remains stationary during the 
change in position of Hnks D and E. 

In Fig. 21, the driver A rotates in the direction L, the pawl P is 
driven by A , through the contact of E at X, the pin B having merely 
moved the pawl about its axis D at the beginning of motion. The 
pin D, which carries the pawl, is itself carried by the sliding block E 
and the extended end of D also passes through an arm of the casting 
G, which latter is an easy fit on the shaft. When A begins to rotate 
in the direction R, the block E remains stationary, while the pin C 
advances against the projecting finger on the pawl P, causing it to 
rotate about D, thus bringing the point above the ratchet teeth. 
At this instant face H of sliding block E comes against the face Y, 
which rotates all the parts except the ratchet as far as the stroke is 
set, the reversal at the end of the stroke again bringing the pawl 
into engagement, and driving the ratchet. 

In Fig. 22, the arm A carrying the pawls PP is made to oscillate 
about the shaft 5 by a variable throw crank used for changing the 
length of feed. The ratchet disk R, which is keyed to the shaft, 
carries on its hub a split hub // to which is riveted a piece of sheet 
steel D, in which are cut slots YY at an angle of about 45 deg. with a 
radial line; into the pawls are driven pins XX. 

In feeding, the arm A moving in a counter-clockwise direction 
causes the pins XX in the pawls PP to ride down in the slot YY, 



278 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



(^ 



^-%^ .^. 






2= Number of Bolts 







-F 



BelatloD of Eej 
Bolt In all Cases 

10 Shaft and aboTQ 
to have 2 KejB 
I10°aiart 



-D--M 

Table i6. — Cast-iron Flanged Shaft Couplings 
Dimensions in inches 



B 



F \ G \ H 



M 



li 

If 



2\ 
2h 
2\ 

3 

3i 

35 
3i 

4 
A\ 

Ah 

S 

52 

6 
6i 

7 

1\ 



9 

9s 



9l 

9t 

l&l 

III 

12 

I2f 
I3i 

145 

iSi 

I6J 

l8 

195 

21 

23 

245 

26 
27 
29 

31 
325 

34 
36 
37i 
39 

41 



I5 
Ij 

2j 
2f 
3 

3l 
3f 
4i 
4i 
4l 

5J 

Si 

6 

6f 

6J 

75 

8i 

9 

9 

10 

Hi 
12 
1 2 -J 
135 
I4i 

IS 

ISi 

16^ 

I7l 

18 



2i 



li 


l^ 


3l 


If 


Ij 


4i 


Il% 


2i 


4H 


iH 


2| 


Si 


lii 


3 


Si 


III 


3f 


61 


iH 


3l 


6i 


iH 


4l 


7f 


2i 


4l 


8i 


2i 


4i 


H 


2A 


Si 


9f 


2i 


Si 


I0| 


2| 


6 , 


lOf 


2| 


61 


III 


2i 


6i 


I2i 


3A 


Ih 


I3i 


3l 


8i 


15 


3M 


9 


l6i 


3 if 


9\ 


175 


4i 


10^ 


19 


4-1 


iii 


20i 


4i 


12 


2lf 


s 


I2i 


22| 


si 


I3i 


24i 


St 


I4i 


25J 


5ii 


15 


26f 


sH- 


isf 


28J. 


6A 


16I 


29i 


6H 


I7i 


30I 


6H 


18 


32i 



II 



lT% 

Ire 



Key 



W\ X 



i 


i 


If 


« 


i 


4 


i 


A 


1 


2 


^ 


^ 


4 


A 


A 


\ 


2| 


ii 


J 


4 


f 


1 


1 


2^ 


i 


1 


S 


i 


1 


1 


2i 


i 


1 


S 


i 


A 


1 


2| 


1 


1 


S 


i 


A 


1 


2j 


i 


f 


s 


i 


A 


i 


3 


i 


1 


s 


3 


A 


1 


3l 




4 


6 


3" 


* 


1 


3l 




! 


6 


1 


1 


1 


3i 




1 


6 


1 


h 


1 


31 




i 


6 




A 


i 


4 


li 


i 


6 




A 


i 


4i 


li 


1 


6 


ji 


1 


i 


45 


li 


i 


8 


If 


ii 


i 


4l 


li 


I 


8 


li 


\ 


i 


Si 


li 


I 


8 


i| 


'rk 


i 


Si 


i| 


li 


8 


ji 


H 


1 


6i 


If 


li 


8 


jS 


1 


1 


6| 


li 


li 


8 


II 


H 


1 


6| 


i\ 


li 


8 


J 3 


lA 


i 


1\ 


i| 


i| 


8 


if 


ij^ 


1 


8 


If 


i| 


10 


If 


lA 


1 


8i 


If 


i| 


10 


li 


lA 


i 


81 


If 


If 


10 


2 


li 


1 


9i 


It 


15 


10 


2 


li 


h 


9l 


i| 


li 


10 


2i 


I5 


2 


loi 


li 


li 


12 


2i 


i| 


-1 


loj 


i| 


If 


12 


2i 


ij 


J 


II 


i| 


If 


12 


2i 



A 



a 



I 

ii 



thus forcing the pawls into engagement with the ratchet disks, 
thereby turning the device as a whole. Upon reversal the pins ride 
outward until they reach the end of the slots and then drag the sheet- 
steel piece, which has an adjusting screw in the split hub, back with 
them. 

In Fig. 23, a wheel, part of which is shown at .4, is given a variable 
back and forth motion, as indicated by the arrows. The friction 
piece B runs in a channel turned through the ratchet teeth, and is 
held by the spring and pin, and operates the pawl by alternately 
coming in contact first on one side and then the other of the conical 
hole at a and b. 

The diinensions of wrenclies may be obtained from Figs. 24, 25 
and 26 and Table 21. In Fig. 24 one-half the bolt diameter is divided 
into four equal parts by lines abed. The point where the circle e, 
representing the inside nut diameter, crosses line a locates the first 
center; the point where line/^, from the middle of one side to the 
corners, crosses line c locates the second center; the point where the 
30-deg. line crosses line c locates the third center, the center of the 
bolt being the fourth center. R may be made equal to the bolt 




Table 17. — Cast-steel Flange Couplings 
Dimensions in inches 





















Horse-power 




Size of 
shaft 


















rating per 
revolution, 
with 33,000 
lbs. torsion 


Approxi- 
mate 
weight 


A 


B 


c 


D 


E 


F 


G 


H 


/ 


in shaft 




I 


2 


A 


2 


I 


Si 


i 


i 




.0102 


10 


li 


2\ 


6 


3 


lA 


si 


i 


i 




.0200 


13 


I5 


2i 


6 


3 


li 


61 


1 


A 




.0346 


20 


i| 


3 


8 


4 


lA 


7f 


1 


A 




.0550 


27 


2 


3i 


8 


4 


lA 


81 


i. 


i 




.0821 


40 


2\ 


4l 


10 


5 


i| 


9i 


i 


i 




. 1604 


53 


3 


s 


12 


6 


lii 


lOi 


I 


1 




.2770 


80 


3l 


6 


12 


6 


2 


12J 


I 


1 




.4400 


130 


4 


61 


14 


7 


2A 


13 


I 


i 




.6670 


160 


Ah 


7i 


14 


7 


2A 


14 


I 


i 


1 


.93SS 


200 


5 


8f 


16 


8 


2| 


ISI 


li 


1 


3 


1.2832 


280 


si 


9 


16 


8 


2| 


l6f 


li 


1 


1 


I. 7081 


320 


6 


9l 


18 


9 


2f 


I7l 


li 


\ 


3 


2.2176 


410 


th 


lah 


18 


9 


2| 


19 


li 


i 




2.8195 


46 s 


7 


III 


20 


10 


3i 


20j 


i| 


i 




3.5210 


59S 


7i 


12 


20 


10 


3i 


2li 


li 


4 




4-3310 


660 


8 


I2f 


24 


12 


3i 


22i 


i| 


1 




5.2570 


840 


8i 


I3i 


24 


12 


3i 


23f 


i| 


a 




6.3050 


96s 


9 


I4i 


28 


14 


3H 


25i 


2 


1 




7.4840 


1315 


10 


16 


32 


16 


4 


27 


2 


1 




10. 2670 


1650 


II 


I7i 


32 


16 


4A 


285 


2i 






13-6650 


I9S0 


12 


19 


36 


18 


4l 


3li 


2i 






17.7410 


2580 


13 


20i 


36 


18 


4H 


33 


2i 






22. 5560 


2970 


14 


22 


40 


20 


s 


3Si 


2i 






28.1730 


3775 


15 


23j 


40 


20 


Si 


37i 


2i 






34-6510 


4320 


16 


2Si 


48 


24 


si 


39i 


3i 


li 


I 


42.0530 


5730 



diameter. Angle wrenches, Fig. 26, are better than the straight 
pattern as they may be used in more confined places. 

Jig and Fixture Details 

Fixture cams may be made self-locking by keeping the rise of the cam 
within the angle of repose. Using a mean value for this angle the 
rise of the cam. Fig. 28, for each 9 deg. of arc is given by the 
formula 

X = .005 d 
in which x = rise of cam, ins., for each 9 deg. of arc 
d = diameter of cam, ins. 

When fixture cams are placed on horizontal axes the handles 
should be so placed that their weight will tend to tighten the cams 
as in Fig. 29, and not as in Fig. 30. 

Suitable clearances between punches and dies for accurate work 
are given in Table 38 by E. Dean {Amer. Mach., May 4, 1905). 
The table relates to the blanking, perforating and forming of flat 
stock in the power press for parts of adding machines, cash regis- 
ters, typewriters, etc. 

In this class of work it is generally desired to make two different 
kinds of cuts with the dies used. First, to leave the outside of the 
blank of a semi-smooth finish, with sharp corners, free from burrs 



MINOR MACHINE PARTS 



279 



H-H->k — J—^ 



i. 



IjWI LatiJ Lanl Udn 



fEU 



U-2iJ->l I 
1 I I 



I ■ I ■ I I 



-H^k- 



iJt 



#+^- 



rqln rrrpi nlfm 




H -i^ ^ 



Table i8. — Clamp Shaft Couplings 
Dimensions in inches 



D 


A 


B 


c 


£ 


F 


G 


H 


^ 


Vi 


R 


S 


Weight, lbs. 


lA 


6 


4i 


2h 


I 


3l 


\ 


li 




A 


if 


1 


i8.7 


lii 


7 


4J 


3 


lA 


4 


■h 


\\ 




A 


lA 


f 


29.0 


in 


8 


Si 


3i 


If 


4i 


i 


2 




A 


lA 


i 


42.7 


lA 


9 


61 


3i 


n 


4J 


1 


2i 




A 


lA 


f 


57.9 


2A 


10 


6i 


3i 


If 


Si 


i 


2i 




A 


lA 


i 


78.9 


2ii 


II 


7i 


4 


ij 


S 


i 


li 


2i 


A 


lA 


f 


94.6 


2^ 


12 


8 


4i 


2 


Si 


i 


Ij 


3 


iJ 


lA 


f 


125.2 


3A 


13 


8f 


4i 


2} 


si 


1 


If 


3i 


iJ 


lA 


i 


149.5 


3A 


14 


9 


4i 


2\ 


6 


i 


Ij 


3i 


H 


lA 


1 


181. 7 


3« 


IS 


9i 


5 


2\ 


6i 


1 


I| 


3i 


H 


lA 


f 


228.6 


3ii 


i6 


lOi 


Si 


2i 


7 


i 


2 


4 


'H 


lA 


I 


277.9 


4* 


i8 


Hi 


6 


3 


7i 


1 


2i 


4} 


H 


lA 


I 


389.8 


4ii 


20 


I2i 


61 


3f 


8} 


i 


2i 


S 


M 


lA 


li 


496.7 



• -A— - — *• 




Table 19. — Leather Link Flexible Couplings 
Dimensions in inches 





Dimensions 


Rating per revolu- 
tion with 400 lbs. 


Max. 
r.p.m. 


Weight 
in lbs. 


Bore 


A 


B 


c 


B 


E 


F 


per sq. in. tensile 
stress in belt 




Kw. 1 H.p. 




\ 


if 


3i 


2\ 


li 


if 


f 


.0012 


.0016 


1800 


2f 


I 


2 


S 


4 


iH 


f 


f 


.0032 


.0043 


1800 


7i 


li 


2i 


6f 


6 


2M 


lA 


f 


.0076 


.0X02 


1800 


IS 


2 


3i 


8i 


8 


3M 


If 


f 


.0149 


.0200 


1800 


27 


2i 


4f 


9i 


10 


4if 


I A 


3 


.0258 


.0346 


1800 


43 


3 


S 


III 


12 


sU 


If 


I 


.0410 


■ osso 


1800 


75 


Z\ 


6 


I2| 


12 


SM 


lif 


If 


.0612 


.0821 


1800 


103 



and with the least amount of rounding on the cutting side. Second, 
to leave the holes and slots that are perforated in the parts as 
smooth and straight as possible, and true to size. The table is the 
result of three years' experimenting on this class of work, and has 
stood the test of three years of use since it was compiled and it 
has worked out to the entire satisfaction of those who have used it. 
The die always governs the size of the work passing through it. 
The punch governs the size of the work that it passes through. 




\*~P-*f G »j 

Table 20. — Laced Leather Flexible Couplings 
Dimensions in inches 



Bore 


Dimensions 


Rating per revolu- 
tion with 400 lbs. 
per sq. in. tensile 
stress in belt 


Max. 
r.p.m. 


Weight 
of com- 
plete 
in lbs. 


A 


B 


C 


D 


£ 


F 


G 


H 


/ 




Kw. 


h.p. 


I 


2 


5 


3 


lA 


H 


2A 


\ 


f 


.0032 


.0043 


1800 


8 


li 


3 


8} 


4 


iH 


8 


3A 


f 


A 


.01492 


.02 


1500 


27 


2 


3i 


9i 


5 


2A 


lA 


3if 


i 


i 


.02S8 


.0346 


1500 


39 


3 


5 


iSi 


6 


2if 


If 


sA 


i 


i 


.1196 


.1604 


1200 


IIS 


4 


61 


l8i 


8 


3if 


2A 


si 


I 


i 


.2066 


.277 


900 


189 


5 


8f 


24§ 


10 


4H 


3A 


6if 


li 


f 


.4901 


.6S7 


750 


367 


6 


9f 


3oi 


12 


sM 


4A 


7A 


li 


I 


.9573 


1.2832 


600 


611 


7 


llf 


37 


14 


611 


sA 


8i 


li 


I 


1.654 


2.2176 


450 


1033 


8 


I2f 


43 


16 


7if 


si* 


9f 


If 


i 


2.627 


3.S2IS 


350 


1527 


9 


isi 


49 


18 


SM 


6ii 


9l 


2 


\ 


3.9214 


5.2566 


300 


2201 


ID 


I6J 


49 


20 


9H 


7i* 


9s 


2 


s 


3.9214 


5.2566 


300 


2376 


II 


i8f 


SS 


22 


loH 


8ii 


loA 


2i 


I 


S.S833 


7.4844 


250 


3171 


12 


I9f 


SS 


24 


iiH 


9ii 


loA 


2i 


I 


5.5833 


7.4844 


250 


3439 


13 


21J 


61 


26 


I2if 


loif 


iiA 


2i 


I 


7.6S9I 


10.2669 


200 


448s 


14 


23 


61 


28 


13x1 


nil 


iiA 


2i 


I 


7-6591 


10. 2669 


200 


4831 



Table 21. — Forged Wrenches 

See Figs. 24, 25 and 26 for notation. 
Dimensions in inches 



D 


L 


ri 


2^2 


'£Z 


51 


52 


1 


S 


3 


1 


1 


3 


5 


2 


8 


4 


8 


4 


8 


5 


6^ 


3 


1 


1 


15 


11 


8 


8 


4 


8 


16 


16 


3 


8 


1 


5 


3 


li 


3 


4, 


2 


16 


16 


4 


7 


«i 




5 


3 


-1 


13 


8 


92 


2 


16 


16 


I4 


16 


I 


II 


f 


1 


1 
4 


If 


7 
8 


li 


12I 


5 
8 


1 


1 
4 


li 


if 


li 


14 


3 
4 


1^ 


5 
16 


If 


I 


If 


iSl 


1 
4 


i^ 


1^ 


l| 


^h 


il 


17- 


3 
4 


1 
2 


3 

8 


l| 


li 


if 


18 


\ 


\ 


5 


2 


IT^ 


if 


19 


7 
8 


9 
16 


3^ 


2\ 


li 


i| 


20 


1 


9 
16 


7 
16 


2i 


lA 


2 


21 


I 


S 
8 


\ 


2f 


If 



In blanking work the die is made to the size of the work wanted 
and the punch smaller. In perforating work the punch is made 
to size of work wanted and the die larger than the punch. The 
clearance between the die and punch governs the results obtained. 
Fig. 31 shows the application of the table in determining the 



280 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 




d= .iD+.^ in. r=.2D — iin. 

Fig. i6. — Hand wheels. 









Fig. 19. — Machine tool knobs. 




D= . I3SL+^ in. d= . iL+i in- 

Fig. 17. — Ball cranks. 





Fig. 18. — Machine tool handles. 



I'iG. 23. Fig. 22. 

Figs. 20 to 23. — Silent pawls. 

clearance for blanking or perforating hard rolled steel .060 in. thick. 
The clearance given in the table for this thickness of metal is .0042, 
and the sketch shows that for blanking to exactly i in. diameter this 
amount is deducted from the diameter of the punch, while for per- 
forating the same amount is added to the diameter of the die. For 
a sliding fit make punch and die .00025 to -0005 in. larger; and for 
a driving fit make punch and die .0005 to .0015 in. smaller. 

The most satisfactory plug cock known to the author is the Westing- 
house construction, Fig. 32, which, almost from the beginning, has 
been a standard feature of air brake equipment. In view of its 
entire success there and the universally recognized defects of the 
common construction, it deserves general adoption. The handle 
is placed on the small end of the plug, pressure on the large end 



MINOR MACHINE PARTS 



281 



I 




ft 






Fig. 24. 


\r 


^T^ 










7^^^^ 


^ 










i \ 


•Vih^ '''' 




- 






— hV 


- L 


i^ 


-Ei 









— r"- 


-J. 






1^ 


j 


T?l^ «H 




-^H-J 


V 








Fig. 26. 



Figs. 24 to 26. — Dimensions of forged wrenches. 



holding the parts in place and automatically taking up 
wear. The taper is 2 ins. per ft. measured on the 
diameter. For frequent service (rock drill throttle 
valves) the author has used the blunter angle of 35 ins. 
per ft. measured on the diameter in order to obtain 
greater ease of movement. In air brake service security 
against movement is of course essential. Note that 
there must be a hole through the end of the plug to 
admit pressure to the large end, or the plug will not 
seat itself. 



Forming Tools 

When two or more diameters are required on a circular 
forming or cutting-ofi tool to produce corresponding sizes 
on the work, the difference in diameters of the tool is 
less than the difference on the work, because the cutting 
edge of the tool is dropped below its center to provide 
the necessary clearance. The relative difference in- 
creases as the smaller diameter decreases. The method 
of calculating the diameters of a tool is as follows: 

In a right-angle triangle. Fig. 35, the short side B equals 
the amount the cutting edge is below the center of the 
tool; the hypothenuse A equals the radius of the tool. 
Find the long side C, which is the horizontal distance 
from the cutting point to the vertical center line. 

From this dimension as a constant, subtract half 
the difference in diameters of work D. Take the remain- 
der E as the long side of a new triangle. Fig. 36, using 
the same short side 5, as before, and find hypothenuse 
F or the new radius, which, doubled, gives the cor- 
rected diameter G for the tool, Fig. 37. 

{Continued on page 28%second column) 





0>^> 



Fig. 28. 



Fig. 29. 
Correct Way 



Fig. 30. 
Wrong Way 

Figs. 28 to 30. — Self-locking fixture cams. 



Blanking 



Diameter of tap = D 
b = 24Z) for double ended wrench 
b = i6D for single ended wrench 
l = D di=D-^m. d2=D+^ia. 
r = D+r^'m. a = D+iin. 
Fig. 27. — Tap wrenches. 




Punch 



Fig. 31. — Location of allowance in blanking and punching. 



282 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Tool 




WC-H I 



0.H5 



NO.OO 




'TT 



_X- 



NO.O 




0.(875 




DimSDlAM. AMT. TOADD 



DIFF. INDIAM. AMT, TO ADD 



DIFF.INDIAM. 



AMT. TO ADD 



DIFF.INDIAM. 



AMT. TO ADD 



0.050 — -_ — 0.0005 



0.1 00 



0.1 50 



O.SO0 



0250 



0.300 



O.J 50 



0.4 00 



0.4 50 





'— 


-O.OOI 
0.0015 




'— 






0;0025 
-0.005 
0.0055 








7— 


O.O045 
-0.005 


d 


0.0055 
-0.00 6 


/ 


^-. 



0.0 50 



0.100 ■ 



0.1 50 



0.200 



0.2 50 



0.300 =: 



0.3 50 




0.40O 



0.4 50 



0.500 



0.550 



o.eoo — -_-. 



0.650 



0.700 



0.0005 



■O.OOI 



0.0015 



-J 


" 


- 0.002 
0.0025 


1 


- 




-f 


- ■ 


- 0.003 


-^ 


-— 


0.0035 


— 


- 




___ 


- 




~~: 


: 


0.0045 


. ~ 


- 




- 




li 


- 


- o.ocrs 


_; 


r- 


0.0055 


^ 


- 


- 0.006 


-E 


'— 


0.006§ 


-r 




- COOT 


- 






-i 


E— 


0.0075 


-_ 


_ 


- O.OOS 


-z 


'— 


0.0085 


—. 




- 0.009 


n 


I— 


0.0095 



Fig. 37, 





0- 








— 


- 


0.050 


-9 


^ 


0.100- 


^ 


I 


0.150 


_^ 


: 






- 




- 


0.200- 




I 




-^ 


I 


0.250 


-^ 


- 


0.500- 


-IE 


: 


0.550 




'— 


0,400- 




'— 


0/450 




: 


0.500- 




: 


0.550 








—z 


- 




- 




O.600 - 


: 


■" . ( 


0.650- 


-^ 


2 


0.700- 


-^ 




0.750 


^ 


r < 


asoo- 


^ 





0.001 
0.001 5 

If 0.002 

^ 0.002 5 
0.00 5 
0.0035 
0.004 
0.0045 
0.0 0'5 
0.0055 
0.006 
0.0065 
0.00 7 
0.007 5 
0.008 
0.0085 
0.009 
0.0095 
0.010 
0.0105 
0.0 II 
0.0115 
0.012 
0.0125 
0.013 



CO ^^ 



"^ 



Fig. 39. 



Fig. 36. 



0,050 


.. - 


't- 


0.100- 


—. 


't- 


0.150 


A~ 



200 




- 




: 


0.250 




: 






: 


0300 




: 


0.5 5 




f- 


0.400 


\- 


0>»50 




[_ 




-i 


z 


O.50O 




[_ 






: 


0ii50 
0.600 • 




E— 


0.65 




i_ 






L. 






z ■ 


O.^SO 


3 


\ 










-^ 


1— 


0.850 




\ 



0.900- 



0.350 



1.000 



1.050 



1.100- 



1 150 



1.200- 



0.000 5 
O.DOi 

0.0015 

0.002 

0.0025 

0.003 

O.O035 

0.004 

0.0045 

0.005 

0.0055 

0.006 

O.O065 

0.007 

0.0075 

0.006 

O.0O85 

0.009 

0.0095 

0.010 

00105 

0.011 

0.0115 

0.012 

0.013 
Q.014 
0.01.5 
0.016 

- aoi7 

- 0.018 

- 0.019 
-0.020 
-0,02f 

- 0.022 

■ 0.025 

■ 0.024 
0.025 
0.026 
0.027 

• 0.028 
0.029 



1.250 



Fig. 38. — Allowances for differences of diameters of forming tools. 



MINOR MACHINE PARTS 



283 





Fig. 32. — The Westinghouse plug cock. 




IH^ 



i& 



-I-2-t- 






J 



Li^fl-^m^ 



A = radius of cutter, 

B = distance of cutting edge below center of the tool, 

C = distance of cutting point from vertical center of tool = \/^ ^— B ^, 

Z) = half the difierence of the diameters of work, 

E = C -D, 

F=-\/E^+B\ 

G = 2^ = corrected diameter. 

Fig. 38, by C. E. Braman {Amer. Mach., Dec. 5, 1912) has been 
calculated from the above equations and laid out for the Brown and 
Sharpe automatic screw machine circular tools. 

The use of the chart is best shown by an example. Required a 
form tool for the piece of work shown in Fig. 39 to be made on the 
No. 2 automatic the outside diameter of the tool being 3 ins. 




Depth of washer below top of foundation = 50X diameter of bolt. 

(The washers are usually square.) 
A =diam. of bolt+J in. (for |-in. bolts and smaller). 
A =diam. of bolt+l in. (for i-in. to 2|-in. bolts). 
A =diam. of bolt+t in. (for 2f-in. and larger). 
B = width of nut or head across flats + i in. to | in. (square 

nuts or heads usually used on lower ends of bolts). 
C = 8Xdiam. of bolt. 
-^ = -37SXdiam. of bolt (not over i in.). 
£ = .5 Xdiam. of bolt. 

Fig. 34. — Foundation bolt washers. 



The difference in the diameters of work, .375 and .625 = .250; 
the apparent second diameter of the tool = 3 — .250 = 2.750; the 
amount to add to the apparent diameter corresponding to the differ- 
ence .250 is found from the chart. Opposite .250 on the scale under 
the heading Diference in diameter is given Amount to add = .0038. 
The corrected second diameter = apparent diameter plus allowance 
= 2.750+. 0038 = 2. 7538. The difference in diameters of the work 
.375 and I =.625; the apparent third diameter of the tool = 3 — .625 
= 2.375. The amount to add to the apparent diameter correspond- 
ing to the difference .625 = .0109. 'The corrected third diameter 
= apparent diameter plus allowance = 2.375-|-.oi09 = 2.3859. 



Fig. 33. — Rack for bar stock. 



284 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 22. — Malleable Iron Cross Knobs 




B 


c 


D 


E 


F 


G 


T-5-" 

I32 


A" 


\" 


^" 


_9_" 
16 




J 11" 

I 16 


7 '/ 

32 


9 '/ 

16 


5 " 

16 


9 " 
16 


5 '/ 

16 


2" 


5 '/ 
16 


11'/ 
16 


3// 

8 


1" 


t" 


2^" 


11 '/ 
32 


3" 

4 


^" 


3" 

1 


Zff 


2^" 


1" 


1" 


3" 

8 


1" 


f" 



z? 



7 // 
16 

1" 
2 

5" 
8 

9 // 
■16 
5" 



Table 23. — Handles — Straight 




Rough 
diam. 



JL." 
16 

1" 
2 

1" 
2 

1" 
2 
9 '/ 



Finish 
diam. 



T^ 

JL'' 



5 


c 


D 


. 9 // 


19// 


- 31'/ 


2l6 


32 


I32 


iV 


1 


2f" 


31" 


3" 

4 


3" 


4^" 


7// 
8 


5" 
38 


If 




If 


5 


I 


4 


-3// 


^11 


^ 3" 


Si 




44 



Table 24. — Handles — Bent 
G 




A" 
4 


-24 


A' 


'-I8 


f" 


-16 


A' 


'-14 


\" 


-13 


A"-I2 


6" 
8 


-II 


3" 
4 


-10 



5 


c 


Z) 


£ 


F 


1" 
2 


1 // 

2 


5" 
8 


1// 

2 


H" 


4" 


¥' 


1" 


i" 


15" 
16 


1" 
5 


2 


511 
8 


*" 


if" 


3" 

4 


3" 

4 


l" 


H" 


lA" 


3" 

4 


i" 


l" 


H" 


lA" 


l" 


1" 


li" 


3" 
4 


li" 


l" 


in 
8 


li" 


3ff 


I}" 


l" 


lit 
8 


li" 


i" 


Ij" 



2t 
^3" 



45 



Table 25. — Single and Double Handled Wrenches 




Single handle 



If 
/>i" 



C 



3* 



D E 



5 " 

16 
5 " 



1" 3" 



£r K 



Double handle 



25 

-3" 



D E F G\H 



1 n 
16 
7 // 

16 

1" 

2 

5" 



1" 
2 

1" 
2 

1" 
2 

i" 



is: 



Table 26. — Malleable Iron Knobs 




1 . 



u-e- 



Size of hole 



7 " 
i« 
1" 



4 


B 


c 


il" 


\" 


t" 


li" 


3ri 

4 


3" 

8 


li" 


i" 


1" 


li" 


f" 


3" 

8 


2" 


lA" 


1" 


2" 


lA" 


1" 
2 


2" 


lA" 


*" 


2^' 


Ij" 


5" 
8 


21" 


li" 


1" 



D 



1" 

1" 






_^ Clearancei j 1 



.t-L — 



-U ...L 

Clearance I ^ 

' — V. — J — rr 




Table 27. — Two Standard Punches 
Dimensions in inches 



Table 28. — Round Dies 
Dimensions in mches 







a 




J 


c 


d 


e 


/ 


« 


Clearance 


1 


A 


A 


1 
4 


i 


2| 


f 


i§ 






.005 


i 


A 


A 


5 

32 


f 


2| 


f 


li 


i 

8 


1 
2 


.003 


A 


7 

32 


i 


A 


3 

8 


2§ 


5 

8 


i4 






.003 


s 

32 








f 


2| 


1 


I* 






.003 


i 








i 


2i 


1 


i4 






.002 


f 








H 


3f 


I 


i| 






.cos 


f 








ii 

16 


3f 


I 


li 






.005 


M 








H 


3l 


I 


li 






.005 


f 








11 

16 


3l 


I 


li 






.005 


A 


H 


f 


M 


4 


2i 


1 


i| 


1 


1 


.003 


A 








h 


2* 


5 

8 


i4 


i 


f 


.003 


3 


to A 






h 


3 


1 


i4 






.003 



a 


h 


d 


/ 


e 


J 


c 


\ 


A 


A 


7 
T2 


f 


3 

* 


1 


a+A 


1 
4 


3° 


i 


9 

32 






1 


1 


1 


a+A 


4 


3° 


A 


4 


A 


A 


4 


4 


3 

4 


a+A 


1 
4 


3° 


7 

32 


4 


9 

32 


5 

16 


7 

8 


7 

8 


1 


o+A 


4 


3° 


^ 


5 


41 


A 


I 


I 


i 


a+A 


3 

8 


3° 


41 


4 






I 




1 


a+A 


3 

8 


3° 


M 


A 


19 

32 


s 

8 


I 




7 

8 


a+A 


4 


3° 


14 


44 






I 




7 
8 


a+4 


4 


3° 


e 


i 


If 


13 
16 


I 


15 


4 


a+4 


4 


3° 


fi 


1 






I 


i4 


4 


a+4 


4 


3° 



MINOR MACHINE PARTS 



285 



>' — M- 




1 -m-»i 

1 
/ 








V 


'll^W 111 


-^ 






lllUilllll 


V 







D 


Thrd. 


L 


r 


/l 


m 


H 


M 


s 


1 

i 




1 


3 

4 


1 
4 


3 


1. 

4 


7 
10 


1- 

•^ 32 




1 


3 

4 


5 
10 


1 
8 


10 


1 

? 


1^ 


3 

e 


li 


1 


3. 

8 


1 
8 


3_ 

8 


8 


2 


7 
IC 


2 


l| 


7 
16 


3 

10 


X 

10 


11 

10 


2| 


1^ 


2 


li 


1 

2 


3 
10 


1 

2 


13 
10 


2i^ 



Table 29. Collar Head Jig Screws 




D 


Thrd. 


H 


L 


r 


1 

4 


M 

0. 


.s 
w 


3 

s 


1 


3 
4 


6 
10 


2_ 
10 


1 


3 
4 


3 

8 


1 
2 


li 


1 



Table 32. Winged Jig Screws 




D 


Thrd. 


L 


/i 


/ 


C 


H 


s 


1 
4 


1 




1 
4 


1 

8 


32 


1 
4 


ll 


10 




5 
10 


1 

8 


J3 
04 


5 
10 


^h 


3 
8 


1-| 


8 


1 

8 


17 
64 


3 

8 


li 


7 
10 


1} 


7 


3 
16 


S 
10 


7 
16 


It 


1 
2 


I7 


j_ 


3 
16 


11 

32 


1 
2 


2 



Table 34. Square Head Jig Screws 




D 


Thrd. 


Z, 


w 


d 


3 
10 




"2 


.032 


I 
10 


1 






3 


T 


"2 




.040 


32 










_5_ 
10 


a 
w 




.057 


3 

32 










3 






1 


1 


8 


1 




10 


8 














1 


5 


1 












10 




64 


8 


1 


1 
1 2 


5 


1 


2 




04 


8 



Table 30. Headless Jig Screws 



l«--7/-*l 




—h 



T 



D 


Thrd. 


L 


h 


S 


11 


1 
4 


"2 

a 

'0 

w 

A 


m 


3^ 
4 


3 

8 


li 


1 


5 

16 


1 


T 


l| 


9 
10 


3 

8 


1 


9 
10 


ifo 


5 

b 



Table 33. Nurled Head Jig Screws 




D 


Thrd. 


L 


h 


m 


s 


fl- 


r 


I 
4 


1 

1 

vx 

1 


3 
4 


10 


3 
10 


li 


3_ 
4 


i_ 
4 


5 

16 


I 


3 

8 


7 

32 


li 


7 
8 


10 


3 

8 


1 


10 


l_ 
4 


iT^ 


1 


3 

8 



h-H^-l 



h*-— 7i-H*- 



/RT 



i 






-T— 



iilllr 9 



£> 


Thrd. 


iJ 


A 


L 


s 


r 


pr 


_6_ 
10 


-s 

en 
-a 
a 

w 

§• 
.a 
w 


16 


6 

8 


li 


2i 


li 


ifo 


3 

8 


3 

8 


11 
10 


H 


2^ 


li 


1^ 


10 


10 


4 


1-8 


2^ 


ik 


1^ 


1^ 


J_ 


3 
4 


1| 


2i 


li 


lf6 



Table 31. Locking Jig Screws 



■yJ}J 1< C_ 




A 


B 


c 


Z) 


E 


G 


No. 52 


1 
4 


9 

10 


1 

4 


7 
10 


_9^ 
10 


No. 30 


10 


¥ 


X 

4 


2 


5 

8 


No. 12 


3 

8 


"8 


10 


9 
10 


11 
10 


1 

4 


Jl_ 


11 

10 


To 


11 

10 


13 
10 


To 


_9_ 

10 


3 

4 


To 


3 

4 


T_ 
3 


3 
3 


-« 


3 
4 


3 

8 


13 
10 


15 
10 


_7_ 
10 


11 

10 


13 
10 


3 
8 


7 
3 


1 


2 


3 
4 


T 


7 
10 


13 
16 


1^ 


9 
10 


13 
10 


7 
8 


10 


1 


^\ 


8 


7 
8 


Ij 
10 


1 

2 


1^ 

•^10 


ifo 


11 

10 


15 
10 


1 


1_ 


, 1 

■"■ 8 


^\ 


3 
4 


^h 


1 


9 
10 


1 
^ 4 


Ifo 


13 
10 


li 


ifo 


9 
16 


1 i 
10 


li 


7 

8 


It 


li 


? 


, 7 
■^10 


ll 


10 

10 


1^ 


^h 


s" 


j_l_ 


1^ 


1 


If 


li 


11 

16 


1 ' 
•^10 


1! 



Table 35. Nurled Head Jig Screws 



Table 36. Loose Bushings for Jigs 



286 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Standard Punches, Dies and Punch Holders 

By A. C. Claire {Amer. Mach., July 4, 1912) 
U 6- 



J- tn -5- 

32 10 15 




h 6- 

Table 37. — Punch Holders 
Dimensions in inches 



b 


C 


d 


e 


f 


g 


h 


J 


k 


; 


2\ 


IT^ 


h 


J 


i 


i 


1 




60° 


1 
1 


2\ 


lA 


h 


i 


f 


i 


1 


3 

4 


60° 


1 

4 



16 

16 



a 


b 


c 


(i 


e 


/ 


^ 


A 


y 


k 


I 


m 


. 





Atoi^ 


2i 




li 


3 

8 


H 


1 

8 


3 

8 


5 

8 




1 
2 


20 






A tot 


2i 




H 


1 


H 


i 


A 


3 

4 




5 
8 


20 






itoA 


2h 




H 


1 


3 

4 


i 


i 


li 




1 


10 






^to A 


3 




f 


5 
8 


M 


i 


f 






i 


20 






3 

4 


3 


11 

16 


M 


1 
4 


li 


1 
4 


li 


l| 




li 


10 






I 


3 


H 


H 


3 
8 


li 


1 
4 


i| 


if 




li 


10 








Table 39. — Holder for Large Punches 
Dimensions in inches 



a 


\b\ c \d\e\f\g\h\j\k\l\m\ n \o\p\q\r\ s \+\u 


lilliAl il i| h\ 2I fl flAliil fl 3o°lA|Al jiAli" 12 thr'dl ll f 



Table 38. — Clearances for Punch and Die for Different 
Thicknesses and Materials 



Thickness of 
stock, in. 


Clearance for 

brass and soft 

steel, in. 


Clearance for 

medium rolled 

steel, in. 


Clearance for 

hajd rolled 

steel, in. 


.ore 


.0005 


.0006 


.0007 


.020 


.cor 


.0012 


.0014 


.030 


.0015 


.0018 


.0021 


.040 


.002 


.0024 


.0028 


.050 


.0025 


.003 


•003 s 


.060 


.003 


.0036 


.0042 


.070 


•003 s 


.0042 


.0049 


.080 


.004 


.0048 


0056 


.090 


.0045 


.0054 


0063 


. 100 


.005 


.006 


.007 


. no 


■0055 


.0066 


.0077 


. 120 


.006 


.0072 


.0084 


.130 


.0065 


.0078 


.0091 


. 140 


.007 


.0084 


.0098 


■ISO 


.0075 


.009 


• oios 


.160 


.008 


.0096 


.0112 


.170 


.0085 


.0102 


.0119 


.180 


.009 


.oro8 


.0126 


. 190 


.0095 


.0114 


•0133 


.200 


.010 


.012 


.014 



Miscellaneous Small Parts 




Table 40. — Small Knuckle Joints 
Dimensions in inches 




Pin 


A 


B 


c 


D 


E 


F 


G 


H 


Pin 


A 


B 


c 


1^ 


E 


F 


G 


H 


7 


if 


1 

8 


1 
4 


f 


A 


A 


A 


1 


3 

8 


S-40 


i 


i 


A 


i 


A 


s 

64 


9 

32 


A 


A 


s 
s 


S-40 


A 


5 

16 


31 
32 


A 


7 

32 


a 


1 3 

16 


1 


8-32 


A 


A 


f 


li 


A 


A 


3 

8 


A 


19 

32 


^ 


8-32 


3 

16 


1 


lA 


1 
4 


1 
4 


3 
4 


I 


5 
8 


10-32 


A 


1 

8 


1 

2 


lA 


i 


i 


a 


1 


3 

4 


I 


10-32 


i 


* 


lA 


A 


A 


7 

8 


lA 


3 
4 


i-20 


1 
4 


i 


1 


lA 


A 


A 


1 
2 


A 


1 


lA 


F20 


5 

16 


5 
8 


iH 


1 


3 

8 


I 


if 


7 
8 


A 


A 


S 
8 


i 


ili 


1 


A 


1 


f 


I 


If 


5 
16 


1 


1 


2i 


7 

16 


1 

2 


ii 


If 


. 1 
■l 16 


f 


f 


1 


1 


2i 


7 

16 


A 


3 
4 


* 


li 


If 


3 

8 


A 


7 
8 


2A 


1 
2 


16 


lA 


2 


-.1 
'■i 


A 


7 

16 


1 


I 


2A 


i 


1 
4 


1 


A 


lA 


2 


7 
16 


i 


I 


2| 


9 
T6 


f 


if 


2i 


If 


i 


^ 


I 


li 


2f 


9 
16 


A 


M 


5 

8 


If 


2i 


1 
2 


5 

8 


If 




1 


a 


iH 


2f 


T 5 
■■■8 


1 


1 


l| 


if 




3 

4 


7 

16 


lA 


H 


iH 


2f 


1 


3 

4 


if 




i 


ft 


2A 


3 


If 


3 
4 


3 

4 


If 


i| 




7 
8 


i 


i| 


15 
16 


2A 


3 


3 
4 


1 


l| 




I 


lA 


2A 


3I 


2i 


7 

8 


7 
8 


l| 


2i 




I 


A 


iH 


lA 


2A 


3i 


f 


I 


2i 




li 


lA 


2M 


4 


2| 


I 


I 


2i 


2| 




li 


1 


i| 


lA 


2il 


4 





MINOR MACHINE PARTS 



287 



TABtE 41. — S. A. E. Yoke and Eye Rod End Standards 



Dimensions in inches 




Adjustable Yoke Ends 



A 


B 


D 


E 


G 




li 


A 


i 


A 


A 




li 


i 


1 

2 


A 


1 

i 




4 


A 


a 


a 


A 






3 


11 


_L 


3 




I2 


8 


16 


16 


8 




li 


7 


13 


1 


7 




16 


16 


2 


16 




,3 


1 


15 


9 


1 




if 


2 


16 


16 


2 







:fl=3 



Plain Yoke Ends 

-A- 




A 


£ 


c 


£» 


£ 


F 


G 


r 


11% 


5 

16 


11 

16 


I 


A 


A 


A 


A 
32 u. s. 


2 


1^ 


1 


li 


9 
32 


f 


1 
i 


1 
4 

28 ALAM 


2i 


i 


I 


lA 


11 
32 


3 

4 


A 


A 
24 ALAM 


2§ 


5 

8 


li 


if 


A 


1 


f 


24 ALAM 


'i 


M 


li 


i| 


i 


I 


A 


A 
20 ALAM 


3 


if 


lA 


If 


A 


li 


1 

2 


20 ALAM 



A 


£ 


c 


D 


£ 


F 


G 


li 


3 
16 


i 


7 

16 


A 


7 
16 


A 


If 


1 
i 




f 


^ 


f 


f 


2 


A 


If 


f 


fi 


i 


A 


2| 


1 

8 


lA 


E 


A 


7 

8 


f 


2i 


A 


-1 


I 


i 


1 


A 


2i 


1 
2 


T 5 
■>■ 8 


If 


9 

T6 


If 


4 



Table 



42.— Round and Hexagon Head Studs for Cam Rolls, Levers, Etc. 
Dimensions in inches 
Hanu Engineering Co., {Amer. Mach., June 6, 1912) 




■< £,5— 



'r< — d-i X 





h--di-*i 




Nom. size 


1,4 


O3 


Zi 


D 


L, 


d. 


ai 


Thread 


1 
4 


If 


A 


0.377 


0.249 


A 


H 


A 


i X24 


A 


lA 


3 

16 


0.408 


0.311 


f 


1 
2. 


A 


i X24 


1 


Iff 


A 


0.439 


0.374 


1* 


A 


A 


AX18 


7 

16 


iH 


3 

16 


0.470 


0.436 


ft 


1 


A 


f X16 


f 


lA 


A 


0.502 


0.499 


I 


ft 


7 

32 


AX14 


S 

8 


li 


3 

16 


0.564 


0.623 


27 
32 


1 


i 


1 X13 


3 

4 


T 25 
I32 


A 


0.627 


0.748 


H 


1 


1 
4 


AX12 


f 


iH 


A 


0.690 


0.873 


lA 


lA 


S 
16 


f Xl2(ll) 


I 


2f 


1 
4 


0.752 


0.998 


If 


lA 


A 


i Xi2(io) 


li 


2f* 


9 

32 


0.877 


1.248 


lA 


if 


i 


I X12 (8) 


li 


2i 


5 

16 


1.002 


1.498 


lA 


if 


f 


li X12 (7) 


li 


3A 


H 


I. 127 


1.748 


if 


2A 


A 


ih X12 (6) 


2 


3A 


3 

8 


I ■ 152 


1.998 


iH 


2i 


1 
2 


if X12 (5) 



Nom. size 


U 


a2 


X2 


D 


L, 


di 


ai 


Thread 


1 
4 


li 


5 

64 


0.282 


0.249 


9 

T6 


B 


A 


i X24 


5 

16 


lA 


_5_ 
64 


0.313 


0.311 


1 


i 


A 


i X24 


f 


lA 


3 
32 


0.3300.374 


32 


9 

16 


3 

16 


AX18 


A 


ifi 


A 


0.3600.436 


23. 
32 


1 


A 


1 X16 


h 


iH 


7 

64 


0.376 


0.499 


3 
4 


M 


A 


AX14 


4 

8 


Iff 


1 
8 


0.423 


0.623 


a 


f 


1 
4 


i X13 


f 


ifi 


A 


0.470 


0.748 


15 
16 


r 


i 


AX12 


f 


ifi 


A 


0.518 


0.873 


lA 


lA 


A 


f Xl2(ll) 


I 


iff 


ii 


0.564 


0.998 


If 


lA 


A 


f X 12(10) 


li 


2fi 


fl 


0.658 


1.248 


lA 


if 


3 

8 


I X12 (8) 


If 


2ff 


ff 


0.752 


1.498 


lA 


i| 


3 

8 


li X12 (7) 


If 


2ff 


if 


0.845 


1.748 


if 


2A 


A 


if X12 (6) 


2 


3-h 


19 
64 


I. 189 


1.998 


2A 


2f 


1 
2 


if X12 (s) 



288 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



}«-ii-» 



l-tdi 



m 




Table 43. — Cam Rollers 
Dimensions in inches 



Nom. size 


z> 


Xl 


fll 


d2 


^ 


i 


1 
1 


f 


h 


A 


1 


A 


A 


13 
32 


3 

32 


32 


I 


f 


1 


1% 


^ 


19 

32 


li 


1^ 


7 

16 


M 


A 


f* 


li 


\ 


* 


1 
2 


\ 


1 


if 


1 


5 
8 


A 


^ 


If 


if 


r 


3 

1 


1 


A 


lA 


i| 


\ 


\ 


H 


11 

64 


lA 


2i 


I 


I 


1 


A 


I*i 


2f 


li 


li 


7 

8 


7 

3T 


ifi 


2i 


I* ' 


Ij 


I 


1 

4 


iM 


3I 


If 


If 


li 


A 


25^ 


3l 


2 


2 


li 


A 


2M 


4f 




Table 44. — Cast-iron Rock Arms 
Dimensions in inches 



Nom. size 


z» 


c 


Xi 


Tap 


H 


7 


i 


1 

4 


f 


f 


i X24 


5 

16 


1 

2 


A 


5 
16 


f 


H 


i X24 


_5_ 
16 


5 

8 


f 


f 


i 


7 
16 


AX18 


A 


H- 


7 

T6 


A 


I 


M 


1 X16 


5 

16 


H 


1 

2 


i 


lA 


i 


AX14 


A 


7 

8 


S 
8 


1 


lA 


A 


5 X13 


a 




f 


f 


li 


5 

8 


AX12 


1 


lA 


1 


1 


If 


H 


1 Xl2(ll) 


M 


If 


I 


I 


2 


3 

i 


f Xl2(l0) 


7 

T6 


lA 


li 


li 


2f 


i 


I X12 (8) 


1 

2 


^7 
15 


l| 


I^ 


2| 


I 


Ij XI2 (7) 


^ 


2i 


If 


If 


3i 


i| 


i| X12 (6) 


A 


2f 


2 


2 


3i 


li 


if X12 (5) 


1 


3 



_^ 


--C- 


-" 










-t 
f 



(o) 



Table 45. — Collars 
Dimensions in inches 



Nom. size 


z> 


c 


i: 


Taper pin 

No. 


Set screw 


i 


1 
4 


i 


f 






A 


A 


f 


M 






f 


3 

8 


f 


A. 






A 


A 




H 






i 


* 


, 1 

^ 16 


* 


o-iA 




f 


5 

8 


lA 


. A 


2-1 A 




f 


3 

4 


-1 
^2 


f 


3-i4 


f X20 


1 


7 
8 


If 


H 


4-if 


i X20 


I 


I 


2 


f 


S-2 


AX18 


li 


li 


2f 


i 


6-2I 


AX18 


i| 


l| 


2I 


I 


7-2f 


f X16 


If 


If 


3i 


If 


8-3i 


J X12 


2 


2 


3f 


If 


8-3f 


i X12 




Table 46. — Cast-iron Yoke Ends 
Dimensions in inches 



Nom. size 


ii 


B 


c 


D 


£ 


ii' 


G 


H 


I 


/ 


i 


1 


A 


5 

8 


1 
4 


. 1 

4 


f 


7 

8 


5 
16 


if 


f 


A 


M 


li 


3 
4 


5 

16 


1 

4 


ff 


M 


A 


lA 


f 


1 


A 


f* 


7 
8 


1 


1 
4 


A 


if 


A 


lA 


H 


A 


If 


23 
64 


I 


16 


i 


fl 


If 


5 

16 


If 


16 


1 


i 


3 
8 


lA 


f 


i 


f 


I 


A 


lA 


f 


f 


9 

16 


ft 


lA 


f 


1 
4 


II 


., 7 
132 


ff 


If 




f 


1 


A 


if 


f 


A 


il 


iff 


f 


if 


lA 


1 


ii 


M 


If 


f 


A 


7 
8 


ifl 


fl 


2A 


If 


I 


3 

4 


i ■ 


2 


I 


A 


15 
16 


lA 


A 


2f 


lA 


li 


f 


9 

T6 


2f 


li 


ff- 


lA 


iM 


1 
2 


2I 


if 


4 


I 


1 


2| 


If 


3 

8 


li 


2 


1 
2 


3f 


2i 


If 


If 


if 


3i 


if 


fl 


if 


2'32' 


16 


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4 


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3 



By the Hanu Engineering Co. {Amer. Mach., June 6, 1912) 



MINOR MACHINE PARTS 



289 




Fig. 40. — Fluted reamers. 



r<G^ 



K-H-M 



4 Flutes 




Table 


47-— 


Series Taps 


FOR Acme and Square Threads 


Size A 


B 


c 


D 


E 


F 


G 


H 


I 


I 


35/64" 


Vz' 


K2' 


3H° 


3M" 


1/2" 


?i6" 


5*" 


Me' 


2 


1%2'' 


1^2' 


1/2' 


3=4" 


3K" 


}^" 


Me" 


5/*" 


Me' 


3 


=/i' 


3%4'' 


K2" 


3?i' 


3H" 


;-«" 


%6" 


=/*" 


Me" 


I 


^1/64" 


=>r64' 


3764" 


33/4" 


33/*" 


}^" 


3/4" 


W 


Me" 


■>4 

2 


«44" 


H" 


3%4' 


33/4" 


33/*" 


}^" 


3/4" 


H" 


Me" 


3 


3/4" 


•Me" 


3764° 


33/4" 


33/*" 


^" 


3/4° 


3/" 


Me" 


I 


=/4' 


2^2° 


2)^2" 


4° 


33/4" 


72" 


13/6" 


%" 


72" 


2 


iMe" 


*%4' 


2^2" 


4" 


33/4" 


72" 


iMe" 


%" 


H" 


4!^' 3 


%° 


"/^4° 


21/^2" 


4" 


33/4" 


72" 


iMo" 


'/*" 


72" 


I 


"/64' 


= ^4" 


5M4" 


4H" 


4'/4° 


72" 


15/6" 


i" 


5*" 


2 

■>/-//T 


8^4" 


7,^" 


6^4" 


4'/^" 


4'/4" 


72" 


15/6" 


t" 


5*" 


D'BL 3 


l" 


I'Me' 


"/64" 


45^" 


4K4" 


72" 


15/6" 


I" 


5*" 


I 


6/64° 


"764' 


55/64" 


4%" 


4W 


72" 


1" 


iH" 


iHe" 


iW 2 


1H2' 


is/te" 


55/6^ 


43/4"- 


m" 


J-^" 


l" 


I}*" 


iMe" 


3H" 3 


l5«4' 


I1/04' 


55/64" 


43/4" 


aW 


}^" 


l" 


ii^" 


iMe" 


4 


iW 


iHe" 


55,64" 


43/4" 


43/*" 


Vz" 


x' 


iH' 


iMo' 


I 


I13/64'' 


i'/^' 


IW" 


5H" 


45*" 


5/*" 


I^" 


I?*" 


w 


iH" 2 


I%2' 


l3/6" 


IW 


SH' 


45*" 


5/*" 


ll/*" 


I?*" 


■'A' 


D'BL 4 


12144" 


Il%4" 


xM" 


sH" 


45*" 


5/*" 


I}*" 


1 3/*" 


w 


i?r 


iMo" 


iW 


sH" 


4=/*" 


5/*" 


ll/*" 


1?*" 


w 












Table 


48.- 


— Counterbores 












A 


B 


c 


D 


E 


F 


G 


H 


J 


K 


L 


M 


N 





I 


5* 


iH 


Me 


3V* 


7 


.572 


H 


M 


I 


I?*2 


M 


.220 


1I/64 


%3 


2 


% 


I3/* 


iMe 


3}.* 


774 


.572 


J4 


3/* 


i!-* 


lj*2 

iMe 


Me 
J* 


.253 


3/6 


M 


3 


% 


172 


iMe 


3% 


iVi 


.778 


5/6 


Me 


ii/* 

174 
1 72 
13/ 


.253 


3/6 


1/4 


4 


I 


l5/* 


% 


3% 


8H 


.778 


5/6 


Me 


iMe 


3/* 


.348 


?*2 


1H2 


S 


iH 


I'/* 


I 


3'/* 


9H 


.778 


5/6 


Me 


iiMe 


Me 


.348 


1704 


1^2 


6 


iH 


21/ 


iW 


3^/* 


9^ 


.778 


Me 


Me 


2 


W 


.442 


lI/*2 


Me 





P 





R 


s 


T 


U 


V 


w 


X 


Y 


Taper of 
shank 


Taper of 
guide peg 


I 


5/*2 


iMe 


m 


M2 


1I142 


3/* 


?*2 


5/*2 


33/ 


5/*2 


No. 2 Morse 


I in 16 


2 


Me 


iMe 


i5* 


%2 


1IM2 


3/* 


%2 


5/2 


354 


5/*2 


No. 2 Morse 


I in 16 


3 


'/*2 


iH 


l3/ 


%2 


ii7ie 


3/* 


%2 


5/*2 


33/ 


Me 


No. 3 Morse 


I in 16 


4 


1/4 


l3/* 


1% 


f* 


iiMe 


Me 


15/62 


1%4 


5 


Me 


No. 3 Morse 


I in 16 


S 


Me 


1 5* 


2}* 


3/* 


2He 


Me 


15/2 


Vm 


5 


Me 


No. 3 Morse 


I in 16 


6 


11^2 


iiMe 


2T* 


3/* 


2\h 


Me 


15/2 


1764 


S 


%2 


No. 3 Morse 


I in 16 



19 



PRESS AND RUNNING FITS 



The tolerances and allowances suitable jor running jits formed the 
subject of a report by the British Engineering Standards Committee, 
rendered in 1906. This report was based on an exhaustive investiga- 
tion, neaxly 800 pieces of work from 13 engineering workshops 
having been measured in order that the final recommendations might 
fairly represent commercial work. The recommendations of the 
Committee were presented in both tabular and graphic form, the 
latter reproduced, by permission, in Fig. i. 

The Committee define tolerance as " a difference in dimensions pre- 
scribed in order to tolerate unavoidable imperfections of workman- 
ship"; and allowance as "a difference in dimensions prescribed in 
order to aUow of various qualities of fit." 

The Report also says in part: 

" For general engineering practice, the Committee have laid down 
three classes of workmanship, viz.: F'irst class; second class; third 

class For special cases in which a very high degree of 

accuracy is required, the Committee have laid down a class of work- 
manship having 'extra fine tolerances and allowances.' This class 
is carried up to 3 ins. in diameter and is intended for cases in which 
extreme accuracy is necessary. 

" For running fits the Committee are of the opinion that, wherever 
possible, the shaft should be the element more nearly approaching 
the true dimension, and allowance be made on the hole according to 
the class of fit required. The tolerances on the shaft are negative 



based on the assumption of perfect surfaces and it seems reasonable 
to expect that the stresses given by it would be experienced with 
ground and reamed surfaces but not with tooled surfaces. The error 
in the latter case, however, would be on the safe side. It would also 
seem best to taper the plug one -half the allowance in order to avoid 
the scouring out of the hole at the entering end, an action that leads 
to the poorest grip at the shoulder where the best is needed. This 
tapering of the shaft, however, is not commonly done. 

The character of the lubricant used is well known to have consider- 
able influence on the forcing pressure. The lubricant used by the 
Lane and Bodley Co., who supplied most of the data from which the 
charts were deduced, was linseed oil and white lead. 

Suitable pressures to aim at are 5 tons per in. of diameter of 
plug for cast-iron hubs on steel shafts and 10 tons for steel hubs on 
steel shafts. 

Taper Press Fits 

Taper press jits have jound favor, among leading manufacturers 
because of their simplicity and surety. The taper is small — usually 
3^6 in- Pfir ft- — 3-nd does not in the slightest endanger the security 
of the work. The travel, or distance, the plug is forced into the ring 
by the pressure is a more satisfactory and practical criterion of the 
surety of a taper fit than the pressure itself. For this reason satis- 
factory records of taper press fits may be had even when they are 



in order that it may never exceed its true dimensions made with a screw or knuckle-joint press. 

Limit gages adapted to such a system may conveniently be referred The members of a taper fit may be placed together and the surfaces 

to as applying to a 'Shaft Basis.' The reverse system may be termed corrected to any degree of accuracy desired, thereby eliminating the 

a 'Hole Basis,' and allowance is then made on the shaft necessity of making delicate micrometer measurements. The lubri- 



"In those cases where it is found necessary to adopt the 'Hole 
Basis,' the tolerances specified for shafts and holes respectively may 
still be employed, and the standard allowance applied to the shaft 
instead of to the hole, the minimum diameter of the hole being accu- 
rately its nominal diameter." 

Straight Press Fits 

The allowances for and the tangential fiber stresses in the hubs due 
to straight press and shrink fits may be obtained from Figs. 2 and 3 
by Prof. A. Lewis Jenkins {Amer. Mach., Mar. 4, 1915) which 
provide for all common combinations of materials. These charts 
are the results of a critical study of data from several hundred fits 
which have been accumulated during many years by the Lane and 
Bodley Co., the Laidlaw-Dunn-Gordon Co. and others, combined 
with mathematical analysis, which it is unnecessary to repeat here. 
The use of the charts is explained below them. 

The work from which the charts were deduced was of customary 
workmanship, that is, turned shafts and bored holes. For ground 
shafts and reamed holes much smaller allowances must be used — not 
over one-half those suitable for turned shafts and bored holes. 
Similarly, turned shafts in reamed holes or ground shafts in bored 
holes should have three-fourths the allowances suitable for turned 
shafts and bored holes. The effect of ground shafts and reamed holes 
in increasing the pressure required for a given allowance is well 
established by experience and should not be ignored. It is obviously 
due to the perfection of the surfaces and of the resulting contact and 
the elimination of the hills and valleys of work made with cutting 
tools. The measurements of turned and bored work being made 
over the high spots, ignore the influence of these irregularities. In 
view of these irregularities the author has doubts about the accuracy 
of the chart for the tangential hub stress. This chart is necessarily 



cant on a taper fit acts very effectively, and the possibility of scoring 
the surfaces in pressing is slight compared with the danger when 
making straight fits. The members of a taper press fit are easily 
centered, and accurate alignment is obtained at the beginning of the 
pressing operation. 

Straight press fits have the following objections: They are difficult 
to measure accurately in fitting and difficult to lubricate satisfactorily 
when of considerable length; the surfaces of the members are likely 
to be scored in pressing; the operation requires extreme care on the 
part of the operator in starting and while pressing on, and the crite- 
rion of good design and workmanship is the total pressure required. 
This can only be determined with any degree of accuracy by using 
a hydraulic press. 

Professor Jenkins has extended his analysis and examination of 
data from shop experience to taper press fits {Amer. Mach., Feb. 17, 
1916). These data show a much greater variation in the pressure 
required than was found in making a similar analysis of straight fits. 
This is accounted for by the fact that the data on taper fits cover a 
greater variety of work, were taken from a greater number of sources 
and have greater personal and instrument errors, and the kind of 
lubricant used has a greater effect on the coefficient of friction for 
taper than for straight fits. 

In particular the lubricant has been found to have such a great 
influence over the pressure requited in making taper press fits that 
the tonnage cannot be depended upon as a criterion. 

The lubricant is trapped between the surfaces and if it is not too 
thin, it is practically impossible for it to be squeezed out or scraped 
off. There is some tendency for the lubricant to be scraped off near 
the ends of a taper fit, but even over the small portions of the length 
at these places the conditions are no more severe than exist through- 
out the complete length of a straight fit. When a heavy paint made 

{Continued on page 2g6, first column) 

290 



PRESS AND RUNNING FITS 



291 




5 6 7 

Diameter, Ins. 

Fig. I. — British standard tolerances and allowances for running fits. 



292 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



7" 'suj 'ii^Sua^ 



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PRESS AND RUNNING FITS 



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PRESS AND RUNNING FITS 



295 



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296 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



of linseed oil and white lead is used as the lubricant, the average 
tonnage required for a taper fit is between one-half and two-thirds 
that required lor the average straight fit of the same allowance and 
proportions. 

The allowances and the tangential fiber stresses in the hubs due to taper 
press fits may be obtained from Figs. 4 and 5 which are by Professor 
Jenkins and give the results of his analysis. Owing to the variation 
in the pressure required because of the effects of the different kinds 
of lubricant used, the values found for the pressure required may be 
considerably greater or less than the actual 
pressures. They represent average values when 
a lubricant of white lead and linseed oil is used. 
As the taper fit system is used at the works of 
the Westinghouse Machine Co., a useful simpli- 
fication has been effected by a slight modifica- 
tion of the taper, due to J. B. Thomas, chief 
inspector of the works {Amer. Mach., Aug. 4, 
1904). The change in the taper is from re or 
.0625 to .06 in. per ft., or .005 in. per in., and 
from \ or .125 to .120 in. per ft., or .01 in. per 
in. measured on the diameter. The result of the 
change in the smaller taper is seen in Table i of 
the diameters at each successive inch of length 
of a hole of 10 ins. diameter at the large end, 
the dimensions in the first column being car- 
ried to four places, while in the second they are 
carried to three, where they stop. 



a result that, with the taper of Xi in. per ft., can be found only by 
calculation. 

Fig 6 shows a form of gage for large taper holes which Mr. Thomas 
prefers to fuU plugs. They are much lighter than full plugs and 
with them the holes can be gaged independently on different diam- 
eters and irregularities in the holes detected. Fig. 7 shows Mr. 
Thomas's method of gaging the largest holes, the angle of taper be- 
ing greatly exaggerated. At the left of the hole is a carefully made 
strip of steel with an upturned end and a row of holes down the 



D 



C 



Fig. 6. 










































to Tjt 00 M to 
r;* ^ M (M CO CO ^ 

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000 00 00 
Allowance 



OJ CO CO 



Fig. 7. 
Figs, 6 and 7. — Gaging taper holes. 

The slight and otherwise unimportant change 
in the taper will be seen to lead to round 
figures for each inch of length, the figures in the 
second column being much more easily read from 
the micrometer than those in the first, while, by 
subtracting from the large diameter five times as 

many thousandths as the piece hss inches of pic. 8.— The General Electric Go's practice in allowances for sliding, press and shrink fits. 

length, the small diameter is obtained directly, 

center, spaced i in. apart. The strip is used in connection with an 
inside micrometer, as shown. The measurements are not made 
perpendicular to the axis, but as indicated by the dotted arc. One 
end of the micrometer being located by a hole in the strip, the 
other is manipulated precisely as though the hole were straight, 
the frequent spacing of the holes in the strip permitting the hole 
to be gaged for uniformity of taper. The dimension thus gaged at 
the large end is the one given on the drawing as the true diameter — 
the microscopic difference between the dimension as called for and 
as made being of no importance. 

The allowance for pressing home is .01 in. on all diameters from 10 
to 30 ins. 



Table i. — Diameters at Each Inch of Length of Taper Press 

Fits 



Taper ^ in. per ft. 


Taper . 06 in. per ft. 


10 


10 


9 . 9948 


9-995 


9.9896 


9-990 


9.9844 


9-985 


9-9792 


9.980 


9.9740 


9-975 


9.9688 


9-970 



PRESS AND RUNNING FITS 



297 



The standard Westingkouse lubricant is i lb. of white lead to i| 
pints of linseed oil. 

One advantage of the taper fit is that the plug may be entered in 
. its seat and the two compared directly, whereas,- with parallel fits, 
the comparison can be made with gages only. Thus compared, the 
distance remaining for pressing home forms the best possible check 
on both pieces. Thus, with the taper of .005 in. per in. and an 
allowance of .01 in. for pressing, the plug should enter the hole 
within 2 ins. of home, or, more generally, tte distance by which the 
parts should not go home, when they are assembled without pressure, 
should be i in. for each five thousandths of pressing allowance. 

The practice of the General Electric Co. in allowatices for sliding, 
press and shrink fits is given in Fig. 8, by John Riddell {Trans. 
A.S.M. E., Vol. 24). Mr. RiddeU said: 

"There are many things to be taken into consideration in laying 
out these tables and diagrams: First, the relation of length of bore 
to diameter. In our case the length of hubs of armature spiders 
is sometimes several times the diameter; but the actual bearing 
surface is about equal in length to the diameter, on account of re- 
cesses in the hub. Second, the outside diameter of hub. My dia- 
gram is laid out on the basis of the hub being twice the diameter 
of the shaft. Third, the nature of the materials. Fourth, how and 
where the parts are to be assembled; if they are to be assembled where 
a suitable hydrostatic press is available, more allowance can be made 
than if the parts are to be put together by the use of bolts and straps. 

"My diagram is based on actual experience extending over a 
number of years, and is eminently satisfactory. 

"There are five curves shown as follows: The left-hand one on the 
minus side of o Hne shows allowances for sliding fits. I mean by this 
such fits as are not loose or free like a running fit, but one that will 
just slide without any perceptible play. The next curve is on the 
right or plus side of the o line and shows exactly the same allowances 
for tight fits, for parts with light hubs, such as commutator shells, etc. 
The third curve gives somewhat greater allowances, and is used for 
steel hubs. The fourth is for our regular armature spiders having 
solid cast-iron hubs. The fifth shows the amount we have found to 
be correct for shrinkage fits, and for such heavy articles as couplings 



avoidable variation due to the wear of the reamer, the variation from 
standard diameter for the various kinds of fits being made in the shaft. 
This variation is, however, not positive, but is made between limits 
of accuracy or tolerance. Taking the case of a press fit on a 2-in. 
shaft, for example, we find that the hole — that is, the reamer — is 
kept between the correct size and .002 in. below size, while the shaft 
must be between .002 and .003 in. over size. For a drive or hand 
fit the limits for the hole are the same as for a press fit, while the 
shaft in the former case must be between .001 and .002 large and 
in the latter between .001 and .002 small. 

Table 6 gives in the same way the allowances for parallel running 
fits of three grades of closeness. The variations allowed in the holes 
are not materially different from those of the preceding table, but 
the shafts are, of course, below instead of above the nominal size. 

Table 7 relates to a feature of the Hunt Company's practice, 
where the preferred practice with press fits is to make them taper, 
the taper used being the Hunt standard of j^ in. in diameter for 
each foot in length. With fits of this character the usual practice 
is reversed, the variation in diameter being in the hole, while the 
shaft is kept to standard size. The holes are made with standard 
reamers, which are maintained at the standard taper, and in each 



Table 4. — Brown & Sharps Mfg. Co.'s Practice in Tolerances 
AND Allowances for Shafts Rough Turned Preparatory 
for Grinding 



Size 


Not go on 


Go on 


Size 


Not go on| Go on j 


Size 


Not go on 


Goon 




Ins. 




Ins. 1 


Ins 


1 


.383 


.387 


a 


• 9455 


.9495 


li 


1.S08 


I. 512 


A 


• 4455 


• 4495 


I 


1.008 


I .012 


lA 


I.S70S 


1.5745 


i 


.508 


• 512 


lA 


I. 070s 


1.0745 


If 


1.633 


1.637 


vs 


.5705 


• 5745 


ij 


1. 133 


I. 137 


lH 


1.6955 


1.699s 


1 


• 633 


.637 


lA 


I. 1955 


I. 1995 


ij 


I. -758 


1 .762 


a 


.6955 


.6995 


li 


1.258 


I . 262 


iH 


I. 820s 


I. 824s 


5 


.758 


.762 


lA 


I.320S 


1.3245 


il 


1.883 


1.887 


lA 


.8205 


• 824s 


If 


1.383 


1.387 


i« 


1.9455 


I.949S 


i 


.883 


.887 


lA 


1-4455 


I . 4495 


2 


2.008 


2.012 




Fig. 9. — C. W. Hunt Co.'s gage for taper press fits. 



These allowances for press fits of armature spiders are for assem 
bhng in the field where equipment of limited capacity must sometimes 
be used. They are, therefore, smaller than the allowances that are 
customary in such work as engine cranks and crank pins. 

The practice of the General Electric Co. in allowances and tolerances 
for journal fits is given in Table 2. 

The practice of the Brown and Sharpe Mfg. Co. in allowances and 
tolerances for ground fits is given in Table 3, by W. A. Viall {Trans. 
A.S.M.E., Vol. 22). 

The practice of the Brown and Sharpe Mfg. Co. in allowances and 
tolerances for shafts rough turned preparatory for grinding, is given in 
Table 4 by W. A. Viall {Trans. A.S.M. E., Vol 32). 

The practice of the C. W. Hunt Co. in allowances and tolerances 
or fits is given in Tables 5, 6 and 7 {Amer. Mach., July 16, 1903). 

Table 5 gives all the particulars for press, drive and close or hand 
fits for parallel shafts ranging between i and 10 ins. in diameter. 
The holes for aU parallel fits are made standard, except for the un-. 



case are sunk into the work to a point determined by Table 7 and 
defined by an adjustable stop gage, which abuts against a machined 
face on the work. A plug gage, shown in Fig. 9, is ground to the 
Hunt taper and to the exact diameter at the zero point A. It is also 
graduated at intervals of xs in. of its length as shown. A taper 
of xg in. per foot is, very closely, toVo in. for each fs in. of length, 
and each division on the scale thus represents very nearly ttsV^ in. 
difiference in diameter. One of these intervals is called a "P" and 
is so entered on the drawings. 

AU shafts for taper fits are turned to within plus or minus jsV^ 
in. of the nominal size at the large end of the taper. The taper 
reamer is then sunk in the hole to such a depth that the hole at 
the large end is small by an amount indicated by the table. Thus 
for a 2-in. press fit the plug gage must enter the hole to such a 
depth that its large end registers between the 6 P and 7 P mark, 
indicating that the hole is between joV^ and to'ott small. The parts are 
then pressed together until the true sizes match — that is, in the case 
of the 2-in. fit, the parts would be pressed between jV and tV in. 

In case the shafts and wheels thus fitted are not driving members, 
no key is used, the grip of the press fit being found to be all suflicient. 
In case they are driving members, the shaft is keyseated for one 
or more Woodruff keys, the key being placed in position before the 
parts are pressed together and being entirely hidden when the work 
is done. 

In all cases the tables apply to steel shafts and cast-iron wheels or 
other members. In the right-hand columns of the tables the formulas 
from which the allowances are calculated are given, and from which 
the range of tables may be extended. 



298 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 2. — General Electric Co.'s Practice in 
AND Tolerances for Journal Fits 



Allowances running fits — high speed, heavy pressure and rocker shafts 





Journal 


Bearings 


Axle 
for 
mo1 


linings 
ry- 


a 

c a 

il 

'•3 




Horizontal j 


Vertical 


Step 


,ors 


Max. 

diam., 

ins. 


H 
n 5 1. 

CI S 


Min. 
bore, 
ins. 


sa 

> <a 
(U > 

e - m 
S '- 

^ 


Min. 
bore, 
ins. 


pa 

(U > 

5 •- 


Min. 
bore, 
ins. 


|a 

^ 


Min. 

bore, 

ins. 


a c 

Va 


f 


.375 


.0005 


.377 


.001 


.376 


.001 


.3755 


.0005 


-380 


.004 


§ 


.500 


.0005 


.502 


.001 


.501 


.001 


.5005 


.0005 


-505 


-004 


i 


.625 


.0005 


.627 


.001 


.626 


.001 


.6255 


.0005 


-630 


.004 


1 


.750 


.0005 


.752 


.001 


.751 


.001 


.7505 


.0005 


.755 


.004 


i 


.875 


.0005 


.877 


.001 


.876 


.001 


.8755 


.0005 


.880 


.004 


I 


I .000 


.0005 


1.002 


.001 


I. 001 


.001 


I . ooos 


.0005 


1.005 


.004 


li 


I. 125 


.0005 


1. 128 


.001 


1. 127 


.001 


1 . 126 


.0005 


1. 130 


.004 


li 


1.250 


.0005 


I. 253 


.001 


1.252 


.001 


1. 251 


.0005 


1.255 


.004 


ij 


1.500 


.0005 


1.503 


.001 


1.502 


.001 


1. 501 


.0005 


1-505 


.004 


If 


1. 750 


.0005 


1.753 


.001 


1.752 


.001 


1-751 


.0005 


I-7SS 


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2 


2.000 


.0005 


2.003 


.001 


2.002 


.001 


2.001 


.0005 


2.005 


.004 


2i 


2.250 


.0005 


2.253 


.001 


2.252 


.001 


2.251 


.0005 


2.255 


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2i 


2.500 


.0005 


2.503 


.001 


2.503 


.001 


2.501 


.0005 


2-505 


.004 


2| 


2.750 


.0005 


2.754 


.002 


2.753 


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2.7515 


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2-755 


.004 


3 


3.000 


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3.004 


.002 


3.003 


.002 


3.001s 


.0005 


3-005 


.004 


31 


3 500 


.001 


3.S04 


.002 


3-504 


.002 


3.S015 


.0005 


3-507 


.004 


4 


4.000 


.001 


4 -005 


.002 


4.004 


.002 


4.002 


.001 


4-007 


.004 


4i 


4.500 


.001 


4.505 


.002 


4.504 


.002 


4.502 


.001 


4-509 


.004 


S 


5.000 


.001 


5.006 


.002 


5-005 


.002 


5.0025 


.001 


5-009 


.004 


Si 


5. 500 


.001 


5.507 


.002 


5-505 


.002 


5.503 


.001 


5-511 


.004 


6 


6.000 


.001 


6.009 


.002 


6.005 


.002 


6.003 


.001 


6. Oil 


.004 


7 


7 .000 


.001 


7. 01 1 


.002 


7.006 


.002 


7.0035 


.001 


7.012 


.004 


8 


8.000 


.001 


8.012 


.003 


8.006 


.003 


8.004 


.002 


8-013 


.004 


9 

10 

II 
12 
13 

14 

IS 

16 
17 
18 
19 

20 

21 
22 
23 

24 
25 

26 
27 
28 
29 
30 

31 
32 
33 
34 
3S 
36 


9.000 
10.000 

I I . 000 
12.000 
13.000 
14.000 

I 5. 000 

16.000 
1 7 . 000 
18.000 
19.000 
20.000 

21.000 
22.000 
23 . 000 
24.000 
25 .000 

26.000 
27 .000 


.001 
.0015 

.0015 
.0015 
.0015 
.0015 
.0015 

.0015 
.0015 
.0015 
.0015 
.0015 


9013 

10.014 
II. 015 


.004 
.005 

.005 
.005 
005 


9.007 
10.007 

1 1 008 


.004 
.005 

.005 
.005 
.005 
.005 
.005 

.005 
.005 
.005 
.005 
.005 

.005 


9.004s 
10.005 

II-O05S 
12 -O06 


. 002 












.002 






12 008 


.002 






13.016 
14.016 
IS. 016 

16 016 


13.009 
14.009 
15.010 

16 010 


13- 0065 

14.007 

15-0075 

16- 008 


002 






.005 
.005 

.005 
.005 
.005 
.005 
.005 

.005 
.008 


.002 












.002 






17.018 
18 018 


17. Oil 
18 oil 


17.008 
18 008 








002 






19.018 


19.012 


19.008 


.002 








.002 






21 0l8 


21.013 
22.013 
23.013 
24.013 


21 .008 


.002 












.002 








23.020 
24.020 
25.020 


.008 


.005 


23.008 


.002 








008 


.005 


24.008 


.002 






.003 

.003 
.003 
.003 
.003 
.003 

.003 
.003 
.003 
.003 
.003 
.003 


.008 
























27.022 
28 022 


008 














008 














29.000 
30 . 000 

31 .000 
32.000 
33.000 
34.000 
35-000 
36.000 


29.022 
30.022 

31.022 
32.024 
33-024 
34-024 
35-024 
36.024 


.008 














008 














008 




























010 














010 




























.010 





















Table 3. — Brown and Sharpe Mfg. Co.'s Practice in Allow- 
ances AND Tolerances for Fits 
running fits — ordinary speed 

To ^-in. diameter, inc 0002510.00075 Small 

To I -in. diameter, inc 00075 to .0015 Small 

To 2 -in. diameter, inc 0015 to .0025 Small 

To 3|-in. diameter, inc 0025 to .0035 Small 

To 6 -in. diameter, inc 0035 to .005 Small 



To ^-in. diameter, inc 0005 

To I -in. diameter, inc 001 

To 2 -in. diameter, inc 002 

To 35-in. diameter, inc 003 

To 6 -in. diameter, inc 004S 



to 


-001 


to 


-002 


to 


.003 


to 


-004s 



to -0065 



sliding fits 

To 5-in. diameter, inc. 00025 to .0005 

To I -in. diameter, inc 0005 to . 001 

To 2 -in. diameter, inc 001 to . 002 

To 35-in. diameter, inc 002 to .0035 

To 6 -in. diameter, inc 003 to . 005 



standard fits 

diameter, inc Standard to . 00025 

0005 



To l-in 

To I -in. diameter, inc Standard to 

To 2 -in. diameter, inc Standard to 

To 35-in. diameter, inc Standard to 

To 6 -in. diameter, inc Standard to 



.001 

,0015 



Small 
Small 
Small 
Small 
Small 

Small 
Small 
Small 
Small 
Small 

Small 
Small 
SmaJl 
Small 
Small 



driving fits — FOR SUCH PIECES AS ARE REQUIRED TO BE READILY 
TAKEN APART 

To |-in. diameter, inc Standard to . 00025 Large 

To I -in. diameter, inc 00025 to .0005 Large 

To 2 -in. diameter, inc 0005 to . 00075 Large 

To 35-in. diameter, inc 00075 to .001 Large 

To 6 -in. diameter, inc 001 to . 0015 Large 



DRIVING FITS 

To 5-in. diameter, inc 0005 to . 001 

To I -in. diameter, inc 001 to . 002 

To 2 -in. diameter, inc 002 to . 003 

To 35-in. diameter, inc 003 to .004 

To 6 -in. diameter, inc 004 to . 005 



To 



FORCING FITS 

5-in. diameter, inc 00075 to . 0015 



to 


.0025 


to 


.004 


to 


.006 


to 


.009 



To I -in. diameter, inc 0015 

To 2 -in. diameter, inc 0025 

To 35-in. diameter, inc 004 

To 6 -in- diameter, inc 006 

SHRINKING FITS — FOR PIECES TO TAKE HARDENED SHELLS 
AND LESS 

To 5-in. diameter, inc 00025 to . 0005 

To I -in. diameter, inc 0005 to . 001 

To 2 -in. diameter, inc 001 to . 0015 

To 35-in. diameter, inc 0015 to . 002 

To 6 -in. diameter, inc 002 to . 003 



Large 
Large 
Large 
Large 
Large 

Large 
Large 
Large 
Large 
Large 

IN. THICK 

Large 
Large 
Large 
Large 
Large 



SHRINKING FITS — FOR PIECES TO TAKE SHELLS, ETC., HAVING A THICK- 
NESS OF MORE THAN f IN- 

To |-in. diameter, inc .0005 to .001 Large 

To I -in. diameter, inc 001 to . 0025 Large 

To 2 -in. diameter, inc 0025 to . 0035 Large 

To 35-in. diameter, inc 0035 to . 005 Large 

To 6 -in- diameter, inc 005 to . 007 Large 

GRINDING LIMITS FOR HOLES 

To |-in- diameter, inc Standard to . 0005 Large 

To I -in. diameter, inc Standard to .00075 Large 

To 2 -in. diameter, inc Standard to . 001 Large 

To 35-in. diameter, inc Standard to .0015 Large 

To 6 -in. diameter, inc Standard to .002 Large 

To i2-in. diameter, inc Standard to . 0025 Large 

The limits given in the table can be recommended for use in the manu- 
facture of machine parts to produce satisfactory commercial work. These 
limits should be followed under ordinary conditions. Special cases should 
always be considered, as it may be desirable to vary slightly from the 
tables. 



PRESS AND RUNNING FITS 



299 



C. W. Hunt Co.'s Practice in Allowances and Tolerances 
Table 5. — Limits to Diameters of Parallel Shafts and Bushings (Shafts Changing) 



Diameters 


I in. 


2 ins. 


3 ins. 1 4 ins. 


S ins. 


6 ins. 


7 ins. 


8 ins. 


9 ins. 


10 ins. 


Formula 


Press fit 


Shaft 
Shaft 
Shaft 
Hole 


{ 
{ 
{ 

{ 


+ .001 
+ .002 

+ . 0005 
+ •0015 

— .001 

— .002 

+ .000 

— .002 


+ .002 
+ .003 

+ .001 
+ .002 

— .001 

— .002 

+ .000 
— . 002 


+ .003 
+ .004 

+ .0015 

+ .O02S 

— .001 

— .002 

+ .000 
— . 002 


+ .004 
+ .005 

+ .002 
+ •003 

— .002 
-.003 

+ .000 
-.003 


+ .00S 
+ .006 

+ .0025 
+ •0035 

— .002 
-.003 

+ .000 

— .003 


+ .006 
+ .007 

+ .003 
+ .004 

— .002 

— .003 

+ . 000 
-•003 


+ .007 
+ .008 

+ •0035 
+ ■ 0045 

-.003 
— . 004 

+ .000 
— . 004 


+ .008 
+ .009 

+ .004 
+ .005 

-.003 

— . 004 

+ .000 

— .004 


+ .009 
+ .010 

+ . 004s 
+ -OOSS 

-.003 

— .004 

+ .000 

— .004 


+ .010 
+ .011 

+ .00S 
+ .006 

-.003 

— .004 

+ .000 

— .004. 


+ (.001 d+.ooo') 


Drive fit 


+ (.001 d+.ooi) 
+ (.0005 d+.ooo) 




+ (.0005 d+.ooi) 


Hand fit 












All fits 

























Table 6. — ^Limits to Diameters of Parallel Journals and Bearings (Journals Changing) 



Diameters 


I in. 


2 ins. 


3 ins. 


4 ins. 


5 ins. 


6 ins. 


7 ins. 


8 ins. 


9 ins. 


10 ins. 


Formula 


Close fit 


Shaft 


{ 


-.003 
— .005 


— .004 

— . 006 


— .005 

— .007 


— .006 
-.008 


— . 007 
— .009 


-.008 
— .010 


— .009 

— .011 


— . OIO 

— .012 


— .011 
-.013 


— .012 

— .014 


— (.OOI d+.002) 




— (.OOI d+.oo4) 


Free fit 


Shaft 


{ 


-.008 
— .011 


— .009 

— .012 


— .010 
-.013 


— .011 

— .014 


— .012 
-.015 


-.013 
— .016 


— .014 
-.017 


-.015 
-.018 


— .016 

— .019 


-.017 
— .020 


— (.OOI d+.oo7) 




— (.OOI d+.oio) 


Loose fit 


Shaft 


{ 


— .023 
-.028 


— .026 
-.031 


— .029 
-•034 


-.032 
-•037 


-•03s 
— .040 


-.038 
-.043 


— .041 
-.044 


-.044 
-.049 


-.047 
-.052 


-.050 
-•OS5 


— (.003 d+.020) 




-(.003 d+.o25) 




Hole 


/ 
\ 


+ .000 
— .002 


+ .000 
— .002 


+ .000 
— .002 


+ .000 
— .002 


+ .000 
-■003 


+ .000 
-.003 


+ .000 
-.003 


+ .000 
— .004 


+ .000 
— .004 


+ .000 
— .004 




AH fits 















Table 7. — Limits to 


Diameters of 


Taper Shafts 


AND Bushings (Holes 


Changing) 




Diameters 


I in. 


2 ins. 


3 ins. 


4 ins. 


5 ins. 


6 ins. 


7 ins. 


8 ins. 


9 ins. 1 10 ins. 


Formula 


Press fit 


Hole 1 
Hole 1 
Hole { 


-5P 
-6 P 

— i P 

-I P 

+ 
-I P 


-6 P 
-7P 

-I P 

-2 P 

+0 
-I P 


-7 P 
-8 P 

-I P 

-2^P 

+ 
-I P 


-8 P 
-9P 

-2 P 

-3P 

+0 
-I P 


-9 P 
-0 P 

-2|P 

-3^P 

+0 
-I P 


-10 P 
-ir P 

-3 P 

-4 P 

+0 
-I P 


-IIP 
-12 P 

-3lP 
-4^P 

+ 
- I P 


-12 P 
-13 P 

-4 P 

-s P 

+0 
-I P 


-13 P 
-14 P 

-4^P 
-5^P 

+0 
- I P 


-14 P 

-isP 
-s P 

-6 P 

+0 
-I P 


-(Pd+4P) 


Drive fit 


-(Pd+S P) 
-(i Pd+o) 


Hand fit 


-(i Pd+P) 
+0 




-P 



BALANCING MACHINE PARTS 



Balancing Rotating Parts 

Two states of perfect balance must be distinguished — standing or 
static and running or dynamic balance. Standing balance insures 
running balance in the case of thin disks but not in the case of long 
drums, multiple-throw crank shafts or similar pieces. 

The method of obtaining standing balance by means of a rolling 
mandrel supported on a pair of straight-edges is too well known to 
need description. It is adequate for many cases but is not sufficiently 
delicate for high speeds. 

The importance of accurate balance in high speed machinery is 
shown by the fact that a weight of i oz. rotating at 3600 r.p.m. at 
I ft. radius produces an unbalanced centrifugal force of 276 lbs. 

Greater sensitiveness than that of the common parallels is char- 
acteristic of the fixture shown in Fig. i, by the L. A. Goodnow 
Foundry Co. (Amer. Mack., June 16, 1910) by whom it is used for 
balancing fly-wheels. It consists of two large, turned cast-iron 
cones, slightly truncated, through which passes an eye bolt having 
a pivot point projecting downward within the eye, and a large Unk 
threaded through the eye, having a bearing for this pivot joint. 

The turned fly-wheel is supported in a horizontal position, held 
by the two cones entering the bore from either side. Because of 
the point of suspension at the top, the fly-wheel is free and can take 
any position, depending upon whether it is in balance or out of bal- 
ance. If it is out of balance, that fact is easily determined by a 
spirit level on the edge of the rim balanced by an equal weight op- 
posite. Weights are then applied to bring it into a truly horizontal 
position. After this has been done the weights are weighed and a 
line is scratched on the inside of the rim indicating the point where 
weight should be applied and its amount. 

Another standing balance apparatus of high sensitiveness is shown 
in Fig. 2, by P. Fenaux {Amer. Mack., July 2,0, 1908). Although 
giving standing balance only, it appears to be adequate for small 
drum-shaped pieces revolving at high speed and it was, in fact, 
designed for the small armatures of electrically driven phonographs. 

The apparatus consists of a base A with two supports B for the 
axle C. The supports are of hardened tool steel and of such shape 
that the knife-edges of C bear on two points only. The balancing 
part is formed of two flanges D connected by C and the counterweight 
E, of such weight that it will balance the armature of the smallest 
weight. The armature is placed in the notches of the flanges. The 
pin F is used for noting the position when balancing. One of the 
flanges is lengthened by a rod, threaded and ended by a point. 
This point comes in front of an index fixed on the plate A. To- 
ward the end, on each side of the rod G comes a small rubber stop 
H. The upper one is fixed on a rather stiff spring K; the lower one 
on a spring L supported by a long spring M. On the rod are screwed 
three nuts. 

The centers of the armature and of D are above the edges of the 
knives, thus bringing the center of gravity of the system above the 
points of support, so that the smallest difference on one side or the 
other will produce a large movement of the point of G. The arma- 
ture to be balanced is put in the notches with a slot in line with the 
top of the pin F. The lower stop is lowered and the point is brought 
in line with the index by moving N, one of the nuts. Then the 
armature is moved half a turn. If in this new position the point 
has a tendency to go under the index, that means that the side of 
the armature next the pin is too light, and the side first placed there 
is to be drilled. Leaving the nut N as it is, one of the two others is 



moved, so as to bring the point again in line with the index, and this 
movement indicates, by comparison, to what depth the hole or holes 
must be drilled. If instead of coming down the point had pushed 
against the spring K, the reverse operation would have been 
performed. 

The principle of Fig. i has been developed by the Westinghouse 
Machine Co. into the highly sensitive and accurate apparatus 
(patented) shown in Figs. 3-10 {Amer. Mach., July 13, 1911). 




Fig. I. — A sensitive standing balance fixture. 



P3 



r- 



pj~3 



BJpiHiiiiaQasrilimfjfe 



C-Jt 





Fig. 2. — A sensitve standing balance apparatus. 

This machine is used for giving a running balance to the rotors 
of Parsons steam turbines, its application for this purpose being due 
to the fact that if a long drum be divided into elementary disks by 
planes perpendicular to the axis and each slice be given a standing 
balance, the drum made up of the assembled disks will be in both 
standing and running balance. The rotor of the Westinghouse 
turbine is so divided, the disks being sufiiciently thin to give an 
entirely satisfactory result. Theorectically, the customary balancing 



300 



BALANCING MACHINE PARTS 



301 




Fig. 3. — The Westinghouse balancing machine. 




fits the opening in E on one axis, it may be slid along the axis at 
right angles across E, by means of the two adjusting screws K, as 
most clearly shown in Fig. 6. Beam E rests upon knife-edges C, 
which in turn rest in self- aligning sockets in blocks F; these latter 
rest upon the main supports. H is z. counterweight on rod G which 
is rigid with D. 

The operation of the machine is shown in Figs. 7 to 10 inclusive. 
Assuming that the turntable and all the parts connected with it 
are first properly balanced, so that the upper surface of the turn- 
table will remain accurately in a horizontal plane during a complete 
rotation, the disk to be balanced is placed upon the turntable, most 
carefully centered with reference to the spindle and properly clamped 
in position. Four points, A, B, C and D, are located upon the 
periphery of the disk at 90 deg. from each other, at the same radius 
from the center of the spindle, and the hanging counterweight is 
so adjusted that the combined apparatus located upon the knife- 
edges will oscillate very slowly, indicating that the center of gravity 
of the combined mass be just below the plane of the knife-edges. 
The spindle socket is now moved along the beam by means of the 
adjusting screws until the beam is balanced in the horizontal posi- 
tion. This will bring the point X, which indicates the position of a 
vertical line passing through the center of gravity, into the vertical 
plane in which the knife-edges are located, as in Fig. 7. 

Next, the turntable is turned 180 deg., so as to bring the point X 
into the position represented in Fig. 8, and thus out of alignment 
with the knife-edges, in which position the beam wiU be deflected 
from its previous condition of balance. Sufficient weight should 
now be added at some point, as at D, to bring the beam again into 
the position of horizontal balance. The amount of weight added at 
this point D we may represent by n. 

The turntable is now turned 90 deg. and the beam moved by 
means of the adjusting screws until it is brought into the horizontal 
position, when the point X will be in the position indicated in Fig. 
9. The turntable is now given another movement of 180 deg. to 
the position indicated in Fig. 10 and weight added at some point, 
as at C, sufficient to bring the beam again 
into the horizontal or balanced position. The 
amount of weight added at the point C may 
be represented by n'. It remains now to 
locate the line EF which lies in both the 
geometric center and the plane of the center 
of gravity. This may be done by determining 
the angle 9 which is made by the lines EF 
and AC. 

By equating moments about the axes and 
throwing out small factors which would not 
materially affect the result, the following ex- 
pression is obtained: 

Tan. d = —,' 
n 

in which n' must be the greater weight. 

Then, the weight necessary to be added to 
point E or to be taken away from point F 
equals: 



\m Knife Edges 
C A 

Fig, 



>D 
Fig. 7. Fig. 8. Fig. 9. Fig. 10. 

Figs. 4 to 19. — Details of the Westinghouse balancing machine and of the 
method of balancing. 



straight-edges would give the same result, but actually they are not 
sufficiently sensitive. 

Fig. 3 shows a disk section in process of testing for standing bal- 
ance. The balancing machine consists of a turntable A, Figs. 4, 5 
and 6, so mounted as to rotate on spindle B in socket D. Socket D 
is supported by flanges resting upon open beam E and, while it closely 



n' being the greater of the two weights. 

The object of shifting the turntable so as 
to bring the center of gravity over the knife- 
edges is to secure just double the effect of the 
faulty balance when the turntable is turned 
180 deg. This is indicated in the above formula by the factor \. 

Final balancing of the turbine disks or sections is obtained by 
drilling at the points found by this method and in accordance with 
Table i giving the depths of holes of various diameters to remove 
certain weights of metal. This table is obviously of equal applica- 
tion to any other standing balance apparatus. 



302 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 




Fig. II. — The Riddell balancing machine. 



The horizontal position of the turntable is determined by the 
pointer A and scale B, Fig. 3, the pointer being made to oscillate 
equally on each side of the zero point in the manner common with 
delicate chemical balances, thus eliminating even the small friction 
of the knife-edges. 

The Riddell balancing machine of the General Electric Co. is shown 
in Fig. II. It also acts upon the principle that if a collection of 
disks on the same shaft are individually in standing, the assembly 
will be in running balance. 

The disk a to be balanced is supported on the pieces be which, in 
turn, rest on the ball step bearing d and the plate e. When the 
balancing operation is not in progress the plate e rests on three set- 
screws, one of which is shown at /. Extending up into the apparatus 
is a grease cylinder g, which is connected by suitable piping with a screw 
plunger h. At the upper end of the grease cylinder is a flat-topped 
plunger i on which rests the ball-ended screw j. By turning the screw 
plunger h, the plunger iis made to rise, thereby lifting the screw j and 
all the connected parts, including the piece to be balanced, free from 
the setscrews /. This obviously leaves the piece to be balanced free 
to assume an inclined position in accordance with its lack of balance. 

The direction of the unbalance is shown by the movement of the 
multiplying index k, which plays over a polished plate I. To insure 
the true centering of the piece to be balanced with the plate I, the 
ball-ended screw _/ drops, when the parts are lowered on the setscrews 
/, into a conical recess at the top end of the cylinder g. The use of the 
ball step bearing d is to facilitate the checking of the indications by 
trying the balance with the parts in various positions. 



Table i. — Depth of Drilling Necessary to Remove Given Weights When Balancing Machine Parts 



Intermediate weights may be found by adding together the depths of hole corresponding to the different weights that go to make up the whole. 
Example: 

Suppose the piece is out of balance — 27 .4 oz. and it is convenient to use a i-in. drill. 

The depth corresponding to 20 oz. = s.3oins. Depth of Hole 

The depth corresponding to 7 oz. = 1.85 ins. 

The depth corresponding to 0.4 oz. = 0.10 ins. 

Total is 7.25 
Deduct for point of drill o. 10 in. 










Brass 










Steel 








Cast-iron 






■Weight, 




Depth to drill 


in ins. 




Depth to drill in ins. 


Depth to drill in ins. 


oz. 


i-in. drill 


1 drill 


i drill 


f drill 


J drill 


I-in. drill 


f drill 


J drill 


1 drill 


J drill 


I-in. drill 


f drill 1 J drill 


1 drill 


i drill 


50 


13.25 

10.60 

7 -95 










14.19 

11.36 

8.52 

5 68 


25.26 
20.20 

iS.i6 








15-43 

12.36 

9. 27 


27.50 

22.00 
16. 50 








40 


18.86 

14. 16 

9.44 

4.72 




















30 




















20 


5. 30 








22.73 
11.38 






6. 19 


10.94 

5. 47 


24.75 
12.38 






10 


2.6s 


10.60 


18.86 




2.83 


5. 05 


20.21 




3 08 


22 .02 




9 


2.38 


4.24 


9^54 


16.97 




2.55 


4.55 


10.23 


18.18 




2.78 


4.95 


II. 12 


19.81 




8 


2. 12 


3.77 


8.48 


15.08 




2.27 


4.04 


9.08 


16.16 


36.37 


2.47 


4.40 


9.90 


17.60 


39.60 


7 


1.8s 


3.30 


7.42 


13- 20 


29.71 


1.98 


3^54 


7.96 


14.14 


31.83 


2.16 


3.86 


8.66 


15.41 


34.65 


6 


1.S9 


2.83 


6.36 


II. 31 


25.46 


I. 71 


3^04 


6.82 


12. 12 


27.28 


1.86 


3.31 


7.43 


13.20 


29.72 


5 


1.32 


2.36 


5.30 


9.43 


21 .22 


1. 41 


2^54 


5.68 


10. 10 


22.73 


1.54 


2.75 


6.18 


II .00 


24.75 


4 


1.06 


1.88 


4.24 


7^54 


16,96 


1. 14 


2.01 


4.55 


8.08 


18.18 


1.24 


2.20 


4,96 


8.80 


3;9.8o 


3 


.79 


1. 41 


3^i8 


5^65 


12. 72 


.85 


I. 51 


3^41 


6.06 


13.63 


.93 


1.65 


3.71 


6.60 


14.8s 


2 


.S3 


• 94 


2 . 12 


3^77 


8.48 


.57 


I. 01 


2.28 


4.04 


9.08 


.62 


1 .09 


2.48 


4.40 


9.90 


1 


.26 


• 47 


1.06 


1.88 


4.24 


.28 


.50 


I. 14 


2.02 


4.55 


.31 


.55 


1.24 


2 . 20 


4.96 


.9 


.24 


.42 


■ 95 


1.69 


3^81 


.26 


.45 


1.02 


1.83 


4.08 


.28 


• 49 


I. II 


1.98 


4.34 


.8 


.21 


.37 


.85 


1.52 


3^39 


.22 


.40 


.91 


1.62 


3.64 


• 25 


.44 


.99 


1.76 


3.96 


■ 7 


.18 


■33 


•74 


1.33 


2.96 


.19 


.35 


.79 


1.4 


3.17 


.22 


.39 


.87 


1.54 


3.46 


.6 


.16 


• 28 


.63 


1. 14 


2.54 


.17 


.30 


.67 


1 .22 


2.72 


.19 


.33 


• 74 


1.32 


2.97 


■ S 


.13 


• 23 


• 53 


.95 


2. 12 


.14 


.25 


.57 


I. 01 


2.28 


. .15 


.28 


.62 


1. 10 


2.48 


•4 


. II 


•19 


■43 


.76 


1 .69 


.11 


.20 


.45 


.81 


1.83 


. 12 


. 22 


.50 


.88 


1.98 


• 3 


.08 


.14 


.31 


• 57 


1.27 


.09 


.15 


.33 


.61 


1.37 


.09 


.17 


■ 37 


.66 


1.49 


.2 


.05 


■ 09 


.21 


.38 


.85 


.06 


. 10 


.22 


.41 


.91 


.06 


. II 


■ 25 


.44 


-99 


. I 


.02 


■ OS 


. II 


.19 


.42 


.02 


.05 


. II 


.20 


.45 


• 03 


.06 


. 12 


.22 


.50 


Deduct for 


. 10 


.07s 


.05 


.04 


.03 


. 10 


.075 


■05 


.04 


.03 


. 10 


.075 


.05 


.04 


.03 


point of 
































drill. 






























. 



Required depth of hole 7.1S ins. 



BALANCING MACHINE PARTS 



303 



The problem of balancing the rotating parts of high speed machinery 
involves three fundamental distinctions. The first of these relates 
to the shapes of bodies to be balanced which are classified as thin 
disks and long drums. 

Ideally and theoretically, a thin disk is one of infinitesimal thick- 
ness, but, since such bodies are not used in machine construction, 
a thin disk is here to be understood as one whose thickness parallel 
with its axis is inconsiderable in comparison with its diameter. The 
thinner it is, the more it approaches the theoretical ideal, the thick- 
ness admissible in any case depending on the conditions, especially 
the speed and the accuracy of balance required. The actual thick- 
ness admissible cannot be defined in exact terms. 

The terms thin disk and long drum, however, are not satisfactory, 
because under them are included bodies of skeleton forms which have 
the same properties as regards balancing, but which frequently do 
not suggest the terms under which they are classified. Thus, a 
revolving spider, like the arms of a pulley with the rim removed, 




Fig. 13. 



c^^e^o 



Fig. 14. 



Fig. 12. 



^^ 



ypj 



(mjn\ 



itij 



Fig. is- 
Figs. 12 to 15. — Thin disks and long drums. 

"Fig. 12, is, for the present purpose, classified as a thin disk, and, if 
all the arms but two are removed, as in Fig. 13, this still remains 
true. The term must even be applied to a pair of heavy balls ab, 
connected by a pair of light arms, Fig. 14. Little as such a construc- 
tion suggests a thin disk, it at least conforms to the definition that 
the' thickness parallel with the axis is inconsiderable in comparison 
with the diameter of the circle of revolution. 

The term long drum must also be stretched to include bodies which 
are far from being drums, the extreme case being, perhaps, an auto- 
mobile crankshaft. Fig. 15, which, while not suggesting a -drum, 
nevertheless has the same properties as regards balancing and is 
hence included in the classification. 

The second fundamental distinction is that between standing and 
running balance — that is, balance when at rest and when in motion. 
These conditions have other names, as gravity or static, for standing, 
and centrifugal or dynamic, for running balance. Thin disks which 
are in standing are also in running balance but this is not necessarily 
true of long drums, in which latter no attempt at correcting standing 
unbalance has any assured value in correcting running unbalance, 
and it may and often does make matters worse instead of better. 
With one real and another apparent exception, a body in balance at 
one speed is in balance at all speeds, with, however, the proviso that 
the body does not change its form because of the stresses set up by the 
centrifugal forces of its various parts. C. H. Norton has discovered 
that four-throw automobile crankshafts, when unsupported at the 
center, will spring as much as J^e in. at 1200 r.p.m. and hence show 
unbalance at high speeds and balance at low. Such action is not, 
however, an exception to the law, nor is the fact that unbalanced 
bodies vibrate more violently as the speed is increased. 



The real exception to the law is this: There is a certain critical 
speed at which an unbalanced body revolves as though balanced. 
This action, which is conclusively proven by observation, is the 
most curious and obscure property of revolving bodies. 

It is commonly but erroneously believed that a body must be 
balanced for the speed at which it is to run, whereas, the fact is that 
if it is balanced at any speed, except the critical speed, it will be 
balanced at all others. 

This belief is based on the behavior of some rapidly revolving 
bodies, for example emery wheels, which run quietly at the working 
speed but vibrate markedly and even violently at some speed through 
which they pass when slowing down from the working speed to a state 
of rest. This action is due to synchronism between the revolution 
time of the wheel and the natural period of vibration of the support. 
At this synchronizing speed a slight unbalance gives rise to a marked 
vibration although insufiicient to produce an appreciable efiect at 
other speeds. 

The third fundamental distinction is that between free and con- 
strained rotation. A freely rotating body is one which is without 
supports and hence free to do whatever it desires. The earth and 
other heavenly bodies are examples of absolutely freely rotating 
bodies, whUe a spinning ball, hung from a twisted string, is, to all 
intents and purposes, free so far as the plane of rotation is concerned. 

A body in constrained rotation is one which is supported by shafts 
and bearings. Constraint is, however, relative. A small flywheel 
on a short, stiff shaft, is highly constrained, while a heavy wheel on 
a long, flexible shaft is lightly constrained. Absolute constraint is 
impossible because no support is without some elasticity. If its 
shaft is flexible enough and its speed high enough, a body in con- 
strained rotation will behave as though free, changing from the 
action of a constrained to that of a free body at the first critical 
speed mentioned above — that at which an unbalanced body behaves 
as though balanced. 

A free body rotates about an axis passing through its center of gravity, 
or about a gravity axis, for short. A moment's reflection will show 
that this implies that the prominent, or high, side of such a body is 
the light side. Constrained bodies (provided the constraint is 
sufficient) do just the opposite; the high side of such bodies being the 
heavy side. 

This statement is, however, true in a general sense only. Observa- 
tion shows that the highest spot lags behind the heavy side by an angle, 
which increases with the speed. Observation further shows that at 
sufficiently high speeds (that is, above the critical speed) the action 
is reversed, the light side running high in constrained rotation at 
such speeds, as it does in free rotation at all speeds, while the lag, 
measured from the light high side, becomes a lead. 

Experiment shows the lag to increase with the speed and the speed at 
which the angle of lag equals 90 deg. is the critical speed at which 
the change takes place, and at higher speeds the light side runs 
high. At these higher speeds the increasing tangential resistance 
leads to continual increase of the lag which, measured from the light 
side which is now high, becomes a lead. 

The balancing of long drums involves an application of the principle 
of couples, by which term is meant two equal and opposfed forces 
acting in parallel directions but not in the same straight line. A 
couple has no single resultant and can be balanced only by another 
couple having the same product of force and arm, that is to say, the 
same moment. 

An infinite number of combinations of force and arm may produce 
the balance, the only requirement being equality of their product 
with that of the first couple. Moreover, the balancing couple may 
be applied anywhere in the plane of the first couple. 

Let Fig. 16 represent a light strip of wood with equal weights ab 
attached to it by cords, of which one runs over a sheave. The bar 
is under the influence of a couple and, representing the magnitudes 
of the forces by the lengths of the arrows ab, it is obvious at a glance 
that equal and opposite forces cd will balance the couple. These 



304 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



forces may, moreover, be applied anywhere on the bar, as at ef, 
their distance apart being the same, with the same result while, if 
the forces be halved and their arm doubled, as at gh, balance will 
still be unchanged, and this will be true for any couple wherever 
applied, provided the product of its force and arm be equal to that 
of the applied couple. 

In Fig. 17 a and h are two balls of equal weight mounted by radial 
arms on a revolving shaft and at equal radial distances. As the 
system revolves the balls generate equal centrifugal forces which 
form a revolving couple. Nevertheless, the center of gravity of the 
two balls lies in the center line of the shaft and hence, when at rest, 
the entire system is in balance. This is an illustration of the condi- 
tion of a body which is in standing balance but running unbalance. 



bO 



e 

t 



Tl 



U 



\d 



\f 



Qa 



Fig. 16. 



c 






d h 

? ? 


3 
9 


-V/Z/A 






9^ 1 


1%^ 




0' 


1 

1 
1 






V//A 


c 


) 


9 


kt 
I 


& FlG. ] 


J. 





Fig. 18. Fig. 19, Fig. 20. 

Figs. 16 to 20. — The action of long drums. 

Precisely as in Fig. 16, the balancing of the system involves the 
application of an opposing couple having the same moment as the 
disturbing couple. If we introduce two weights ci of the same weight 
and at the same radius as ah, the balance is obviously accomplished. 
These added weights are analogous to the balancing forces cd of 
Fig. 16, and as various substitutes were found for cd, of Fig. 16, so 
may substitutes be found for cd, of Fig. 17. The balance weights 
may be increased and the radius reduced, as at ef (the increased 
weights being shown by drawing the balls larger) ; the original weights 
may be rhoved bodily lengthwise of the shaft, as at gh; the weights 
may be smaller if placed further apart, as at ij, or larger if placed 
closer together, while additional variations may be made by combin- 
ing those already made and changing simultaneously aU three vari- 
ables, weight, radius and distance apart. In short, an indefinite 
number of arrangements of the balance weights are possible. 

The conditions of Fig. 17 appear in slightly disguised form in Fig. 
18, in which the weighted arms are replaced by the disks having equal 
heavy spots at ab. The construction of Fig. 18, like that of Fig. 17, 
is in standing, but not in running, balance, the only important 
difference between the two being that the cause of the running un- 
balance is apparent to the eye in Fig. 17, but not in Fig. 18. It will 
be observed that the disks of Fig. 18 are individually out of both 



standing and running balance. If they are removed and placed 
individually in standing balance and then replaced, the condition 
leading to running unbalance wiU be corrected. When the construc- 
tion is such as to permit this to be done, this principle is frequently 
used to produce running balance, the Westinghouse and Riddell 
balancing machines being applications of this principle. 

In Fig. 19 the disks of Fig. 18 are replaced by a drum having equal 
heavy spots at db, but the conditions are unchanged. The drum, 
like the disks, is in standing balance but running unbalance. If, in 
place of the heavy spots, we assume the presence of light spots due 
to sponginess, we will have conditions that are more common in 
practice and which lead to the same result, the denser material oppo- 
site the spongy spots producing the excess of centrifugal force, which 
leads to running unbalance. 

The drum of Fig. 19 represents the conditions which confront us 
when balancing any long drum. Remember that the body is in 
standing balance. The running unbalance is produced by centrifu- 
gal force. To generate centrifugal force motion is necessary, and 
it would, therefore, seem to be safe deduction that no possible test 
which can be applied to the body when at rest can detect the cause 
of the unbalance or find a remedy for it. 




Fig. 21. — The Norton running balance machine. 

Not only is the detection of running unbalance impossible by a 
test of the body at rest, but, if a long drum is initially in both stand- 
ing and running unbalance, the correction of the former may increase 
the latter. Such correction is, in fact, nearly as apt to do harm as 
good. The conditions under which harm is done are shown in 
Fig. 20. Of the four balls attached to the shaft by light equal 
weight arms, a, b and c are of the same weight, while the weight of 
d is equal to two of the other balls. The system is plainly out of 
balance both standing and running, the latter due to the centrifugal 
force of the excess weight of d — that is, a weight equal to that of 
one of the other balls. If correction be made by halving the weight 
of d, both standing and running unbalance will be corrected. Stand- 






BALANCING MACHINE PARTS 



305 



ing balance may, however, be made equally as well, and is just as 
apt to be made by removing weight a. If this be done, the result, 
when the system is made to revolve, is that the centrifugal force of 
the excess weight of d is unchanged, whUe a new unbalanced centrifu- 
gal force of b at the other end of the shaft has been added, correc- 
tion of standing unbalance having increased the running unbalance. 
It will thus be seen that no attempt at bringing about standing balance 
of long drums has any assured value. 

The balancing of long drums is expeditiously carried out by 
the running balance machine of the Norton Grinding Co., Fig. 21 
{Amer. Mack., Dec. 16, 1909) and by it the theoretical principles of 
running balance have been experimentally proven {Amer. Mach., 
Aug. II, 1910). This machine is the first organized attempt to attack 
the problem as primarily one of running balance and to discard all 
plans of attack through the channel of standing balance. This it 
does by providing means for recording, interpreting and correcting 
the indications given by unbalanced revolving bodies in motion. It 
is, moreover, applicable to those numerous cases in which the slicing 
of the revolving member is structurally impossible. It is also the 
first balancing machine to introduce the principle of light constraint, 
thereby leading to clear indications at moderate speed and to a com- 
paratively low critical speed. 

The piece to be balanced is carried at each end on four rollers 
which are mounted in suitable cradles and carried on the upper 
ends of inverted pendulums. The lateral motion thus provided is 
limited by rubber disks through which the pendulum rods pass. 
Multiplying vertical pointers, plainly shown, are so connected with 
the rods as to vibrate with them and to magnify the vibration to 
the eye. Adjustable scriber points are provided, the markings being 
made more distinct by coating the shaft with red paint. The 
machine includes an electric motor together with a friction disk drive 
by which the speed may be varied and the direction of motion be 
reversed for reasons that have been explained by Mr. Douglas. 

Among other things the machine has demonstrated that high- 
speed rotating parts should be so designed that they will not distort 
from centrifugal action, if rotated free at any speed at which they 
may run in use, and thus destroy a state of running balance. 

In the use of this machine the technique of balancing long drums has 
been greatly simplified and this technique may be applied to extem- 
porized apparatus. 

The first difficulty which the machine surmounts is that due to 
the indeterminate value of the lag, because of which the high spot 
does not correctly indicate the heavy spot. The machine is revers- 
ible in its direction of rotation and, the piece to be balanced being 
driven in opposite directions and at the same speed, the high spots 
are marked. The lag being the same in both cases, the heavy spot 
lies on the diameter which equally divides the angle between them. 
There remains, however, a second difficulty due to the fact that, if 
the speed be below the critical speed and the heavy side high, the 
heavy spot is at the same side of the body as the marks while, if the 
speed be above the critical speed and the light side high, the heavy 
spot is at the side of the body opposite the marks, and the behavior 
of the body does not indicate which condition obtains: 

The method of solving this problem is shown in Figs. 22 and 23 
which show an automobile crank shaft which has first been placed 
in running balance and then thrown into unbalance by attaching the 
weight a to one of the crank cheeks in order to show to the eye the 
position of the heavy spot. The marks made when the piece runs 
below the critical speed are shown in Fig. 22 in which arrow heads 
have been placed at the middle of their lengths and pointing in the 
direction of rotation when the marks were made. The arrow heads 
will be seen to point toward one another and toward the heavy spot. 
Similarly the marks made when the piece runs above the critical 
speed are shown in Fig. 23, in which the arrow heads, which again 
point in the direction of rotation but away from one another, 
still point toward the heavy spot. From this we get the rule : 
20 



When the arrows point toward one another the heavy side runs high. 
When the arrows point away from one another the light side runs high. 
In both cases the arrows point toward the heavy spot which lies midway 
between them. 

Because of the low critical speed due to the light contraint, this 
machine is usually operated above the critical speed. The opposite 
practice obtains, however, in the case of bodies which are so badly 
out of balance as to endanger their flying out of the machine if run 
above the critical speed and also in the case of bodies that are so 
flexible as to distort from the centrifugal force if run above the 
critical speed. 

a 




Fig. 22. — Marks made below the critical speed with the heavy side 

high. 




Fig. 23. — Marks made above the critical speed with the light side 

high. 

A more recent machine is that of N. W. Akimofp {Trans. A.S.M.E., 
1916) in which a squirrel cage with longitudinally adjustable bars is 
mounted parallel with and made to revolve at the same speed as the 
piece under test. The cage being initially in balance, its bars are 
adjusted to throw it out of balance, the adjustment being continued 
by trial untU the unbalance of the cage compensates that of the 
piece under test. The resulting displaced bars show directly the 
plane of the unbalance, whUe the amount of the displacement, 
properly interpreted, gives the values of the compensating weights. 
A standard speed is adopted and the frame of the machine is sup- 
ported by a spring of such strength that the period of vibration of 
the whole synchronizes with the standard speed, the result being a 
high degree of sensitiveness and accuracy. 

Balancing Reciprocating Parts 

For the position of the center of gravity of counterweights of 
usual forms see Center of Gravity. 

Reciprocating parts driven by a crank and connecting rod may be 
balanced in the direction of the reciprocation at the expense of unbal- 
ance in a direction at right angles thereto. To do this, consider the 
mass of the reciprocating parts, including aU of the connecting rod, 
as concentrated at the center of the crank pin and calculate its centrifu.- 
gal force. Then a mass, added on a radial line opposite the crank 
pin or subtracted on the side of the crank pin, which wiU generate an 
equal centrifugal force will balance the reciprocating parts in the 
direction of reciprocation. At the same radius as the crank-pin 
center, the weight should obviously equal that of the reciprocating 
parts. At any other radius the weight is inversely proportional to 
the radius — the radius being understood to be that of the center of 
gravity of the mass. 

This mass will give perfect balance in the direction of the recip- 
rocation with a Scotch yoke or slotted cross-head. With a connect- 
ing rod it gives a slight overbalance at one center and a slight under- 
balance at the other, the result being the best that can be obtained. 

The center of gravity of the counterweight, for perfect results, must 
be in line with the center of the piston rod which, in center-crank 



306 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



engines, can be secured by dividing the weight equally between the 
two cranks. In side-crank engines there must be a slight offset 
with a resulting negligible horizontal twisting moment. In slow- 
speed engines it is frequently impracticable to place sufiicient counter- 
weight in the crank disk because of lack of room. Such engines do 
not commonly require balancing but, when necessary, satisfactory re- 
sults have been obtained by placing as much of the counterweight 
as possible in the crank disk and the remainder in the fly-wheel. 
For a applicationn of this plan to large Corliss engines see a paper, 
Counterweights for Large Engines, by Dr. D. S. Jacobus in Trans. 
A. S. M. E., Vol. 26. The final result was a considerable hori- 
zontal twisting moment but a nearly complete stoppage of serious 
vibrations. 

In horizontal engines the provision of a suitable counterweight 
in the crank disc balances the parts in a horizontal direction but it 
introduces a tendency toward vertical vibration equal to the tend- 
ency toward horizontal vibration with no counterweight whatever, 
and this tendency must be resisted by the foundation. 

In vertical engines, on a proper foundation, the perfect balance 
with this arrangement is vertical, where it is not needed, while the 




Fig. 24. — Diagrammatic arrangement of 
crusher parts. 



unbalance is horizontal where it does harm. Such engines are best 
when entirely unbalanced, as that arrangement leaves the unbalance 
vertical where it is resisted by the foundation. 

Reciprocating parts driven by more complex mechanism than 
a crank and connecting rod, may be balanced in the direction of 
the reciprocation, at the expense of unbalance in a direction at right 
angles thereto. The application of the method to a rock crusher 
is thus explained by Prof. O. P. Hood (Amer. Mack., Nov. 26, 1908) : 
The crusher balanced was located in a rock house of a Lake Superior 
copper mine, where the elevation of the crusher above the ground 
led to an actual horizontal vibration of the rock house of .22 in., 
which was reduced to a negligible amount by the method 
described. 

It is now feasible to run crushers of the type described at any 
reasonable speed without the danger of racking the building or re- 
quiring unusually heavy construction to resist useless forces. 

The problem was as follows: 

A mass of approximately 8 tons vibrated 175 times a minute 
through a short path, the nature of the vibration being determined by 
an eccentric revolving at a uniform speed, a pitman, toggle joint 
and swinging jaw. Is the nature of this movement such that a 
rotating weight, properly placed, can balance the inertia of the 
swinging jaw and parts moving with it? The forces which move 
the jaw must react against the frame and building to which it is 
attached and move the mass of these to produce the vibration. To 



find the amount of force it is necessary to find the velocity of the 
moving jaw, from which we can find the acceleration of the mass. 

Fig. 24 shows diagrammatically the arrangements of parts of a 
crusher although very much distorted. Let BB' represent the 
path of the eccentric crank, AB the pitman articulated at A to the 
two links AC and AD. Link AC abuts against the frame at C and 
the point D is constrained to move about H as a, center, or prac- 
tically in the direction ED. As the eccentric revolves, the point D 
(a part of the jaw DH) has a small motion which crushes the rock 
against the frame. The character of this motion is readily found 
by finding the position of D for several positions of B, as shown in 
dotted lines at B'A'D'. An inspection of this diagram shows that 
A moves about C as a center and also relatively about Z>, so that if 
the eccentric circle be drawn AB" FG and arcs AF from C and AG 
from Z? as a center, the distance, as A'K between the two arcs oppo- 
site any point in the circle as B" measures the distance the point 
D has moved, from its inner position. Thus A'K is equal to D'D, 
and this corresponds to the position of the crank shown at B'. 

In the actual crusher under consideration the eccentricity OB is 
I in., AB 37 ins., AC 13 Ins., AD 23 ins., J A 2,2 ins., JE 3 ins. 

The jaw movement and the eccen- 
tric circle are so small that it is well 
to magnify the movement image, as in 
Fig. 25, where the arcs, AF and AG are 
drawn properly in proportion to the 
eccentric circle, which is here marked 
into 10-deg. spaces. Since the eccen- 
tric revolves uniformly these lo-deg. 
spaces are passed in equal times and, 
when plotted in Fig. 26, as horizontal 
distances, represent time. Laying off 
vertically in Fig. 26 the several dis- 
tances similar to A'K for each angle of 
the eccentric, we find the curve OJ, 
whose distance from the axis OX 
represents the jaw movement magni- 
fied. The actual maximum jaw move- 
ment was found to be .484 in. and this 
is represented in the diagram by XJ , 
which measures 1.95 ins.,^ therefore, 
• I in. in hight of the diagram represents 
.248 in. of jaw movement. From the curve OJ we can find the velocity 
at any point, for the velocity is the rate at which the jaw is changing 
its position, and by drawing tangents as LM at the 40-deg. point, then 
the vertical distance MN represents the distance the jaw would have 
traveled while the eccentric was moving the time LN, provided the 
rate had been uniform. The distance LN is taken as any conve- 
nient distance as 150 deg., but is taken the same for each point on 
the curve OJ. Plotting the MN distances with time as the base 
gives the curve OVX, which is the velocity curve of the jaw. The 
scale of this curve can be found from any of the triangles LMN, 
for the time is given by the constant base LN and the space by MN, 
the velocity being equal to the space divided by the time. The 
distance MN measures 1.74 ins., which equals i.74X.248 = .43i in., 
movement of the jaw. It is convenient to assume one revolution 
per second for the speed so that since the base LN = 150 deg., the 
time will be i5o-i-36o = .4i7 sec. 

The velocity when running at 60 r.p.m. equals .431 -t-.4I7 = 1.034 
ins. per sec. at the 40-deg. point, and this is represented by the hight 
MN of 1.74 ins. One inch in height of the velocity curve, there- 
fore, represents 1.034 -f-i.74Xi2 = .o5 ft. per sec. 

To change the velocity of a mass requires a force proportional to 
the amount of change of velocity made in a second, that is, in pro- 
portion to the acceleration which is the rate of change of velocity. 

' The dimensions refer to the original drawing of which the engraving is a 
reduced copy. 



Fig. 25. — Diagram of magnified jaw 
movement. 



BALANCING MACHINE PARTS 



307 



From the velocity curve we can find this acceleration by drawing 
tangents as before. To illustrate, at the 6o-deg. point the velocity 
changes at the rate shown by the tangent L'M' and in the time 
L'N' would change the amount M'N'. Plotting these values of 
M'N' for each of the points of the velocity curve we have the ac- 
celeration curve RK'S. When the velocity is the greatest, and 
before it begins to decrease, the velocity is unchanging for an instant, 
therefore there is no acceleration as shown by the curve RK'S having 
a zero value at K'. Up to this point the jaw has been increasing in 
velocity and, therefore, requiring a force to make the change, but 
after passing K' the jaw velocity is decreasing and it now requires a 
force to stop it. This must evidently be in the opposite direction 
to the first force and is, therefore, shown below the line OX. Since 
the forces required are proportional to the acceleration, the curve 
RK'S must also represent to some other scale the forces tending to 
shake the machine in the line of direction of the movement of the jaw. 

That particular point on the jaw, whose movement has the same 
effect as if all the moving mass was concentrated at that point, is 
called the center of gyration which is measured from the point // 
about which the jaw swings. By computation from the drawings 
of the section of the casting this was found to be 44 ins. from the 
center, and here the velocities would be greater than at D in the 
proportion of 44 to 37, or 19 per cent. more. 

The scale of the curve RK'S can be found from the triangle 
L'M'N', where M'N' = i .9 2 ins. This represents i .9 2 X .05 = .096 ft. 
per sec. as the velocity gained in the time L'N'. One-half second is 
represented by OX, which is 6.28 ins. long. Therefore, L'N' repre- 
sents .254 sec. The acceleration is .096 -f-. 254 = .384 ft. per sec. 
per sec. This is represented by a line M'N' 1.92 ins. long, there- 
fore, each inch in height represents .384 -^ 1.92 = .2 ft. per sec. per 
sec. acceleration. 

The maximum force required is evidently at the beginning of the 
jaw movement when the force is represented by OR. Here at 60 
r.p.m. the acceleration is 47 ins. X -2 = .94 ft. per sec. per sec, which 
at the radius of gyration of the jaw wiU be 19 per cent, more, or 
1. 12 ft. per sec. per sec. The mass moved weighs 16,000 lbs. The 
force required to move this will be 16,000-^-32, equals 500, multi- 
plied by the acceleration 1.12 or 560 lbs. when running at 60 r.p.m. 
The force of acceleration will vary as the square of the velocity, so 
that it is no wonder that several such crushers will shake a building 
when run at 175 r.p.m., as many do, when the shaking force for 
each crusher amounts to nearly two and a half tons. 

From the curve of acceleration it is noted that this force is applied 
as a push to the building at the beginning of the jaw movement, 
growing less in amount until it reverses just before the quarter 
revolution. The force then becomes a pull on the building, reaching 
an amount about 72 per cent, of the maximum push, but keeping 
at it longer. From 180 to 360 deg. the acceleration curve would 
be symmetrical about the line JS so that there would be a push on 
the building for about 46 per cent, of the time, and a pull for 54 per 
cent. An inspection of the curve shows that it does not depart 
widely from that form which would be made by an unbalanced 
weight revolving with the eccentric. Such a curve would be a sine 
curve. 

Laying off OS' equal to XS we will take a point T half way between 
S' and R and assume this as the average force to be balanced, for 
if underbalanced at R it will be equally overbalanced at S. 
The curve TT' is readily laid in as a sine curve. To furnish this 
balancing force we can place a weight in the fly-wheel that can 



give the forces TT', but this must be in such phase with the 
moving jaw that OT shall oppose OR instead of being in phase with 
it as shown. 

We find that there is a convenient place in the fly-wheel of the 
crusher where the counterweight mass will be about 2 ft. from the 
center of the shaft. Here the velocity at i r.p.m. will be 1 2.50 ft. per 
sec, and the force will be equal to the weight times the velocity 
squared divided by 32 times the radius in ft. The force OT scales 
480 lbs., if OR represents 560, so that the weight we seek figures 196 
lbs. Half of this can be put in each fly-wheel in such place that, as 
the jaw begins to move forward the counterweight will begin to move 
back. 

In the diagram Fig. 26, if we combine the curve RK'S and TKT' 
we have the resultant curve WYZ. This shows that the shaking 




Fig. 26. — Graphical solution of the crusher balancing problem. 

forces have been greatly reduced and their alternations have been 
doubled so that the smaller force also has a shorter time to produce 
movement before reversal. 

This resultant XYZ shows that a nearly complete balance could be 
had by adding to the single rotating weight here described a second 
weight rotating at twice the revolutions per minute of the main 
eccentric. If the inertia of this secondary weight was made equal 
and opposed to OW the final resultant would be nearly a straight 
line. The secondary weight would have to be geared to the main 
shaft with a gear ratio of 2 to i, making a complication of parts not 
warranted or needed in the usual crusher installation, but still of 
possible value in extreme cases. 



MISCELLANEOUS MECHANISMS, CONSTRUCTIONS AND DATA 



The Hooke Universal Coupling 

The Hooke universal shaft coupling, Figs, i and 2, does not, when 
used singly, transmit a uniform motion and, when two couplings are 
used to connect offset shafts, they are often so assembled as to double 
the irregularity of a single coupUng, although, when correctly assem- 
bled, the irregularities neutralize each other and give as a final result 
a true, uniform motion. Fig. i shows the correct and Fig. 2 the incor- 
rect arrangement. In the former the yokes of the intermediate 
shaft are in the same plane while in the latter they are at right angles 
to one another. 

It is also necessary that the angles between the intermediate and the 
end axes be equal. This follows as a matter of course if the end axes 
be parallel but otherwise it must be provided for. 

Actual constructions are shown in Figs. 3, 4, and 5. In Fig. 3 
' the offsetting of the pivot pins to permit their passing each other 
introduces a small additional error. In Fig. 4, the bearing bushings 
are clamped in position and also locked by detent pins. In Fig. 5 
the cross takes the form of an external split ring, which holds the 
bearing bushings. The pivots are integral with the forks. The 
exterior of the forks and the interior of the rings are spherical to 
retain the grease. 

In comparing the movements of the two shafts two methods are 
possible {a) The angular velocities or (6) the angular positions of the 
shafts may be compared. The angular velocities of the two shafts 
are given by the equation: 

V cos A 

V "" I - sin M sin ^B 
in which V = speed of driving shaft, 

D = speed of driven shaft, 
A = angle between shafts, 

B = angle of rotation of driving shaft from position of shaft 
A, Fig. I. 



Relative speeds of the driven shaft for various angles between 
the shafts have been calculated from this equation by Earl Buck- 
ingham (Amer. Mack., Jan. 16, 19 13). The results are given in 
Table i and graphically in Fig. 6. 

The relative positions of the two shafts are given by the equation: 
tan C = tan B cos A 
in which A = angle between shafts, 

£ = angle of rotation of driving shaft from position of 

shaft A, Fig. i, 
C = angle of rotation of driven shaft from corresponding 
position. 




Split Ring 

Fig. s. 
Figs, i to 5. — The Hooke universal shaft coupling. 



350 




- 








' 












~ 




































— 


~ 


































































































































































































































































300 






























































































































































^ 
































































I 1 
































































y 
































250 
































\ 






























































^ 
































































































































































































\ 
































































V 






























200 






























/' 


N ^ 






























































r 


V 






























































/ 


s 
































































A 
































































- 
































150 
































*^ \_ 




\ 
































































\ 


































J 


= 


lO- 


_ 




_ 


A 


J. 


^ 




4 






■"~-- 




J 


Nf 




_ 






_ 




_ 


_ 


_ 


_, 


^ 


_ 


100 


-A 


i- 


y 


-t 


-'I 

I 


l)^ 


L 




s' 


8 


^ 




= — - 


: — 




r 


^ 




Sw 




— 


— 




— 


— 


— 


— 


- 


- 










— 




-M 


^ 


•J^ 


/ 


/ 












r 




\ 


\ 


^ 


f^ 


^ 




— 


— 


n 


^ 


r 


^ 










"' 


^ 








/ 




/ 


1 










\ 




\ 




s 




^ 




— , 


^ 




— 


A 


"(^ 






--1 




y' 




/ 




/ 












s 




s 




V 






.^ 






~ 


— 1 


— 






Ti 




1 




/ 






' 












\ 






^ 




s 








- 


IT: 


_ 


» 


50 




1 


A 


? 


'\'^ 




< 






/ 
















\ 






s 








-«. 












^0 


'i 




/ 




















\ 










^ 


















— 






f^* 




\ 
























s 










~ 


— ■ 


— 


^ 


__ 


















-^ 






■^R 


n^ 
























■~ 




^ 














=: 


— 


^ 


— 


_ 


_ 


_ 


-/ 


_ 




^9= 


_ 










_ 


_ 


_ 


_ 






_ 




_ 


_ 


i: 


^ 


— 


- 


U 






L_ 






La 






Is 






Ld 


1= 










1= 


^ 




1= 






L= 




Ui 


Ui 


\m 


Ib 


^ 


aj 



10 20 30 40 60 60 70 80 90 100 110 120 130 140 150 160 170 180 
Angle j3 of Rotation of Driving Shaft 

Fig. 6. 



S 90 ^ 




•S 80 - ^ 


'V 


^ '?" "V 


\ 


« 60 - S 




u«so + jt ^ 


»5.„° ^^ ^v ^ 


1^0 I i^^^ 


1 0,? ^ -- ^.^^ 


2 2** Z ..^^ ^^=v Ss^ 


5 ° / ^^^s,^ 


S ^" -/7^^'" "~"^-"'^5 




3 "HrN4^"oi|-^WiJ2oWfl^ffl^±R^^^y 


' t^^rro^^^^rm 


s 10 ^^^52^ — T-^^^^ji 


> 0^° ^x<:--^e-i yt 


S 20^jp^C:^-^ J_J^ r^ 


^ Qn ^^v>.^=^^^_--'^ t 


•& ^° ^^ \-W 


i^-.„° ^ ^^'X.X,i 


^■3^" ^ ~-~3^I,c 


1-l.d ^\ ^-^SO 


gOQBO ^;^ jp 


S eA S 4>a-«fl m' 


j5 60 - .:> ^ >a ^ 




« 70 ^^ - 


a \ 


1 ^^ \" 


1 901I 111 1 III 1 n^ii II III 1 II Mill IN 



10° 20° 30° 40° 50° 60° 70° 80° 90°100°110°120°130° 140°150°160 170 180 
Angle P of Rotation of Driving Shaft 

Fig. 7. 



Figs. 6 and 7. — Errors of velocity and position of the Hooke universal shaft coupling. 

308 






MISCELLANEOUS MECHANISMS, CONSTRUCTIONS AND DATA 



309 



I 



Mr Buckingham has also calculated the relative positions of the 
two shafts for various angles between the shafts by this equation. 
The results are given in Table 2 and graphically in Fig. 7. 

Table i. — Relative Speed of Driven Shaft — Speed of Driving 
Shaft = 100 



Angle B 








Angle A between shafts 






of rota- 








~ 










tion of 






















driving 


0° 


10° 


20° 


30° 


40° 


50° 


60° 


70° 


80° 


89° 59' 


shaft 

























100 


98. s 


93-97 


86.6 


76.6 


64.3 


50.0 


34.2 


17.4 


.029 


S 


100 


98. s 


94.1 


86.8 


76.8 


64.6 


50.3 


34-4 


17-5 


.029 


10 


100 


98.6 


94-5 


87.3 


77.6 


65.4 


SI. 2 


35.1 


17.9 


.029 


IS 


100 


98.7 


94-8 


88.1 


78.8 


66.9 


52. 6 


36.4 


18.6 


.030 


20 


100 


98.8 


9S.3 


89.2 


80. 5 


69.0 


54-8 


38.1 


19.6 


.032 


25 


100 


99.0 


96.0 


90.7 


82.7 


71.8 


S7.7 


40.6 


21.0 


.035 


30 


100 


99.2 


96.8 


92.4 


85.7 


7S.3 


61.5 


43.9 


22.9 


.039 


35 


100 


99.46 


97. S 


94-9 


88.7 


79.7 


66.4 


48.2 


25.5 


.043 


40 


100 


99.72 


98.7 


96.6 


92.4 


84.9 


72. 5 


S3. 8 


29.0 


.049 


4S 


100 


99.98 


99.8 


98.97 


96.6 


91 .0 


80.0 


61.2 


33.7 


.058 


SO 


100 


100.22 


100.9 


loi.s 


lOI.I 


98.0 


89.3 


71.0 


40.6 


.070 


SS 


100 


100. s 


101.9 


104.0 


105. 9 


106.0 


100.6 


83.9 


49.7 


.088 


60 


100 


100.7 


103.0 


106.6 


III.O 


114. 8 


114. 2 


I0I.2 


63.7 


.116 


6s 


100 


100.9 


103.9 


108.9 


115. 9 


124. 1 


130.2 


124.5 


85.4 


.162 


70 


100 


lOI.I 


104.8 


in. I 


120.6 


133.4 


148.0 


15s. 2 


120.9 


.249 


7S 


100 


IOI.3 


105.4 


113. 4 


124.6 


142.0 


162.7 


194. I 


219.4 


.434 


80 


100 


IOI.4 


105.9 


II4-3 


127.7 


149. 1 


183.4 


238.1 


292.3 


.96s 


8S 


100 


loi.s 


106.3 


II5.I 


129.8 


154.5 


195.5 


276.6 


462.8 


3.83 


90 


100 


101.6 


106.4 


IIS.4 


130. 5 


ISS.S 


200.0 


292.3 


575. 8 


3428.0 




C=H A 



Table 3. — Dimensions and Transmitting Capacities of the 
Baush Universal Joint 



A 


w 


w 


5i" 


H° 


W 


i' 


iH" 


^w 


1%' 


2' 


2Vl- 


3" 


4' 


B 


m' 


2' 


2H" 


2lH6' 


3!4" 


3H" 


3%" 


4/1" 


aW 


sM6' 


7' 


9" 


\0%" 














'A 


H 


I.I 


1.4 


2M 


6J^ 


11 



















the steam on the piston may exceed that of the hand on the lever 
by any desired amount, the effect being a multiplication of the force 
combined with the same absolute control as would be obtained by 
mechanical connection, as through levers or gears. The principle 
has wide application to the auxiliaries for controlling heavy machinery 
and it is best explained by an actual example as set forth by B. V. E. 
Nordberg {Amer. Mack., June 26, 1913) in the following description 
of the steam actuated brake of a large Nordberg hoisting engine: 

A floating lever may be defined as a lever none of whose centers is 
fixed. Fig. 8 shows a brake as used in connection with a large hoist. 
The drum of the hoist has a separate brake drum a bolted rigidly 
to it, and a brake shoe set against the brake drum by means of a 



Table 2. — Difference in Angular Motions of Shafts 



Angle A between shafts 



30 



40 



50" 



60° 



70" 



80° 



89° 59' 



Angle C of rotation of driven shaft 
































( 


D 




5 


4-55-28 


4-42- 


4-19-58 


3-50- 3 


3-10-15 


2-30-18 


1-42-50 


0-52-14 








10 


9-51- 4 


9-24-29 


8-40-56 


7-41-33 


6-27-59 


5- 2-18 


3-27- 4 


1-45-14 


0- 


O-IO 




IS 


14-46-56 


14- 7-58 


13- 3-52 


11-35-58 


9-46-20 


7-37- 


5-14-10 


2-39-50 






4-» 


20 


I 9-43- II 


18-52-54 


17-29-43 


15-34-47 


13-10- 4 


10-17-51 


7- S-46 


3-37- 


0- 


0-23 


bO 


25 


24-39-57 


23-39-45 


21-59-26 


19-39-27 


16-41- 8 


13- 7-27 


9- 3-41 


4-37-46 








30 


29-37-18 


28-28-52 


26-33-54 


23-51-31 


20-21-38 


16- 6- 8 


11-10-13 


5-43-30 


0- 


0-35 




35 


34-35-20 


33-20-39 


31-13-57 


28-12-31 


24-13-54 


19-17-43 


13-28- 3 


6-55- 






«AM 


40 


39-34- 7 


38-15-20 


36- 0-19 


32-43-57 


28-20-27 


22-45-33 


16- 0-47 


8-17-24 


0- 


0-50 





45 


44-33-41 


43-13- 9 


40-53-37 


37-27-13 


32-43-56 


26-33-54 


18-52-53 


9-51- 4 






p 


50 


49-34- 3 


48-14-12 


45-54-17 


42-23-39 


37-27-13 


30-47-23 


22-10-33 


11-41-31 


0- 


I-I2 




55 


54-35-12 


53-18-31 


SI- 2-36 


47-34-15 


42-33- 6 


35-31-47 


26- i-s8 


13-55-41 






K> 


60 


59-37- 7 


58-26- 


56-18-36 


52-59-44 


48- 4-12 


40-53-36 


30-38-33 


16-44-22 


0- 


1-45 


V 


65 


64-39-44 


63-36-28 


61-42- 


58-40-13 


54- 2-28 


46-59-49 


36-15-27 


20-25-29 






< 


70 


69-42-59 


68-49-38 


67-12-15 


64-35-10 


60-28-47 


53-56-51 


43-13- 9 


25-30-20 


0- 


2-45 




75 


74-46-45 


74- 5- S 


72-46-14 


70-43-16 


67-22-16 


61-48-47 


51-55-24 


32-56-45 








80 


79-50-56 


79-22-36 


78-29-30 


77- 2-34 


74-39-37 


70-34-29 


62-43-36 


44-33-41 


0- 


5-40 




85 


84-55-24 


84-40-51 


84-13-53 


83-29- 5 


82-14-57 


80- 4-30 


75-39- 4 


63-15-35 


0-II-2S 




90 


90 


90 


90 


90 


90 


90 


90 


90 







The Floating Lever 

The floating lever is a device for the indirect, graduated control 
of large forces through the initial application of small ones. The 
small force, for example that of the operator's hand, is applied to a 
hand lever which, through the interposition of a floating lever, is 
made to control the movement of, for example, a piston under 
steam pressure, the movement of the piston following that of the 
hand lever as though mechanically connected with it. The force of 



dead weight b. The brake shoe is released by the auxiliary steam 
cylinder c, which has no other function. In case the steam pressure 
should suddenly give out, the brake would be automatically set. 

The steam cylinder has a valve placed at its bottom, admitting 
steam below a piston, when turned in one direction, and allowing 
the steam to escape when turned in the opposite direction. In 
order to make the piston respond to a slight motion of the hand lever 
d, this valve must set line on line and have no lap. Directly above 
the steam cylinder is a distance piece which allows the stufiing-boxes 



310 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



of the steam and oil cylinders to be adjusted. The oil cylinder above 
has a valve in the center which puts the upper and the lower parts 
of it in communication with each other, and is set to open and close 
with the steam valve below by means of a rod. The function of the 
oil cylinder is to lock the whole mechanism in place when the weight h 
has assumed the desired position. 

Above the oil cylinder is the crosshead guide and crosshead which 
takes the end of the piston rod, and is connected to the weight b 
through a rod. The crosshead eliminates any tendency to bend the 
piston rod that might result from the swinging of the point of attach- 
ment to the weight lever e. Over the crosshead guide is the floating 
lever /. The point g of the floating lever is attached to the weight l 
so that its motion corresponds to that of the weight. The point li 
is attached to the steam and oil valves of the thrust cylinder c. The 
operator's hand lever d is connected to the point i. . 




^Steam Inlet 



Fig. 8. — The floating lever. 

For the sake of clearness, let us assume that the motion of the 
thrust cylinders is to take the steps outlined below. As the operator 
moves the lever d from the position x to the position y, the floating 
lever is moved to the position indicated by the dotted line, pivoting 
about the point g. Raising the point h opens both the steam and the 
oil valves. This admits steam under the piston, and as it rises it 
takes the weight b with it, pushing the point g of the floating lever 
to the position s. In doing so, the floating lever moves down until 
the point h is reached, when the steam and oil valves close and all 
motion ceases. 

In reality, the motion does not take the decided steps outlined 
above. It is clear, of course, that as soon as the steam valve opens 
the least bit, the steam piston will move, forcing the point g up, 
immediately closing the steam valve again. If the valve is turned 
in the other direction, which is effected by moving the hand lever d 
away from y, the steam in the cylinder escapes, being forced out by 
the weight h. The point g on the floating lever follows up the weight 
and tends to close the valves again, and resists any further falling 
of h. It can now be plainly seen that when the operator moves the 



hand lever d, the weight b immediately follows. When watching a 
thrust cylinder in operation, it is very difficult to detect any motion 
at the point h, so closely will the weight b follow the hand. 

It wiU also be noticed that the oil valve moves exactly with the 
steam valve; in other words, both are opened and closed at the same 
time. When once the weight b has closed the oil valve, the mechan- 
ism is solidly locked, permitting of no motion until it is again opened. 
This prevents any overtraveling of the thrust pistons and ultimately 
the dancing of the weight b, which might otherwise happen when an 
elastic medium like steam or air is used for the operation of the 
auxiliaries. 

The final result is that the brake shoe is applied with a graduated 
pressure, precisely as though connected mechanically with the 
hand lever. 

Velocity and Force Relations in Linkwork 

All motions in a plane may be regarded as turning motions — motion 
in a straight line being regarded as a turning motion about a center 
at an infinite distance. In the case of puUeys, gears and similar 
parts. Figs, g and lo, the motion is about fitxed centers, while in 
many other cases the centers themselves move. A case in point is 
a vehicle wheel rolling upon the ground. Fig. ii, in which the center 
about which the wheel turns at any instant is its point of contact a 
with the ground, this point moving forward with the wheel and trac- 
ing the line ab. This point a is called the instantaneous center. 
Similarly a circle a, Fig. 12, rolling on a second stationary circle 6 
turns at any instant about the point of contact c of the two circles 
which point c is the instantaneous center. "' 

These very simple cases assist to an understanding of the somewhat 
difficult conception of motion about a moving center. Certain 
principles of obvious truth as applied to turning about fixed centers 
are equally, although less obviously, true as applied to turning about 
moving centers. Thus referring to the revolving disk, Fig. 9, it is 
obvious that at any moment, the direction of motion of any point b 
of the disk is that of the perpendicular be to the radius ab. Therefore, 
if the direction of motion is known, we have the first of these prin- 
ciples: 

(A) The center of motion of any moving point is located in a line 
drawn through the point and perpendicular to the direction of motion. 
If the point moves in a straight line this statement still holds true, 
with the proviso that the center of motion is at an infinite distance. 

Other principles are: 

(B) The velocities of all points at the same distance from the 
center of motion are equal. 

(C) The velocities of points at different distances from the center 
are to one another as those distances. 

From {B) and (C) we obtain a graphical method of finding the 
velocity of any point from the known velocity of any other point of a 
revolving disk as follows: 

(D) The velocity of point b, Fig. 10, being represented to scale by 
the length of the line be, to find the velocity of point d, draw the 
arc de from the center a, draw ac and, through e, ef parallel to be, 
and ef is the velocity of d. 

Let Fig. 13 represent a set of four links jointed as shown, the link 
ab being held stationary as in a vice and as indicated conventionally 
by the short inclined lines leading from it. Link ab represents the 
frame of actual mechanisms. 

In this piece of linkwork links ad and be turn about the pins a 
and b and our first consideration is the character of the motion of 
link cd. The pin c is a part of both links be and cd. Its motion is 
determined by the fact that it is part of be, the direction of that 
motion being the arrow ee perpendicular to be. As it can have but 
one motion, its direction of motion when considered as part of link 
ed must also be this arrow ee. By principle (A) the center of its 
motion as part of link cd must be somewhere in a line bf through c 
and perpendicular to ce. Applying the same reasoning to pin d, 
its center of motion when considered as part of link cd must lie 



MISCELLANEOUS MECHANISMS, CONSTRUCTIONS AND DATA 



311 



somewhere in a line c/ through d and perpendicular to dg. Now taneous center, we obtain the path abed traced by the instantaneous 

the link cd being a rigid body, it moves as a whole and aU points on it center corresponding to line ah, Fig. ii, and ch, Fig. 12. 
must turn about the same point. Since that point for one end of it Suppose, now, we have given the velocity of point e of link ad, 

lies in line 6/ and for the other end of it in line af, the single point Fig. 15, and wish to find the velocity of point/ of link he. Represent- 



I 




Fig. 12 



Fig. 13. 
Figs. 9 to 14. — The instantaneous center. 





Fig. 17. 




s- 



Fig. 19. 



Figs. 15 to 19. — Velocity relations in link work. 



about which the entire link, for the instant, turns must be at the ing the velocity of point e by the length of the line eg and applying 

intersection / of these lines. This point is the instantaneous center principle (D), we find the velocity of point d to be the length of the 

of link ed corresponding to point a of Fig. 11 and point c of Fig. 12. line dh. Drawing the arc di from the instantaneous center y, we 

By drawing the parts in a series of positions as in Fig. 14 and then obtain the point i. If this point were carried by a short arm ih 

drawing a smooth curve through the various positions of the instan- projecting from link ed, it would, by principle (B), have the same 



312 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN . 



velocity as point d. Laying down il equal to dh and again applying 
principle (D), through the triangular construction from instantaneous 
center j, we find the velocity of pin c to be cm. Repeating the con- 
struction again from center b, we find the required velocity of / 
to be fn. These simple constructions are aU that are required in 
similar cases. 

The constructions so far shown wiU give the velocity relation be- 
tween any two points at any selected position of the mechanism. 
In a shaper mechanism, for example, the complete study of the motion 
makes necessary the laying down of a velocity diagram in which the 
velocity of the cutting tool at various points is plotted vertically 
to scale along a horizontal line the length of which represents a coi^i- 
plete stroke. It is necessary only to repeat the construction shown 
in connection with Fig. 15 for a sufi&cient number of positions to 
obtain these various velocities for the velocity diagram. These 
repeated constructions involve the drawing of a good many lines, 
but the constructions being simple repetitions of one another, these 
numerous lines do not lead to mental confusion if the principle is 
understood. 

It should be noted that in many mechanisms the lines leading to 
the instantaneous center become parallel at certain positions and do 
not meet, and in other positions near this one these intersections fall 
beyond the limits of the drawing board. The former case does not 
lead to uncertainty, as it shows that the velocity of the connected 
pins c and d, of Fig. 15, are identical and call for no construction. 
The velocity curve on each side of this point is usually quite flat and 
does not need closely spaced points for its determination, such points 
as can be found by constructions that fall within the limits of the 
drawing board being sufficient. 

The simple constructions shown give a complete exposition of the 
method, but additional illustrations are desirable for those to whom 
it is new. 

Fig. 16 represents the common crank, connecting-rod and cross- 
head mechanism of a steam engine, and by the construction shown 
we obtain the velocity of the crosshead compared with that of the 
crankpin. Recalling the proviso of principle (A), we draw the 
vertical through the center of the crosshead pin, extend the crank 
and find the instantaneous center a of the connecting-rod. Letting 
be represent the velocity of the crankpin, we draw ac. From a 
as a center and through the crosshead pin center, we draw the arc 
de and find point e, which, were it a point of an arm ef of the con- 
necting-rod, would, by principle {B), have the same velocity as the 
crosshead pin. Drawing eg, its length is that velocity. 

Fig. 17 represents the mechanism of an oscillating steam engine, 
the block at the right oscillating on its trunnions, whUe the inclined 
link slides through it. The construction for finding the velocity of 
this sliding link as compared with the crankpin is given in the illus- 
tration, but is so similar to the last one that detailed description 
seems unnecessary. 

Fig. 18 shows the Whitworth quick return motion. The radius 
arm ac is the constant length, constant speed, first-motion radius 
arm of the Whitworth motion, be, carrying the slide, being the vari- 
able length, variable-speed radius arm and ab being the eccentricity. 
Adding link de and slide e, we have the mechanism complete. 

The radius arms ac and be here revolve about fixed centers, and when 
comparing their velocities we have no instantaneous center to con- 
sider. Let the velocity of pin c of arm ac be represented by the line 
cf. The velocity of this pin in the direction perpendicular to arm 
J>e is eg found by constructing the triangle cfg. The velocity of pin 
d is di found from eg as shown. To obtain the velocity of the tool 
slide e we find the instantaneous center h of the connecting-rod de 
and draw hi. The arc ej drawn from h gives the point j having the 
same velocity as e, this velocity being jk. 

Fig. 19, which differs from Fig. 18 in the proportion of its parts 
only, shows the oscillating arm quick return motion found on most 
shaping machines. 

The line ab of Fig. 18 has been lengthened and the fixed-length 



radius arm ae shortened and placed at the top instead of the bottom; 
that is, the mechanism has been turned bottom side up. The con- 
struction for the velocity of the tool slide scarcely differs and is 
lettered in the same order, cf being the constant tangential velocity of 
the crankpin, eg its velocity perpendicular to the swinging arm and 
di the velocity of pin d. The instantaneous center of the connecting- 
rod de is at h, and the velocity of the tool slide is jk. 

The Force Relation 

The velocity relation having been found, the force relation is given 
at once by the principle of virtual velocities, which tells us that the 
forces exerted are inversely as the velocities. 

That is, referring to Fig. 15, 

force at / velocity of e 



or 



force at e velocity of/ 

, velocity of e 



force at /= force at eX 



velocity of / 



An illustration is found in the force relation of the common toggle 
joint. Fig. 20, in which a known weight is attached at a and it is 
desired to find the resulting thrust at b. The velocity with which 
the weight descends, which need not be known but is assumed as 




Fig. 20. 



Fig. 21. 



Figs. 20 and 21. — Force relations in link work. 

represented by any convenient length of line as ac, is resolved by 
the triangle aed into tangential and horizontal components, of which 
the former is ad. The corresponding velocity of pin e is ef. The 
instantaneous center of link be is g. The point having the same 
velocity as b is k, and the velocity of h and of b is hi. Now we have : 

force at b velocity of a 



weight at a velocity of b 



or 



force at J = weight at ay.—r-r 

A more complex case of force relations is found in the double 
toggle joint of a compressed air riveting machine. Fig. 21. At o is 
the piston pin, the cylinder and piston being indicated. Link ab 
joints with be, c being fixed to the frame. Link bd connects pin b 
with the slide to which the rivet closing die is attached. 

The piston, as in the last case, is assumed to move at any con- 
venient velocity represented by the line ae. Drawing af perpendicu- 
lar to ae and extending be, we find / the instantaneous center of ab. 
The arc ag locates point g which, were it attached to ab by the arm 
gh, would move with the same velocity as the piston. Laying down 
gi equal to ae and drawing if and bj, we find bj the velocity of point 



MISCELLANEOUS MECHANISMS, CONSTRUCTIONS AND DATA 



313 



b. Next, we find the instantaneous center of bd at k. The arc dl 
gives the point I which, were it attached to bd by an arm Im, would 
have the same velocity as d, and drawing In, we have that velocity. 

It should be noted that we may also find a second point o having 
the same velocity as d, but the velocity op found from it is obviously 
equal to In. In point of fact there are always two such points, on 
opposite sides of the instantaneous center. Were the arc de of Fig. i6 
extended to a complete circle, it would give a second intersection 
with ab beyond the instantaneous center. In Fig. i6 the second 
point is beyond the limits of the drawing, while in Fig. 21 it is within 
them, and that is all the difference there is between the two cases. 

The force relation is given by the equation: 



pressure on rivet = pressure on piston X' 



In 



The Geneva Stop 

The designing of the Geneva stop is shown in Fig. 22 and explained 
as follows by E. Kwartz {Amer. Mach., June 8, 1911): 

Referring to the illustration, the driving roller E is shown leaving the 
star wheel, after having turned the latter through part of one revolu- 
tion; or in the position of entering the star wheel for driving, if the 
direction of rotation is reversed. The round part HKQ, which 
may be cast in one with the crank disk O, is in position to lock the 
star wheel until the roller enters at G; or releasing it until the roller 
leaves at G, depending upon the direction of rotation. Part of the 
circle HKQ is cut away at the left for clearing the arms of the star 
wheel. 

In using this movement, the designer may either determine the 
number of slots he wants in the star wheel, which will limit the rela- 
tive time of operation and the dwell of shaft A during one revolution 
of shaft 5; or he may settle approximately the relation between operat- 
ing time and locking time on shaft B, which will limit the number of 
slots in the star wheel. 

Let iV = number of slots in star wheel. Examining the drawing 
it will be seen that angle A must always be 90 deg. in order that 
pin E may enter the slot properly, and 



Angle « = 



360 

N 



(«) 



To anyone familiar with geometry, it also is plain that 
' a-l-i3 = i8o deg. 



from which 



:i8o— a 



ih) 



Again consulting the drawing, it may be seen that the smallest 
number of slots with which it would be possible to operate the star 
wheel wovdd be three. The greatest number of slots possible depends 
upon the diameter of the star wheel and the size of the slots. If the 
slots could be considered infinitely narrow, their number might be 
infinite. Thus the theoretical limits for number of slots lie between 
3 and infinity. Of course, the largest number possible in practice 
will not be very great, but the probabilities are that this limit will 
not often have to be reached. 

Suppose that the number of slots is determined, and one desires 
to find the relation between operating time and locking time during 
one revolution of shaft B. By formula (a) angle ^3 expresses the 
operating time in degrees. If it is desired as a fraction of one revolu- 
tion of shaft B, call this fraction Bt. 



Then, 



and from (a) 



Bt=—-7-, but from (b) 
360' ' 

/? = i8o-q: 



Therefore, 



180 — 



Bt=- 



360 

N 



360 



'^l-~ AT 

2 N 



From formula (c) we may obtain 



N-- 



\-Bt 



w 



id) 



r~3 ~^" 



but. 



This formula will give N only approximately, unless the answer 
should be a whole number. As N is the number of slots, it can, of 
course, be only a whole number, and Bt eventually must be made to 
correspond. 

Examples. — Assume, first, that number of slots N is determined; 

360 
say iV = 6. Then from formula (a) , « = -7- = 60 deg. From formula 

(b), /? = 180— 60 = 120 deg.; and from formula (c), Bt = i—\='\ 
revolution of shaft B, for operating time. 

Changing the conditions, assume Bt = \; that is, \ turn of shaft B 
is desired to operate the star wheel. Then, from formula {d), 

iV = rz:Y = 4 slots required in the star wheel. It would seem logical 

that four slots would give \ operating time, but this does not hold 
for all fractions, as the next example will show. 

Assume Bt=\. Then from formula {d), N=^ — 
~ 2 

as N can be only a whole number, we wiU have to be satisfied with 
either three or four slots in star wheel and take B what it comes for 
this number. 

In most cases where this device would be used, one probably 
would start out by deciding upon a certain center distance, if this 
is not already fixed; then construct a semicircle DEFC upon this 

a 
center distance and lay out an angle DCE = — Connect Z)£ and 

2 

draw line GD, extending it toward H, which will make an angle 
QDH = ^, limiting the ends of the clearance cut QH of the locking 
circle. 

Radius r of the locking circle is somewhat a matter of choice. 
As a standard, the nearest sixteenth of an inch to the result obtained 
from expression: r = DE — ild, d being the diameter of driving roller 
may be taken. The shape of the clearance cut is found by tracing 
one arm of the star wheelon one piece of tracing cloth and the crank- 
pin roller, center of crank disk and circle HKQ on another piece of 
tracing cloth. Fasten these pieces with pins to the drawing board 
with their proper center distance, placing the crank-pin tracing on top, 
and rotate together, tracing the arm of the star wheel in different 
positions on the crank-disk tracing while turning over center D. 
Draw a curve QH that will clear these marks. The curve may be 
cast with a considerable clearance, but the end points H and Q should 
be located properly on the legs of angle ^3, as described. 

a 



Distance Z)£ = radius of crank circle = DC sin 



To find the 



extreme radius of star wheel, make an accurate layout, and scale 



distance CP: or calculate thus: CE = DC cos — ■ 
' 2 



Assume diameter 



of driving roller = (f. Then, 



CP 



=Vf 



+(CEy 



If an accurate layout is made, the calculating of CP is not necessary 
It will be seen from the illustration that the number of slots may 
not be very much less than 6 before the crank disk wiU interfere 
with the hub of the star wheel. When such becomes the case, the 
crank disk will have to be placed on the opposite side of the star 
wheel. 



314 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Rock Arms and Link Work 

The length of rocker arms to divide equally the side vibration of the 
connecting link may be obtained from the formulas 

a= 

4C 



a+Vo2-62 



the notation being as in Fig. 23. 



and 



L=l+ 



AB :E0 :: CD : FO. 



Also, 

This gives a simple formula for computing the length of arms which 
will give an equal vibration on each side of the central line of 
motion. 

A problem in link work, which occurs in the layout of Corliss valve 
gears, together with its solution, by E. H. Berry {Amer. Mach., 
Aug. 13, 1908), is shown in Figs. 25 and 26. 




Fig. 22. — Designing the Geneva stop. 




Fig. 23. — The length of rocker arms. 

Vibrating levers are frequently required to transmit a reversed 
motion along two parallel lines, with a given stroke along eacli line 
and a given distance apart. Let AB, Fig. 24, represent the stroke 
and center line of one motion, CD the stroke and center line of 
the other, and EF the vertical distance between them. To find 
the position of the central stud and length of the levers: Lay 
o£f EH — I stroke AB and FN — i stroke CD on opposite sides 
of EF, and draw HN. The intersection of HN with FE is the 
center of oscillation. Draw GK at right angles with HN and G 
and K are the middle and extreme positions of the upper pin. Draw 
MR at right angles with HN and M and R are the extreme and middle 
positions for the lower pin. This gives the length GO for the upper 
arm and RO for the lower. 

Solving the problem mathematically: 



5 
7 



tan a, and — = tan a*. 




Fig. 24. — ^Laying out vibrating levers. 




•B \ Fig. 26. 

Figs. 25 and 26. — A problem in link work and its solution. 

Given the point and the three positions OB, OC, and OD of an 
arm of known length swinging about 0; given the point Q and the 
angles b and c; required the length of the arm QR and the length of 
the link BR. 

Solution: Draw QB and QC as in Fig. 26. With center Q and 
with radius QB draw the indefinite arc BBi with the same center, 
and with -radius QC draw the indefinite arc CCi. Lay off the angle 
BQBi equal to the given angle c, and lay off the angle CQCi equal to 
the given angle b. Find the center T of a circle passing through the 
three points Bi, Ci and D. Then QT is the required length of arm, 
and DT is the required length of link. 

When obstructions interfere with the rise and fall of the end of a 
vibrating lever, it may be made to travel in an approximately straight 
line by the construction shown in Figs. 27 and 28, by A. E. Guy 
{Amer. Mach., Apr. 21, 1898). The lever slides over a guide block 



MISCELLANEOUS MECHANISMS, CONSTRUCTIONS AND DATA 



315 



swiveled to a fixed point and is driven by an osciRating crank arm 
connected to its lower end. Mr. Guy has found that, assuming 
the upper end B, Fig. 27, to be guided in a straight line, when the 
angle 2a, Fig. 27, is as large as 75 deg., and 0£ = about one-third of 
OB, the path of the point E is almost an arc of circle, and for 2 a =90 
deg., which value may be considered as extreme for an ordinary 
lever, the curve coincides with the arc of a circle until near the ends 
E and E', when it bends inward. 

Consequently if the point E, Fig. 27, is made to travel along the 
arc of a circle EAE', the path of point B will be very nearly a straight 
line. It is easy to find the radius of the arc since it passes through 
three points whose positions are known. 

In Fig. 27 the lever is shown at its two extreme positions, BE, and 
B'E'. Draw AE, and at its middle draw the perpendicular DF, then 
DA is the radius of the arc. For many purposes this graphical 
method is sufficiently accurate, but for other cases the following 
equations may be used: 




Fig. 27. 



;E"ig.*28. 



Figs. 27 and 28. — A straight line lever mechanism. 



In Fig. 13 let 



and we have 



tan^ 



BE = a 
OE = b 
DA=r 

b sin a 



and r = 



a(i — cos a) 
h sin a 



(a) 



Since angle a is known, the value of angle will be easily found by 
formula {a) when the sine of 26 will be taken from trigonometrical 
tables and introduced in equation (6). 

The Ball Expansion Drive Stud 

The hall expansion drive stud, Figs. 29 and 30, invented, patented 
and largely used by the Link Belt Co., was by that company pre- 
sented to the mechanical public, without fee or royalty, through 
the American Machinist of Dec. 9, 1909. 

The illustrations show sections before and after driving. The 
rivet or stud is a plain piece of stock having a hole drilled in one end, 
with a chamfer surrounding the hole and then cut off from a bar of 
cold-rolled stock. 

A hard-steel ball, bought at a very low price from the culls taken 
from the balls selected by makers of bearings, and slightly larger than 
the hole in the end of the stud, is dropped into the hole ahead of 
the stud, which is then driven into place over the steel ball, as 
shown in Fig. 30. The chamfered end of the stud aids in closing it 
around the ball. 



The amount of expansion on the lower end of the stud depends 
upon the difference between the diameter of the hole and that of the 
ball. The diameter of the hole in the end of the stud should be about 
three-quarters of the outside diameter, and the depth of the hole in 
the stud should equal the diameter of the ball. Excessive driving 
weakens rather than increases the hold. The depth of the hole in 
the casting should be about twice the diameter of the stud for small 
sizes and i§ times for large sizes, while the difference in diameter 
between the ball and the hole in the stud should be about ^ in. 

Tests have shown that j^- and i-in. studs are about 20 per cent, 
stronger than bolts of the same diameter, and the average grip of 
a |-in. stud is nearly equal to the breaking strength of a bolt of this 
size. 

These studs are greatly superior to screwed-in studs. They have 
been used by the Link Belt Co. with complete success in sizes up to 




Fig. 31. Fig. 32. 

Figs. 29 to 32. — The ball expansion drive stud. 

I in. Table 3 gives the dimensions for small sizes. While the ex- 
perience of the Link Belt Co. has shown complete security of the con- 
struction, the larger sizes may, if desired, be given a security which 
no one can question by the method shown in Figs. 31 and 32, by 
Professor Sweet (Amer. Mach., Jan. 19, 1905). A simple wabble 
drill, Fig. 31, chambers the bottom of the hole while the ball expands 
the steel into the chamber. Fig. 32. 



Table 3. — Dimensions of Expansion Drive Studs 



Diameter of stud 

Depth of hole 

Size of ball 

Diameter of center bore 
Depth of center bore. . . 



In. 


In. 


^ 


i 


f 


1 
2 


i 


^ 


■^ 


5 
¥2 


A 


A 



In. 



Balance Diaphragms 

The effective balancing area of a diaphragm, when opposed to a 
poppet valve with a knife-edge seat, has been worked out by James 
Clark (Amer. Mach., Oct. 27, 1904). Referring to Fig. 33, he finds 
the effective area of the diaphragm to be expressed by the formula: 

effective area =-{R^+Ra-2a^} 

in which 22 = outer radius of diaphragm, ins., 

a = radius of stem connecting valve and diaphragm; ins. 
To make use of the formula, calculate the area to be balanced, that 



316 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



is, the annular area between valve and stem. Assuming, for example, 
the radius of the valve to be f in. and of the stem | in., this area is: 



m'-m-?. 



which is to be equated with the formula for the effective area of the 
diaphragm giving: 

21 



in which P = total effective pressure of the valve against its seating 
surface. 

For conical seated poppet valves the slant height of the inner and 
outer cones are to be treated as r and R. These are to be substituted 
in formula (a) and the value of d found. The cone is then to be devel- 
oped and such part of 5 T^d'' used as the developed cone is of 360 deg. 

Formula (a) applies strictly to non-elastic fluids only; but the differ- 



3 



■2a2) = 7r 



64 



or, R^+Ra-2a^ = :^ 

64 

I 

which, as a = - becomes: 

4 

R 71 

4 64 

which, solved for R, gives 

R = i.i8s or -.935111. 
of which the positive value is the one available. 



I— K^--^ 




Diaphram 



I-. 



■t! 



Required Dia. 

Fig. 33. — Effective balancing area of diaphragms. 

The formula of the preceding paragraph may he 
applied to the balancing of poppet valves with 
other than knife-edge seats by the consideration of 
the following paragraph. 

The effective pressure or equilibrium area of poppet 
valves, when closed, formed the subject of an experi- 
mental investigation by Prof. S. W. Robinson 
(Trans. A. S. M. E., Vol. 4). These experiments 
showed the presence of a creeping film of steam in 
the valve seat, the pressure of which, beginning at 
the pressure of the high-pressure edge, decreases to 
that of the low-pressure edge and acts to partially 
balance the applied pressure. The result of the ex- 
periments was to develop a formula for the area of 
the surface which, multiplied by the applied pres- 
sure, equals the actual closing pressure, this area 
being the effective pressure or equilibrium area of 
the valve. For flat-seated poppet valves the for- 
mula is: 

in which (f = diameter of the equilibrium area, ins., 
r = inside radius of valve, ins., 
i2 = outside radius of valve, ins., 
/>!= pressure on inner area, lbs. per sq. in. abs., 
/»2 = pressure on outer area, lbs. per sq. in. abs. 



llO 



^ 



s. 



o 



o 



o 



o 






o 




o 



o 



© 



o 



o 



o 



o 



o 



o 



o 



o 



o 



o 



o 



o 



o 



o 



o 



I I 



o 



Q 



o 



o 



01 

J 



o 



O! 



o 



o 



o 



iO 



o 



01 



o 



o 



o 



O 



O 



o 



o 



k ' 




O 



O 



O 



[3 



o 



o 



ioj; 



ioi 



S 



^^^ 







Fig. 34. — Dimensions of a section of cast-iron floor plate. 



Surface Plates 



Concii'ete'S W' 

^^. ..■?■.. a..A. 



— T-TV-r-TTT' 
.^'C^ciretK -^ 



wm 



Sand 



^ 



'■//;, 



Saud 



^ "Soiftfrete « ^ t Coriftete/ 



P^ 



Sand 



» ..^..^.sy.^s 



; Coriftete 



// Sand \ 



(a) 



Then 



P = - r.d\p,-p,) 
4 



(J) 



Fig. 35. — Section of a floor plate foundation. 

ences between calculated results from the formula and experimental 
results using steam pressure were small. 

Cast-iron Floor Plates 

The construction of cast-iron floor plates for use with portable 
machine tools, as practiced at the works of the General Electric Co., 
is shown in Figs. 34-35, by John Riddell, who originated this system 
of doing heavy machine work {Amer. Mach., Nov. 28, 1907). Fig. 
34 gives the dimensions of one section, which is planed and grooved 



MISCELLANEOUS MECHANISMS, CONSTRUCTIONS AND DATA 



317 



on the edges, surfaced on top and provided with regularly spaced 
holes for pouring the grouting. 

The holes should be made of a size such that, with rags for packing, 
a piece of pipe about 3 ft. long may be inserted. Pouring the pipe 
full of grout produces a hydrostatic head which forces the grout 
under the plate and forms a solid bed for it. 

Fig. 35 relates to an earlier pattern, when the plates were made 
heavier and less dependance placed on the foundation. It, however, 
shows the character and dimensions of the foundation. Fig. 36 
gives a section of a floor at the same works used for erecting and 
testing but not for machining operations. Various details are shown, 




Section of Ran and Slot Floor Construction 




2 Eails for Eloor Construction 



IO( 





Spreader for 2 Kails 

Ytptaik of Spreader Blocks and Joint of Z Rails 

.^ ^ Ip . r^ 




Method of Locating and Supjpoxting Rails while putting in the Concrete Foundation 
Fig. 36. — Section and details of an erecting and testing floor. 



To get the third radius lay ofi a base line AB, Fig. 39, of any 
convenient length, and divide it into five equal parts by the points 
I, 2, 3 and 4. At one end of this line erect the perpendicular AC, 
equal to the radius AH, and at the other end erect the perpen- 
dicular BD, equal to the radius CG. Connect the upper ends of 
these perpendiculars by the line CD. From point 2 erect the per- 
pendicular 2 E. The length 2 E will be the desired third radius. 
With the compasses set to this radius find, by trial, a center /, 
Fig. 38, from which a curve can be struck which will be just tan- 
gent to the curves struck from the centers H and G. Lines drawn 
from / through H and from G through / wiU determine the meet- 
ing points of the different curves. 
From other centers similarly located 
the remainder of the ellipse is 
formed. In many cases one-half 
the major axis is a satisfactory 
value for the third radius. 

For narrow elhpses the radius AH 
with which the ends are formed 
should be lengthened as follows: 
When the breadth of the ellipse is 
one-half of its length lengthen AH 
one-eighth; if the breadth of the 
ellipse is one-third of its length, 
make AH one-quarter longer; if the 
breadth is one-quarter of the length 
make AH one-half longer. 

Arcs of Circles 

The radius of a circular arc of 
which the span and rise are given, 
may be found as follows: In Fig. 
40, g being half the span h, the rise 
and r the radius, 



I I 
I I 



Joint Spreader 
for 2 Rails 



h 



+h 



ncluding the method of locating and supporting the rails while 
putting in the foundation. If the rails and beams are reason- 
ably straight, no machine work, other than drilling, is necessary. 

Laying out Approximate Ellipses 

The layout of approximate ellipses by four circular arcs may be 
facilitated by the use of Fig. 37, by S. J. Teller {Amer. Mach., 
Feb. 6, 1908). To use the chart, find the length of the major axis 
on one side of the sheet and the length of the minor axis on the 
top or bottom. Follow the corresponding horizontal and vertical 
lines to their intersection and read on the curved lines or by inter- 
polation the radius of the larger arcs. Then find the length of the 
minor axis on one side of the sheet, and the length of the major 
axis on the top or bottom. Follow the corresponding horizontal 
and vertical lines to their intersection and read on the curved lines 
the radius of the small arcs. In some cases it may be found more 
convenient to put in the small arcs by the cut and try method, 
as it is a simple matter to draw in arcs which will connect the 
large arcs and pass through the ends of the major axis. 

The layout of approximate ellipses by eight circular arcs may be 
facilitated by the method shown in Figs. 38 and 39 {Amer. Mach., 
Mar. 18, 1909). 

Lay out the long diameter AB and the short diameter CD, Fig. 
38, crossing each other centrally at F. Construct the parallelogram 
AECF, and draw the diagonal AC. From E draw a line at right 
angles to AC, crossing the long diameter at H and meeting the 
short diameter, extended, if necessary, at G; H is the center, and 
AH the radius for the end of the ellipse; G is the center, and CG 
is the radius for the side. 



To lay off the length of a circular arc on a straight line; Draw the 
chord ah. Fig. 41 , and extend it. Make he = \ah. With c as a center 
strike the arc ad. The length hd of the tangent at b equals the length 
of the arc ah very nearly. If the arc is of 60 deg. the error is a 
little less than toVtt of the length of the arc, the error varying as the 
fourth power of the angle subtended. 

To lay off the length of a straight line on a circular arc: Draw the 
arc tangent to the given line ah, Fig. 42, at a. Make ac=\ ah. 
With c as a center strike the arc hd. The length ad of the arc equals 
the length ah very nearly. The error of this construction and the 
law of its variation are the same as those of the one above. 
Both the above rules are due to Professor Rankine {Rules and 
Tables). 

Circular arcs of large radius may be drawn by the instrument 
shown in Fig. 43. The pencil a is located at the extremity of the 
rise of the arc and knife-edged weights bb are placed at the extremities 
of the chord. The parts being then clamped in position, the pencil 
will trace a true circular arc as shown. 

A compass for circular arcs of large radius, which is not a make- 
shift, is shown in Figs. 44 and 45 by U. Peters {Amer. Mach., 
Oct. 12, 1899). The entire instrument is 25 ins. long and it will 
fit arcs of any radius up to infinity. It consists of the rod A , an 
assortment of metal disks D and Di, drawing pen holder P and 
weight g. Fig. 44. 

By placing on the rod A one disk Di of somewhat smaller di- 
ameter at a certain distance from the other D, it is clear that by 
rolling the instrument over a plane (by means of handle H), on 
the principle of rolling cones, every point of the rod will describe 
an arc of a certain radius. 



318 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



The relations between the desired radius R, Fig. 45, the distance 
a between the disk edges and the diameters d and di of the disks 
are given by the equation : 



d-d 



10 



10 



p 


'// 


7. 


7 


/ 


y 


-^ 


^ 


^ 





/ 


7 / 


//. 


7 








^ 


^ 


8 
7 
6 
5 


m 


/ /"^z 


7, 


/ 


^ 




^ 


-^ 


^ 


m 




7 


x^. 


^ 


^ 




_^_^ 


^ 


m 


\A 




■^. 


"iV^ 




"^ 


.^ 


-- 


'F' 


^ 


y ^^ 




V 


VA^ 




--— 


--^ 




/// 


yy 


^ 








\ 








4 
3 
2 




^ 


^ 


^^ 










__i4__ 





W/y 


^■^ 
















« 






















1 



123456789 10 

.Fig. 37. — Laying out approximate ellipses by four circular arcs. 




2 3 
Fig. 39- 

Figs. 38 and 39. — Laying out approximate ellipses by eight circular 

arcs. 



If the larger disk be of 4 ins. diameter, this becomes: 



{^-dC)R 

or a = 

4 

from which Table 4 is obtained. 

Example. — Required the settings for an arc of 52 ft. radius. Con- 
sulting the table, the third change disk of diameter 3tf ins. should be 

placed on the rod at a distance 0=7- =fl ft. = 9| ins. 

04 




Fig. 40. — Calculating the radius of an arc for a given span and rise 

■h 




Fig. 41. Fig. 42, 

Figs. 41 and 42. — ^Lengths of arcs and straight lines. 




Fig. 43. — Instrument for drawing circular arcs of large radius. 

Table 4. — Disk Diameters and Settings for Large Circle 

Compass 



Radius 


Diameter of change 
disk, ins. 


Distance a between 
steady and change disk 


R in ins. | R in ft. 


edges 


24 to 96 

96 to 384 

384 to 1536 

1536 to 6144 


2 to 8 

8 to 32 

32 to 128 

128 to 512 


3 

3i 

3li 

3H 


a = i R 
a=fs R 
a=h. R 



R-- 



Permissible Cost of Special Shop Equipment 

The justifiable cost of special shop equipment from small jigs and 
fixtures up to large special machine tools is to be found by balancing 
the cost against the saving. If such equipment is to be profitable 
is must return its original cost during its useful life, pay interest on 
this cost and then a profit in addition. A ready method of determin- 
ing the justifiable cost from the estimated life and saving Is found 
in Table 5 by John H. Van Deventer {Amer. Mack., May 13, 1915). 
The use of which is best shown by an example: 

Required the permissible cost of a special tool that will have a 
probable life of two years and is to save thirty dollars per annum. 
In line with the saving and below a probable life of two years we find 
that if the cost is $45.30 the return on the investment will be 10 per 
cent., if the cost is I39.50 the return will be 20 per cent, and so on. 



MISCELLANEOUS MECHANISMS, CONSTRUCTIONS AND DATA 



319 



Table 5. — Profit Return on Investment in Special Shop Equipment of a Given Length op Useful Life 
The table includes 6 per cent, interest on the investment over and above the per cent, given as earned. For greater annual savings than those shown, the 
permissible investment is in direct proportion. Thus for an annual saving of $500.00, a life of two years and a return of 20 per cent, read $657.00 for $65.70 
found in the table for a saving of $50.00 per annum. 



Estimated 


Probable life one year 


Probable life two years 


Probable life five years 


annual 


• Per cent, earned on 


investment 


Per cent, earned on 


nvestment 


Per cent, earned on investmen 


t 


saving 


10 


20 


30 


40 


SO 


10 


20 


30 


40 


SO 


10 


20 


30 


40 


50 


effected 


per cent. 


per cent. 


per cent. 


per cent. 


per cent. 


per cent. 


per cent. 


per cent. 


per cent. 


per cent. 


per cent. 


per cent. 


per cent. 


per cent. 


per cent. 


$10 


8.60 


7.90 


■ 7.30 


6.80 


6.40 


IS- 10 


13-20 


II .60 


10.40 


9.40 


27.80 


21.80 


17.80 


15.10 


13.20 


20 


17 . 20 


15.90 


14.70 


13-70 


12 .80 


30.30 


26-30 


23-20 


20.80 


18.90 


55-50 


43 SO 


35-70 


30.30 


26.30 


30 


25.80 


23.80 


22 .00 


20.60 


19.20 


4S.S0 


39-50 


34-90 


31 -20 


28.30 


83-30 


65.20 


53 so 


4S.S0 


39-50 


40 


34 SO 


31.80 


29.40 


27-40 


25.60 


60-50 


52-60 


46-50 


41 .60 


37-80 


III - 00 


87 .00 


71.40 


60.50 


52 -60 


50 


43.10 


39.70 


36.80 


34-30 


32.10 


75-70 


65-70 


58.10 


52.00 


47-20 


139-00 


108.80 


89-30 


75.70 


65-70 


60 


51.70 


47.60 


44.10 


41.10 


38-SO 


91 -00 


79-00 


69-80 


62.40 


56-60 


167.00 


130.00 


107-00 


91 .00 


79-00 


70 


60.40 


55-60 


SI. 50 


47-00 


45.00 


106.00 


92.00 


81 -40 


72.90 


66.00 


194 • SO 


152.00 


125 -OO 


106.00 


92 .00 


80 


69.00 


63. SO 


58.90 


S4-P0 


SI -40 


121 .00 


105.20 


93-00 


83.20 


75-50 


222 .00 


174.00 


143.00 


121 .00 


105 .20 


90 


77-60 


7X-S0 


66.20 


61 -70 


57-80 


136.00 


118.50 


105.00 


93-60 


85-00 


250.00 


196.00 


160.00 


136.00 


118.50 



Estimated 


Probable life seven and 


one-half years 


Probable life ten years 


Probable life fifteen years 


annual 


Per cent, earned on 


nvestment 


Per cent, earned on 


nvestment 


Per cent, earned on investment 


saving 


10 


20 30 


40 


SO 


10 


20 


30 


40 


SO 


10 


20 


30 


40 


SO 


effected 


per cent. 


per cent, per cent. 


per cent. 


per cent. 


per cent. 


per cent. 


per cent. 


per cent. 


per cent. 


per cent. 


per cent. 


per cent. 


per cent. 


per cent. 


$10 


34-20 


25.40 


20.30 


16.90 


14.40 


38.50 


27.80 


21 .80 


17.80 


15.10 


44.20 


30.60 


23.50 


19.00 


16.00 


20 


68.00 


51.00 


40.60 


33-80 


28.80 


77-00 


55-50 


43-50 


35-70 


30.30 


88.30 


61 .20 


47-00 


38.00 


32.00 


30 


102.00 


76.50 


61.00 


50.60 


43.30 


115.50 


83.30 


65.20 


53-50 


45.50 


132 -SO 


92.00 


70-40 


57.00 


48.00 


40 


136.50 


102.00 


81.20 


67.50 


57.60 


154.00 


III. 00 


87.00 


71.40 


60.50 


177-00 


122.50 


93.80 


76.00 


64.00 


SO 


171.00 


127.00 


103.00 


84.40 


72.00 


193.00 


139.00 


108.80 


89-30 


75.70 


221.00 


153.00 


117.00 


95.00 


80.00 


60 


205.00 


153.00 


122.00 


10 I .00 


86.50 


231 .00 


167.00 


130.00 


107-00 


91.00 


265.00 


184.00 


140 . 00 


114.00 


96.00 


70 


239.00 


178.00 


142.00 


118.00 


101 .00 


270.00 


194 -so 


152.00 


125,00 


106.00 


310.00 


214.00 


164.00 


133.00 


112 .00 


80 


273.00 


204.00 


162 .50 


135.00 


IIS-OO 


308.00 


222.00 


174.00 


143-00 


121 .00 


354-00 


245 .00 


187.50 


152 .00 


128.00 


90 


307.00 


229.00 


183.50 


152.00 


130.00 


348.00 


250.00 


196-00 


160.00 


136.00 


398.00 


276.00 


211 .00 


171 .00 


143 -SO 




Fig. 44. 



!tltl;^^^3^ 



-rr^ 



EiG. 4S. 



Figs. 44 and 45. — Compass for large circles; 



Weight of Solids of Revolution 

The volume and weight of solids of revolution may be found by the 
rule of Guldinus: The volume of any solid of revolution is equal to 
the area of the axial section multiplied by the length of the path 
of the center of gravity of that section. The section must be entirely 
on one side of the axis of revolution. 

With irregular sections (for example car wheels) the determination 
of the area of the section and of its center of gravity is troublesome, 
especially when it is desired to design a body of a given weight. 
For such cases Geo. F. Summers {Amer. Mack, Jan. 15, 1903) has 
devised an experimental method of sufficient accuracy for all practical 
purposes. The section a, Fig. 46, is cut from a piece of cardboard, 
mounted on an accurately balanced stick, and so located on the stick 
that the center of revolution of the body coincides with the pivot c 
of the stick. A balance piece b cut from the same cardboard, of a 
size explained below and with its center line marked, is placed upon 
the stick, a position being found by trial such that the pieces balance, 
when the distance x from the pivot to the center of the balance card, 
measured to a suitable scale, gives the weight of the body of revolu- 
tion. 

According to the rule of Guldinus: 

volume of body = area of section X2Tr>' 



in which f = radius of center of gravity of the section (which need 
not be known). 

weight of body = area of section X2irr»« 

in which wz = weight of a unit volume of the material. The last 
equation may be transposed to read : 

areaXr=^^ XWt 

2Trm 

The left side of this equation is the moment of the left side of 

Fig. 46. If the area of the balance card be made equal to and 

the parts be balanced as explained above, the distance x will repre- 
sent the other factor of the moment, that is the desired weight. 
If area be in square inches and r in inches, then 

area of balance card = sq. ins. 

27rW} 

and 1 in. radius of balance card represents i lb. weight. 

So large a scale as i in. for i lb. would, in most cases, lead to an 
inconvenient or even impracticable length of stick. In such cases 
it is only necessary to reduce the length representing i lb. to any 
desired value and then multiply the area of the balance card to 
correspond. That is to say, if }/[q in. represents i lb., then 



320 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



area of balance card = Xio 

2Trin 

In the case shown in Fig. 46, Hoo in- represents i lb. and the value 
of a; is 6.12 ins., showing the weight of the body of revolution to be 
612 lbs. 

In order to design a body of a given weight an approximate section, 
somewhat too large, is first made, the radius of the balance card is 




+ 



Fig. 46. 



-x-Q.\Z- 



FiG. 47. 

Figs. 46 and 47. 



-Experimental method of finding the weight of 
solids of revolution. 



adjusted to the value for the desired weight and the trial section 
is then trimmed until the pieces balance. The balance card once 
made is used for all pieces of the same material and for the same 
relation of weight to radius of balance card. 

The radial section card may be made to a reduced scale if due 
account be taken of the fact that the weights of similar bodies are 
to each other as the cubes of their like dimensions. If the radial 



scale, i.e., if the drawing scale be >^ size, multiply the scale beam 
reading by 2, if 34 size, by 4, etc. 

In the construction of the apparatus a small glass bead is inserted 
in the stick for a bearing, the support being a fine sewing needle 
driven into the wall with some distance beads to preserve its position 
as in Fig. 47. 

Diameters of Shell Blanks 

When a sample is furnished the best method of finding the diame- 
ter of a shell blank is by comparative weights. A disk of i sq. in. 
area, that is 1.128 ins. diameter, and of the same gage metal as 
the sample is cut out and this disk and the sample are then weighed 
on a fine and accurate scale. The weight of the sample divided by 
the weight of the disk gives the area of the article and of the required 
blank in square inches, from which the diameter of the blank is 
readily calculated. That is 

if a = area of sample and blank, sq. in., 
W = weight of sample in any unit, 
w = weight of disk in the same unit, 
«Z = diameter of blank, ins.. 
W 



and as a=- 



we have 



W 



from which d- 



■-a 

■nd^ 

4 
Tvd^ 

4 
^1.128 



\ w 



{a) 





Figs. 48 to 51. — Graphical method of finding the diameters of shell blanks. 



section card be made to a scale of 3^ full size, its area will be J^ the 
full area and if the section card be located on the stick by the same 
scale its moment -will be 3^ the full moment. If now the balance 
card be made according to the above rule and full size, the scale 
beam reading will give the weight of a body of the same actual 
section as the section card, that is (3^)' = 3^ the weight of the full 
size body. If, however, the balance card be also made to the 3^ 
scale, its area will be 3<4 the full area and its radius — that is the above 
scale beam reading — will be multiplied by 4 and become }iX4 — /4 
the weight of the full sized body. Therefore we have the rule: 
Make the balance card to the same scale as the section card and 
multiply the scale beam reading by the reciprocal of the drawing 



If a sample is not at hand, the rule of Guldinus supplies a graphical 
method which is applicable to any cross-section, provided the shell 
be a surface of revolution. It assumes, as do the previous and 
other methods, that the thickness of the metal is not changed 
during the operation. The rule of Guldinus which is the basis 
of the method is as follows: The area of a surface generated by 
the revolution of a line about a central axis is equal to the length 
of the line multiplied by the circumference of the circle traced 
by the center of gravity of the line. It is thus only necessary to 
find the length and center of gravity of the desired profile to find the 
area of the desired article from which, as it is also the area of the 
blank, the diameter of the blank is easily determined. This method 



MISCELLANEOUS MECHANISMS, CONSTRUCTIONS AND DATA 



321 



as explained by F. Spaekuhl (Amer. Mach., Dec. 4, 1913) is as 
follows: 

If a = area of article and of blank, sq. ins., 

r = radius of center of gravity of generating line from the central 

axis, ins., 
/ = length of generating line, ins., 
i = diameter of shell blank, ins. 
we have, by the rule of Guldinus: 
a = 2wrl 

article equals the area of the blank, we have 



and. 


as the area of th 


rJso 




ird^ 




a = 


4 


or 


2ivrl = 


4 


from which d = 


■-Vm 



ib) 

in which it is only necessary to find r and I in order to calculate d, 
I being readUy obtained from the section of the article and r by the 
following graphical method. Note that I is the length of one-half 
the complete section, that is, its length on one side of the axis, and 
the center of gravity is also that of one-half the section. 

The center of gravity of the entire half section is obtained graphic- 
ally from the centers of gravity of parts of it which are known. 
Taking first, for illustration, the simplest possible case, a flat bot- 
tomed shell, Fig. 48, the center of gravity of the part ah is at its 
center d and that of be is at its center e. At the right make a'h' = ah 
and h'c' = bc, their sum being / of formula (6). Choose the pole a at 
any convenient point and draw a'o, b'o and c'o. Drop perpendiculars 
df and eg from the centers of gravity d and e and of any convenient 
length. Draw /A parallel with a'o,fg parallel with b'o and gh parallel 
with c'o when the radius r oi h is, the required radius of the center of 
gravity of section ahc to be substituted in formula (6) in order to 
find the required diameter of the blank. 

Taking next the flanged straight shell. Fig. 49, locate the centers 
of gravity e,f, g of the sections, construct the polar diagram and drop 
perpendiculars from the centers of gravity as before. In drawing 
the parallels to the rays of the polar diagram, begin with one extreme 
as a'o, drawing hi of indefinite length and parallel with a'o. Follow 
in order with hj parallel with h'o, jk parallel with c'o, and kl parallel 
with d'o, giving r as before. 

In the case of a bevel flanged round bottom shell. Fig. 50, the 
only difference from the preceding case grows out of the fact that the 
center of gravity g of the arc cd lies within the arc and not on it. 
In laying down the polar diagram the length of cd may be obtained 
from a table of circumferences, graphically by Rankine's rule for 
rectifying an arc, which see, or by the slide rule. The center of 
gravity g of the arc is to be found by the rule for that purpose, which 
see. In the case of the most usual arc — a quadrant — we have the 
relation gk = .goo3Xhi, hi being the median radius. Laying down g 
the process becomes the same as that of Fig. 49, the bevel flange 
introducing no new feature. 

Considering finally a more complex case, Fig. 51, locate the centers 
of gravity of the arcs as in the last example, construct the polar 
diagram and find r by drawing parallels to the rays of the polar 
diagram, beginning with one of the extreme rays and following with 
the others in order, as directed in connection with Fig. 49. 

Addition of Binary Fractions 

The addition of binary Jractions is usually made an unnecessarily 
laborious operation. They should be added in essentially the same 
manner as decimals, adding first those with the largest denominator, 
dividing the result by 2, setting down the remainder of i, if there 
is any, and carrying the quotient to the fractions having the next 
smaller denominator. The following illustration will show the 
analogy between the processes of adding these and decimal fractions. 
Let it be required to add the quantities: 

4f +3i+4:^ + ^ + 2^ + ll^-|-^ + 2f = ? 
21 



Beginning with the fractions having the largest denominator — 32 — 
we add them and obtain: 

3+S+9 = H=tV+^ 
the ^ forming part of the answer and the xV being carried to the 
sixteenths, which are next added, thus: 

8+7+9 =*f=Y+A 
the xV forming part of the answer and the -^ being carried to the 
eighths, which are then added, thus: 

i2+3=-¥-=i+i 

the I forming again part of the answer and the i being again carried 
to the fourths thus: 

7 + 1+3=^^ = 2 + 1 
Proceeding as before with the whole numbers we have 
2+4+3+4+2 + 1 + 2 = 18 
and, annexing the several remainders to the final 18, we have the 
answer 

i8 + 3+5 + ig- + 32 = 183^ or i85 + 32' 
the latter being the preferable method of expression on drawings. 

Standard Cross-sections 

Standard cross-sections for drawings in accordance with the rec- 
ommendations of a committee of the A. S. M. E., 191 2, are given 
in Figs. 52 and 53. The author gives them, not because he believes 
in such conventions but because many others do, and if they are to be 
used at all uniformity is obviously desirable. The committee rec- 
ommend that subdivisions of any of the materials shown genericaUy 
in Fig. 52, should be made by taking one of these standard cross- 
sections as a basis and making minor changes, but maintaining the 
general characteristics; or by writing on the standard section the 
name of the material. To illustrate, the committee has subdivided 
concrete into concrete blocks, cyclopean concrete and reinforced 
concrete, as shown in Fig. 53; and also wrought steel into nickel, 
chrome and vanadium steels. 

In the author's opinion the method shown in the lower right-hand 
corner of Fig. 52 is the only one to be encouraged. The others are 
nothing but hieroglyphs which require memorization by all concerned 
or the constant consultation of a key. The hieroglyphs are memor- 
ized by few and why they, with the necessary key, should be pre- 
ferred to self-explanatory English has never been explained. It is 
impossible to make such a schedule complete, as Fig. 52 will show, and 
resort must be made to simple English in the end. Why not use it 
in all cases? The whole plan is a case of system gone to seed. 

Filing Notes and Clippings 

Every engineer finds a systematic plan of filing and indexing notes 
of experience and clippings from technical papers a necessity. No 
technical paper is worth preserving entire, such preservation, in fact, 
soon defeating its own purpose by the bulk of unclassified information 
to which it leads. 

A satisfactory plan should embody the following features: (i) A 
minimum of pasting and indexing; (2) indefinite expansibility; (3) 
notes, clippings and references to books should be kept together; 
(4) all related information should be grouped together. 

Repeated publication in technical papers of methods of filing and 
indexing leads the author to include here his own method {Amer. 
Mach., March 11, 1909). It is an adaptation of the vertical filing 
system, a file box (containing one-half the alphabet) being shown in 
Fig. 54- 

The articles which it is desired to preserve are simply clipped out 
and folded to uniform size. A letter is written at the title of the 
article to indicate its place in the box, the clipping is dropped into 
place, and that is all there is of it. The index letters on the clippings 
are necessary to insure their replacement where they belong after 
consultation. Many articles may be indexed under several heads, 



322 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



and unless the index letter is used an article is sure to be dropped into 
the wrong place at some time in the future. 

Notes are written on sheets of stiff squared paper, cut to the size 
to which the clippings are folded, which are dropped in place among 
the clippings. These sheets also serve for references to books and 




Cast Iron 



Wrought Iron 



Cast Steel 



Wrought Steel 



I ^/, //////// 





Aluminum 




Glass 



Wood 



Water 



Puddle 




,, ,, ,,,, 


■ 1 ■ ■ r 


II 1 


t 1 


1 1 ) 




n-'.j^ 






Concrete 



Brick 



Coursed Uncoursed 
Bubble 



Ashlar 




Rock 



Original Filling 
Earth 



Fig. 32 - Recommended Standard Cross section 



^■■■v.--;!q-,':'.-.-:V.:^,:.. 



Concrete 



l:v<-,v.<rNvi?:'.'.:i'A.-- 
Concrete Blocks 





r"^^ 






;;*v 






'^:;m 



Expanded-Wire or 
Metal Rods 

Reinforced 
Concrete 




Vanadium Steel 



Wrought Steel Nickel Steel Chrome Steel 

Fig. 53- - Typical Subdivisions 
A. S. M. E. standard cross sections. 



for cross references to articles printed on the backs of others. As the 
collection grows, folders are used to segregate matter on various 
subjects and when the box is full its contents are divided between 
two boxes as in the illustration. 



The size of the box should be such as to take the standard 6X9 in, 
page. Pages from standard 9X12 periodicals require a single fold. 

Blue Print Solution 

I oz. red prussiate of potash, 
3j oz. water, 
I oz. citrate of iron and ammonia, 
35 oz. water. 




Index file for notes and clippings. 



Metallic Indicator Paper 

The paper should be sized with glue or paste, and zinc oxide, in 
powder, should be sprinkled on before the size is dry. When dry 
the paper must be pressed or rolled smooth. Running it through a 
photographic burnisher gives the best surface. 

For a pencil, use a brass wire with end rounded and buffed smooth. 



MISCELLANEOUS MECHANISMS, CONSTRUCTIONS AND DATA 



323 




Fig. sS- — American railroad clearances. Official composite cross-section by the American Railway Association. 



PERFORMANCE AND POWER REQUIREMENTS OF TOOLS 



For the carbon content of steel suitable for various cutting tools, 
ese Index. 

Power Constants for Lathe Tools 

The pressure on cutting tools formed the subject of an exhaustive 
investigation by F. W. Taylor and his associates {Trans. A. S. M. E., 
Vol. 28) . Following are Mr. Taylor's general conclusions regarding the 
tangential pressure of the chip on lathe tools when cutting cast-iron. 

{A) The total tangential pressure of the chip on the tool in cut- 
ting cast-iron of the difierent qualities experimented upon varies 
between the low limit of 3 s tons (of 2000 lbs.) per sq. in. sectional area 
of chip for soft cast-iron, when a coarse feed is used, and 99 tons 
per sq. in. sectional area of chip for hard cast-iron, when a fine feed is 
used. 

{B) In cutting the same piece of cast-iron, the pressure of the chip 
on the tool per sq. in. sectional area of chip grows considerably 
greater as the chip becomes thinner, and slightly greater as the cut 
becomes more shallow in depth. The following are the high and low 
limits of pressure per sq. in. of sectional area of the chip when light 
and heavy cuts are taken on the same piece of cast-iron: 

Depth of cut \ in.Xfeed .0328 in.: Total pressure per sq. in 
sectional area of chip, 128,000 lbs. Depth of cutfi in.Xfeed .1292 
in.: Total pressure per sq. in. sectional area of chip, 75,000 lbs. 

(C) The same fact mathematically expressed is that in cutting 
the same piece of cast-iron, the pressure of the chip on the too' per 
sq. in. sectional area of chip grows greater as the thickness of the chip 
grows less in proportion to (thickness of feed) *. 

The pressure of the chip per sq. in. of section also grows greater 
as the depth of the cut grows less in proportion to (depth of cut) 15. 

(Z)) The effect upon the pressure of the chip on the tool of a change 
in the thickness of the feed and the depth of the cut is the same 
for hard and soft cast-iron, and is represented by the same general 
formula, with a change merely of the constant. 

{E) In taking cuts having the same depth and the same feed, the 
pressure of the chip on the tool becomes slightly greater the larger 
the cutting tool that is used. This increase in the pressure follows 
from the fact that the larger the curve of the cutting edge of the tool 
the thinner the shaving becomes. 

Following are the corresponding conclusions regarding the tan- 
gential pressure on the chip when cutting steel: 

{A) The total pressure of the chip on the tool in cutting steel of the 
different qualities experimented upon varies between the low limit 
of 92 tons (of 2000 lbs.) per sq. in., and the high limit of 168 tons 
per sq. in. sectional area of the chip. 

{B) In cutting the same piece of steel, the pressure of the chip 
on the tool per sq. in. of sectional area of the chip grows very slightly 
greater as the chip becomes thinner, and is practically the same 
whether the cut is deep or shallow. The following are typical cases 
illustrating the relative pressures of a thin feed on the one hand and 
a coarse feed on the other: 

Depth of cut ]%■ in.Xfeed .0156 in.: Total pressure per sq. in. 
sectional area of chip, 295,000 lbs. Depth of cut xs in.Xfeed .125 
in.: Total pressure per sq. in. sectional area of chip 257,000 lbs. 

(C) The same fact mathematically expressed is that in cutting 
the same piece of steel, the pressure of the chip on the tool per sq. 
in. of sectional area of the chip grows faster as the thickness of the 
chip grows less in proportion to (thickness of feed) '". 

The pressure of the chip is in direct proportion to the depth of the 
cut. 



{D) Within the limits of cutting speed in common use, the pressure 
of the chip upon the tool is the same whether fast or slow cutting 
speeds are used. 

(£) The pressure of the chip upon the tool depends but little upon 
the hardness or softness of the steel being cut, but increases as the 
quality of the seel grows finer. In other words, high grades of steel, 
whether soft or hard, give greater pressures on the tool than are given 
by inferior qualities of steel. 

(F) The pressure of the chip on the tool per sq. in. of sectional 
area of the chip depends both upon the tensile strength of the steel 
and its percentage of stretch, and increases both as the tensile strength 
and stretch increase; although a higher tensile strength has more 
effect than a large percentage of stretch in increasing the pressure. 

Mr. Taylor considers the most important conclusion resulting 
from his experiments on the pressure of the chip on the tool to be 
that the gearing designed in lathes, boring mills, etc., for feeding 
the tool should be sufficiently strong to deliver at the nose of the 
tool a feeding pressure equal to the entire driving pressure of the chip 
upon the lip surface of the tool. 

The pressures on cutting tools may be determined from Figs, i and 2, 
by H. L. Seward {Amer. Mach., Nov. 16, 1911), which represent 
the formulas developed by Mr. Taylor as follows: 

For steel, P = 2^0,000 DF^^ 

For cast-iron, P = CDiiF^ 

in which P = tangential pressure, lbs., 
£> = depth of cut, ins., 
7^ = feed, ins., 

C = a constant which varies from 45,000 for soft cast-iron to 
69,000 for hard cast-iron. 

Note that in Mr. Taylor's experiments the pressures were measured 
at the tool and do not include the effort necessary to drive the machine. 
Below the charts will be found directions for their use. 

Power Constants for Twist Drills 

The torque and thrust of twist drills formed the subject of extensive 
tests by Dempster Smith and P. Poliakoff {Proc. I. M. E., 1909). 
The results have been plotted by W. T. Sears, Mech. Engr., Niles- 
Bement-Pond Co. (Amer. Mach., Sept. 5, 191 2) whose charts are 
repeated in Figs. 3 and 4 from which the torgue and thrust of twist 
drills within their range may be obtained. The use of the charts is 
explained below them 

The steel experimented upon was of medium hardness— .29 per 
cent, carbon, .625 per cent, manganese. 

Note that in the experiments the torque was measured at the drill 
and does not include the friction of the driving mechanism. 

Experiments on smaller drills in cast-iron only, were made by C. S. 
Frarey and E. A. Adams {Journal Worcester Polytechnic Institute, 
1906). The results for torque, after averaging, are presented in 
Fig. 5 {Amer. Mach., Feb. 14, 1907) together with those for thrust 
(Henry Hess, Amer. Mach., Apr. 25, 1907). The use of the torque 
chart is best shown by an example: 

For a f-in. drill at a feed of 15 thousandths per revolution, raise a 
perpendicular from the intersection a of the size of drill, and the 15 
thousandths feed lines to the observation line. To find the height 
of the intersection, ab may be taken in the dividers and compared 
with the vertical scale of torques, or we may follow one set of diagonals 
to c, and the other to d, and there read 195 Ib.-ins. 
(Continued on page 327, first column) 



324 



PERFORMANCE AND POWER REQUIREMENTS OF TOOLS 



325 



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PERFORMANCE AND POWER REQUIREMENTS OF TOOLS 



327 



-Feed, Ins, per Revolution for End Thrust 
.06 .04 .03 .02 .01 ( 



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Feed, Ins. per Kevolution for Torque and Horsepower 



4 6 



10 12 11 la 

Horsepower 



18 20 22 21 26 



Enter the left, lower scale oifeed per revolution of spindle for tongue and horse-power at the point representing the given feed; trace 
vertically upward to the full curve of the given drill diameter; then horizontally from this intersection to the right, crossing the 
scale torque in lbs. at ift. radius, from which the torque can be read; then continuing to the inclined line of the given speed. From 
this intersection trace vertically downward to the horse-power scale and read the horse-power required. To find the end thrust : 
Enter the left upper scale of feed in ins. per resolution for end thrust with the given feed and trace vertically downward to the broken 
line representing the size of drill; then horizontally from this intersection to the left to the scale end thricst in lbs. from which the 
thrust can be read. To find the cutting speed : Enter the right vertical scale diameter of drill in ins. for cutting speed with the given 
size of drill; trace horizontally to the left to an intersection with the line representing the given speed in rev. per min.; then ver- 
tically upward to the scale cutting speed in ft. per min. and read the cutting speed required. 

Fig. 3. — Torque, end thrust and horse-power of twist drills drilling cast-iron. 



As it is always permissible to use the results of a set of experiments 
somewhat beyond their actual range, the diagram has been extended 
in dotted lines to include drills up to i^-ins. and feeds up to 30 thou- 
sandths — the full line portion of the diagram representing the field 
of the actual experiments, and the dotted portions the extensions. 
The length of ef, obtained by the dividers or by following the diag- 
onals as before, shows the torque for a i-in. drill under a feed of 25 
thousandths to be 476 Ib.-ins. 

In these experiments, also, the torque was measured at the drill. 
The chart for thrust is self-explanatory. 

Power Consumed by Drilling Machines 

A very complete and remarkably concordant series of tests were made 
by H. M. NoERis, Mach. Engr., the Cincinnati Bickford Tool Co. 
{Amer. Mach., Aug. 13, 1914), using i, i^i, ij^, 1% and 2 in. drills 
of the forged twist type and 2, 2^, 2H, 2% and 3 in. drills of the 
flat twisted type, the materials operated on being cast iron, of which 
the chemical composition and physical properties were not deter- 
mined, and machinery steel having the following composition and 
properties : 

Carbon, per cent 18 

Manganese, per cent 54 

Silicon, per cent 09 

Sulphur, per cent 059 



Phosphorus, per cent 085 

Tensile strength, lbs. per sq. in 56,200 

Elastic limit, lbs. per sq. in 34,200 

Elongation in 2 in., per cent 37-5 

Reduction of area, per cent 66 . o 

The power consumed was obtained from the current readings of 
the motor, corrected for the motor losses, the values used being the 
net power absorbed by the drilling machines. The machines used 
were Cincinnati Bickford 6-ft. plain radials, for which Mr. Norris 
deduces the general formula: 

h.p. = r[r.p.m.-348(^)"'"] (a) 

in which c = a factor which includes the machine used, the diameter 
of the drill and the feed, 
rf = diameter of drill, ins. 

For one of the machines used, which was fitted with a ball thrust 
bearing for the drill spindle only, this formula becomes: 

For I- to 2-in. drills of the forged twisted type drilling cast iron with 
back gears disengaged: 

h.p. = .33s<fi-2y-"|^r.p.m.-348(^)''"] (6) 

Machine steel, back gears disengaged : 

h.p. = 1. 218 (^i-^y-ssj^r.p.m. -348 (^) '■"'] (c) 



328 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



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1000 



,01 .02 .03 ,04 .05 

Feed, Ins. per Revolution for Torque and End Xhrust 



Cutting Speed, Ft. per Min. 
100 200 300 




15 20 25 30 
Horsepower 



Enter the left lower scale of jeed in ins. per revohUionjor torque and end thrust at the point representing the given feed; trace ver- 
tically upward to an intersection with the full curve representing the given drill diameter; then horizontally to the right, crossing 
the scale of torque in lbs. at i Jt. radius, from which the required torque may be read; then to the line representing the given drill 
speed and from this intersection vertically downward to the horse-power scale, from which the power may be read. To find the 
end thrust: Enter the scale of feed per revolution ; trace vertically upward to the broken line representing the size of drill. From 
this mtersection trace horizontally to the left to the scale end thrust in lbs. from which the desired thrust may be read. The cutting 
speed is found in the same manner as on the preceding chart for cast-iron. 

Fig. 4. — Torque, end thrust and horse-power of twist drills drilling steel. 



2- to 3-in. drills of the flat twisted type: cast-iron, back gears disen- 
gaged: 

h.p.= .21 <;i. 77^.^4 l^r.p.m. -348 (^)''"] {d) 

Cast-iron, back gears engaged: 

h.p. = .2S2rfi"/-'<[r.p.m.-348(^)"°'] {e) 

Machine steel, back gears disengaged : 

h.p. = i.i2<f"/»^[r.p.m.-348(^)""'] (/) 

Machine steel, back gears engaged: 

h.p. = 1 .34 d^ -/ - [r.p.m. -348 (^) ""] W 

in which (f = diameter of drill, ins., 

/=feed, ins. per revolution. 



For a second machine (the Cincinnati Bickford 6-ft., high speed, 
high power plain radial) having ball thrust bearings for both the 
drill spindle and the reversing gears, the general formula becomes, 
for machinery steel: 



h.p. = .i52 (i?+2.i)<fi-2y-7^[r.p.m.- (^+6.8)1 



m 



in which i? = ratio of gearing between intake shaft and drill spindle, 
the values in this machine being i, 2, 4 and 8, 
remaining rotation as before. 

In order to eliminate the use of logarithms in ordinary cases the 
values for the various exponential expressions given in Tables i 
and 2 were calculated. 

{Continued on next page, second column) 



PERFORMANCE AND POWER REQUIREMENTS OF TOOLS 



329 



Table i.— Values or Exponential Expressions Involving d in 
Formulas (b) to (h) 



Diam. 


^1.26 


(^1-" ■ 


.,n( "■ V" 




^^n<i+-6; 


'A 


•556 


•435 


189- s 


90.3 


M 


.698 


.601 


138.2 


76.4 


Vs 


.846 


.789 


105 .0 


66.6 


I 


1 .000 


1. 000 


81.8 


S9-0 


iVa 


1.158 


1 . 220 


64.9 


53- 


iH 


1.322 


1.484 


52.3 


48. 5 


iH 


1.489 


I-7S7 


42.8 


44.8 


IH 


1.660 


2.050 


3S-4 


41.6 


iM 


1-833 


2.360 


304 


38.9 


iH 


2.013 


2.693 


25-1 


36.6 


iVs 


2.194 


3.041 


21.4 


34-6 


2 


2.378 


3-411 


18.4 


32-9 


^Vs 


2.566 


3-798 


iS-9 


31.2 


2M 


2.756 


4. 201 


13 -9 


30.0 


2% 


2.948 


4.619 


12 . 1 


28.8 


2H 


3-144 


5.062 


10.7 


27-7 


2^ 


3-342 


S-S14 


9-4 


26.7 


2^i 


3 -541 


S-993 


8.4 


25.8 


2% 


3-741 


6.479 


7-S 


25-0 


3 


3 948 


6.991 


6.7 


24.2 



Table 2. — ^Values of Exponential Expressions Involving / in 
Formulas (6) to (g) 



Feed 


f.7. 


/•" ■ 


/ .88 


/.96 


.006 


.02269 


.01946 


.01108 


.00736 


.007 


•02 543 


.02191 


.01270 


.00854 


.008 


.02804 


.02429 


.01428 


.00970 


.009 


.03063 


.02659 


.01584 


.01087 


.010 


.03312 


,02886 


.01739 


.01203 


.011 


■03S53 


•03104 


.01897 


.01317 


.012 


.03789 


•03319 


.02040 


.01432 


.013 


.04021 


•03530 


.02189 


•01547 


.014 


.04248 


•03737 


•02339 


.01661 


.015 


.04470 


.03940 


•02483 


.01774 


.016 


.04689 


.04142 


.02628 


.01888 


.018 


■05115 


•04535 


.02915 


.02114 


.020 


•05530 


.04918 


.03198 


•02338 


.022 


•05934 


.05292 


•03478 


.02563 


.024 


.06329 


■05659 


•03755 


.02787 


.026 


.06715 


.06019 


.04029 


. 03009 


.028 


.07094 


•06373 


. 04300 


.03230 


.030 


.07466 


.06720 


.04569 


.03452 


.032 


.07831 


.07063 


•04837 


.03672 


-034 


.08190 


. 07400 


.05102 


•03893 


.036 


.08544 


•07733 


•05365 


.04112 


-038 


. 08894 


.08062 


.05626 


•04331 


.040 


.09237 


.08386 


.05886 


.04549 



Values calculated from formulas (b) to (g) which agree remarkably 
well with the observed values are given in Table 3. 



Table 3 

Power consumed by a Cincinnati Bickford, 6-ft., regular plain radial drill 
with ball thrust bearing at drill spindle only in driving i- to 2-inch twist 
drills o£ the forged type and 2- to 3-inch drills of the flat twisted type in cast 
iron and machinery steel at various speeds and feeds. The asterisks show the 
back gears to have been engaged. 





Particulars of drills 


Cast iron 


Machinery steel 


Type of 
drill 


Diam. 

of 

drill 


Speed of drill 


Feed per revolution, 
ins. 


Feed per revolution, 
ins. 




Rev. 


Ft. 


.013 1 .018 


.024 


.009 


.013 


.018 






229 


60 


1 .84 


2-37 


2.96 


2,84 


3.92 


S.23 






267 


70 


2.32 


2.98 


3.72 


3-57 


4.94 


6.58 






306 


80 


2.81 


3.61 


4-50 


4^33 


5. 98 


7.96 






344 


90 


3.28 


4.22 


5.26 


5.06 


6.99 


9.31 






382 


100 


3.76 


4-83 


6.03 


5.79 


8.00 


10.66 




iM 


183 


60 


2.17 


2.78 


3.47 


3.33 


4 .60 


6.14 




iM 


214 


70 


2.68 


3-44 


4.29 


4.12 


5.70 


7.59 




iH 


24s 


80 


319 


4. 10 


5. II 


A .91 


6.79 


9.04 




iH 


27S 


90 


3.68 


4.74 


5-91 


5.67 


7.84 


10.4s 




iH 


306 


100 


4.20 


5-39 


6.73 


6.46 


8.93 


11.90 




1 1/4 


IS3 


60 


2.41 


3.09 


3. 85 


3.76 


5 .20 


6.92 




iH 


178 


70 


2 .92 


3-75 


4.68 


4-57 


6.31 


8.40 


Forged ■ 


IW 


204 


80 


3-45 


4-43 


5.53 


5. 40 


7.46 


9.93 




IH 


229 


90 


3 96 


5-09 


6.35 


6 . 20 


8.56 


II .40 




Ij-i 


254 


100 


4-47 


5.75 


7.17 


7 .00 


9.66 


12.88 




iH 


131 


60 


2.67 


3.43 


4.28 


4. II 


5.68 


7.57 




iH 


153 


70 


3-23 


4. IS 


5.17 


4-97 


6.36 


9.14 




m 


175 


80 


3.78 


4.86 


6.06 


S.82 


8.04 


10.72 




iH 


196 


90 


4-31 


5.54 


6 . 91 


6.64 


9.17 


12 .21 




iH 


218 


100 


4.86 


6.25 


7.79 


7-49 


10.34 


13.78 




2 


115 


60 


2.88 


3.70 


4.61 


4.43 


6.12 


8. IS 




2 


134 


70 


3.44 


4.42 


S.52 


5.30 


7.32 


9.76 




2 


153 


80 


4 .01 


5.15 


6 .42 


6.17 


8.52 


11.35 




2 


172 


90 


4-57 


5.88 


7.32 


7.04 


9-73 


12 .96 




2 


191 


100 


5-14 


6.60 


8.23 


7.92 


10.93 


14.56 




2 


IIS 


60 


2.78 


3.54 


4-38 


4-03 


5-71 


6.80 




2 


134 


70 


3.33 


4.24 


5.24 


4.80 


6.83 


9-34 




2 


153 


80 


3-88 


4-93 


6 . 10 


5-55 


7.9s 


10.87 




2 


172 


90 


4.42 


5. 63 


6.96 


6.38 


9.07 


12 .40 




2 


191 


100 


4.97 


6.32 


7.82 


7.17 


10.20 


13.94 




2H 


102 


60 


3.12 


3.98 


4.92 


4.51 


6 .42 


8.76 




2H 


119 


70 


3.73 


4-74 


5.86 


5. 38 


7.64 


10.4s 




2H 


136 


80 


4-33 


5.51 


6.81 


6 .24 


8.88 


12.14 




2M 


153 


90 


4-93 


6.28 


7.76 


7.12 


10. 12 


13.84 




2U 


170 


100 


5.54 


7.04 


8.71 


7.98 


II .36 


15.52 




2H 


91.7 


60* 


4-iS 


5. 28 


6.54 


5.98 


8.50 


11.62 




2H 


107 


70 


4.12 


5-24 


6.42 


S-94 


8.44 


11.54 


F. T. ■ 


2H 


122 


80 


4 76 


6.05 


7-49 


6.86 


9.76 


13 -34 




2H 


138 


90 


5-40 


6.92 


8.56 


7.84 


II . 16 


15.2s 




2H 


153 


100 


6.08 


7.74 


9.58 


8.76 


12.48 


17. OS 




2% 


83.4 


6o* 


455 


5-79 


7.16 


6.54 


9-31 


12 .72 




2% 


97-2 


70' 


S-39 


6.85 


8.48 


7-75 


1 1. OS 


1507 




2H 


III 


80 


5. 19 


6.60 


8.17 


7.48 


10 .65 


I4-5S 




2H 


125 


90 


5-90 


7.50 


9.28 


8.51 


12 . 10 


16.53 




2H 


139 


100 


6,61 


8.40 


10.41 


9 .53 


13 -55 


18.52 




3 


76.4 


60 • 


4^94 


6.38 


7.77 


7.10 


10.10 


13.80 




3 


89.1 


70* 


5^84 


7.42 


9.18 


8.39 


11-94 


16.32 




3 


102 


80 


5.62 


7. IS 


8.85 


8. II 


11-54 


15.76 




3 


114 


90 


6.33 


8.05 


9 .97 


9.14 


12 .98 


17.75 




3 


127 


100 


7.10 


9.03 


II. 18 


10.23 


1455 


19.90 



330 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 




1500 



1000 



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Torque 



Drill Diameter, Ins, 

Thrust 



Fig. 5. — Torqvie and end thrust of twist drills drilling cast-iron. 



Comparing the powers absorbed when drilling cast iron and 
machinery steel Mr. Norris finds for the ratio between them the 
following values: 

At .013-in. feed, i- to 2-in. drills, 2. 13 
At .oi8-in. feed, i- to 2-in. drills, 2.209 
At .013-in. feed, 2- to 3-in. drills, 2.053 
At .oi8-in. feed, 2- to 3-in. drills, 2.204 
Mean 2 . 149 

The value of this mean agrees remarkably well with Mr. Smith's 
value of 2.1 and, in general, the results of the Norris and Smith tests 
show a very satisfactory concordance. 

The values deduced from formula (/j) are given in Table 4 in which 
the gear ratios are shown by the parentheses in the feet per min. 
column — the figures i, 2 and 4 representing the ratios i to i, i to 2 
and I to 4, respectively. This tables gives also the results obtained 
in drilling cast-iron, but these, at this writing, have not been 
formulated. 

The speeds of twist drills according to the latest practice of Mr. 
Norris and of the Henry and Wright Mfg. Co. are given in Table 5. 
For cast iron Mr. Norris's cutting speeds are uniformly 40 ft. per 
min. for carbon, and 80 ft. per min. for high speed steel drills. When 



drilling steel the table presupposes an adequate supply of cooling 
liquid, the cutting speeds for carbon steel drills being in accordance 
with the formula: 

6 



cutting speed, ft. per min. = ^ 



^+38 

ird 



12 

T + 76 

■wd 



and for high speed steel drills: 

cutting speed, ft. per min. = 

in which (i = diameter of drill, ins. 

The object and eSect of these formulas are to reduce the peripheral 
speed with increase of diameter in order to provide for the less efln- 
cient action of the cooling liquid on large drills. 

Power Constants for Milling Machines 

Two very complete sets of tests of the power required for slab 
milling were made by Alfred Herbert, Ltd., and reported by 
P. V. Vernon {The Engineer, 1909), the machine used being of the 
Herbert knee type. The following data are extracted from Mr. 
Vernon's report. The horse-powers are the equivalents of the cur- 



PERFORMANCE AND POWER REQUIREMENTS OF TOOLS 



331 



rent readings and include the motor losses and also a constant loss 
of 1.8 h.p. consumed in driving a jack shaft and countershaft through 
which the power was transmitted. 

Slabbing mild steel, average of 44, 2.52 h.p. per cu. in. per min. 



Table 5. — Speeds of Twist Drills 
Practice of the Cincinnati Bickford Tool Co. 



Slabbing mild steel, minimum, 
Slabbing mild steel, maximum, 
Slabbing cast-iron, average of 38, 
Slabbing cast-iron, minimum, 
Slabbing cast-iron, maximum, 



1.95 h.p. per cu. in. per min. 
3.02 h.p. per cu. in. per min. 
1. 10 h.p. per cu. in. per min. 
.89 h.p. per cu. in. per min. 
1.25 h.p. per cu. in. per min 



(Continued on next page, first column) 



Table 4 



Power consumed by a Cincinnati Bickford, 6-ft., high-speed, high-power, 
plain radial drill, fitted with ball thrust bearings at both the drill spindle and 
reversing gears. 



Drill 


Cast iron 


Machinery steel 


Feet 


Diam. 


Rev. 


.020" 


.030" 


.040" 


.012" 


.016" 


.020" 


per 


of 


per- 


feed, 


feed, 


feed. 


feed. 


feed, 


feed, 


min. 


drill 


mm. 


power 


power 


power 


power 


power 


power 


60 (i) 


H 


306 


2.76 


3.52 


4,20 


2.84 


3-53 


4-15 


70 (I) 


H 


357 


3.36 


4-29 


5. 10 


3-49 


4-32 


5.09 


80 (I) 


H 


.408 


3 98 


5.08 


6.04 


4.12 


5-10 


6.02 


90 (i) 


H 


4S9 


4.60 


5-86 


6.96 


4.76 


5.89 


6.95 


100 (i) 


Yi 


S09 


5.21 


6.64 


7.89 


5. 40 


6.68 


7.88 


60 (2) 


I 


229 


2.88 


3.67 


4-36 


4.01 


4.96 


5.85 


70 (I) 


I 


267 


3 09 


3.94 


4.68 


3-72 


4.60 


5,43 


80 (I) 


I 


306 


3.66 


4.66 


5.54 


4-39 


5.44 


6,42 


go (i) 


I 


344 


4.22 


5.38 


6.39 


5.16 


6.39 


7.54 


100 (i) 


I 


382 


4-79 


6. II 


7.26 


5-79 


7.17 


8,46 


60 (2) 


i!4 


183 


3-10 


3-95 


4-70 


4.21 


5.21 


6,15 


70 (2) 


iH 


214 


3.80 


4.84 


5-75 


5. 17 


6.40 


7-55 


80 (2) 


iVi 


24s 


4-SO 


5.74 


6.82 


5.96 


7.57 


8,93 


90 (i) 


iVi 


27s 


3-95 


5. 04 


5.99 


5-35 


6.62 


7,81 


100 (i) 


iH 


306 


4.49 


5-73 


6.81 


6.08 


7.52 


8,87 


60 (2) 


m 


153 


3.27 


4.17 


4 96 


4-36 


5-39 


6,36 


70 (2) 


l],i 


178 


4.02 


5-12 


6.08 


5.35 


6.62 


7.81 


80 (2) 


m 


204 


4-77 


6,08 


7.23 


6.35 


7.86 


9,27 


90 (2) 


i\<i. 


230 


5-51 


7.03 


8.36 


7.35 


9. 10 


10.73 


100 (2) 


m 


254 


6.27 


7-99 


950 


8.34 


10.32 


12 . 17 


60 (2) 


lU 


131 


3 42 


436 


5.18 


4.48 


S-SS 


6.55 


70 (2) 


m 


153 


4.21 


5. 37 


6.38 


5-53 


6.84 


8.07 


80 (2) 


m 


175 


5-00 


6.38 


7.58 


6.56 


8.12 


9.58 


90 (2) 


m 


196 


5.80 


7.39 


8.78 


7 .60 


9.41 


II . 10 


100 (2) 


m 


218 


6.59 


8.40 


9.98 


8.63 


10.68 


12 .60 


60 (4) 


2 


115 


4.87 


6.22 


7-39 


6.82 


8.44 


9.96 


70 (2) 


2 


134 


4-38 


S.59 


6.64 


5-66 


7 .00 


8.26 


80 (2) 


2 


153 


5.21 


6.6s 


7.90 


6.73 


8.33 


9.83 


90 (2) 


2 


172 


6.04 


7.71 


9. 16 


7.81 


9.66 


II ,40 


100 (2) 


2 


191 


6.87 


8.77 


10.32 


8.87 


10.98 


12.95 


60 (4) 


2H 


102 


5.18 


6,60 


7.84 


6.95 


8.60 


10.14 


70 (4) 


2H 


119 


6.40 


8.16 


9.70 


8.60 


10 .64 


12.55 


80 (2) 


2H 


136 


5. 40 


6.88 


8.18 


6.88 


8.52 


10.05 


90 (2) 


2K 


153 


6.27 


7-99 


950 


7.98 


9.88 


11 .6s 


100 (2) 


2H 


170 


7.13 


9 09 


10,80 


9.09 


II .2S 


13.27 


60 (4) 


2H 


91.7 


5.46 


6.96 


8.27 


7 .06 


8.74 


10.31 


70 (4) 


2K2 


107 


6.76 


8.63 


10.26 


8.75 


10.83 


12.77 


80 (4) 


2y2 


122 


8.06 


10.29 


12.23 


10.43 


12 .91 


15.23 


90 (4) 


2 Mi 


138 


6.46 


8.24 


9.79 


8.14 


10,08 


11 .89 


100 (2) 


2'A 


153 


7.36 


9.39 


II . 16 


9.28 


11.49 


13. ss 


60 (4) 


2H 


83-4 


5.73 


7.30 


6.68 


. 7. IS 


8.8s 


10.44 


70 (4) 


2H 


97-2 


7. II 


9.06 


10.77 


8.87 


10.98 


12.95 


80 (4) 


2U 


III 


8.49 


10.83 


12.87 


10.62 


13.14 


15.50 


90 (2) 


2U 


125 


9.90 


12 .62 


15 .00 


12.34 


15-27 


18.00 


100 (2) 


2% 


139 


7.59 


9.68 


II .50 


9.46 


II. 71 


13.81 


60 (4) 


3 


76.4 


5.96 


7 .60 


9.03 


7.22 


8.94 


10. SS 


70 (4) 


3 


89.1 


7-42 


9.46 


1 1. 25 


8.98 


II . 12 


13-12 


80 (4) 


3 


102 


8.88 


II .32 


13.45 


10. 75 


13.31 


15.71 


90 (4) 


3 


"5 


10.33 


13.17 


15.65 


12 .S2 


15.48 


18.26 


100 (4) 


3 


127 


11.75 


15 -00 


17.82 


14.26 


17,65 


20.82 





Drill speed, r.p.m. j 


Size 


D 


rill speed, r.p.m. 


Size 


High speed 


Corliss steel 1 


High speed 


Corliss steel 


of 


steel drill 


drill 1 


of 
drill 


drill 


■ drill 


drill 


Cast 


Mild 


Cast 


Mild 


Cast 


Mild 


Cast 


Mild 




iron 


steel 


iron 


steel 




iron 


steel 


iron 


steel 


}i 


1222 


1895 


611 


947 


iH 


175 


181 


87 


90 


Me 


978 


1399 


489 


699 


I'Me 


169 


174 


84 


87 


H 


8iS 


IIOO 


408 


550 


m 


163 


168 


81 


84 


Me 


699 


904 


349 


452 


I'Me 


158 


162 


79 


. 81 


Vz 


611 


764 


306 


382 


2 


153 


157 


76 


78 


?i6 ■ 


543 


662 


272 


331 


2M6 


148 


152 


74 


76 


5^ 


489 


S82 


245 


291 


2W 


144 


147 


72 


73 


iHs 


444 


S19 


222 


259 


2K8 


140 


142 


70 


71 


H 


408 


469 


203 


234 


2H 


136 


138 


68 


69 


'Me 


379 


427 


190 


213 


25/6 


132 


134 


66 


67 


Zi 


349 


392 


I7S 


196 


2H 


129 


130 


64 


65 


lji6 


326 


363 


163 


182 


2}U 


125 


127 


63 


63 


I 


306 


336 


IS3 


l58 


2M2 


122 


124 


61 


62 


I Me 


287 


314 


144 


157 


29/i6 


119 


120 


60 


60 


iH 


272 


294 


136 


147 


2H 


116 


117 


58 


58 


I?i6 


2S8 


277 


129 


139 


2IM6 


114 


115 


57 


56 


iVi 


245 


262 


123 


131 


2?i 


III 


112 


56 


55 


l5i6 


233 


248 


116 


124 


21^6 


109 


109 


54 


54 


m 


222 


23s 


III 


118 


274 


106 


106 


S3 


53 


ijie 


212 


224 


106 


112 


21^6 


104 


104 


52 


52 


IH 


204 


214 


102 


107 


3 


102 


102 


51 


51 


iMe 


195 


20S 


98 


102 


3\i 


98 


98 


49 


49 


m 


188 


196 


94 


98 


3H 


94 


94 


47 


47 


I'Me 


181 


189 


90 


94 


3H 


87 


87 


44 


44 



Carbon Steel Drills 
Practice of the Henry and Wright Mfg. Co. 



Size 
of 

drill 


Feed 


Bronze 
Drass, 


C. iron 
ann'ld. 


Hard 
c. iron. 


Mild 
steel. 


Drop 
forg., 


Mai. 
iron, 


Tool 
steel, 


Cast 
steel, 


per 
rev. 


ISO ft. 


8S ft. 


40 ft. 


60 ft. 


30 ft. 


45 ft. 


30 ft. 


20 ft. 


R.p.m. 


R.p.m. 


R.p.m. 


R.p.m. 


R.p.m. 


R.p.m. 


R.p.m. 


R.p.m. 


Me 


.003 




5185 


2440 


3660 


1830 


2745 


1830 


1220 


H 


.004 


^575 


2593 


1220 


1830 


915 


1375 


91S 


610 


Me 


.005 


3050 


1728 


813 


1220 


610 


91S 


610 


407 


H 


.006 


2287 


1296 


610 


91S 


458 


636 


458 


305 


5/1 e 


.007 


1830 


1037 


488 


732 


366 


569 


366 


24s 


H 


.008 


1525 


864 


407 


610 


30s 


458 


30s 


203 


Me 


.009 


1307 


741 


349 


523 


261 


392 


261 


174 


'A 


.010 


1143 


648 


305 


458 


229 


343 


229 


153 


5i 


.011 


915 


519 


244 


366 


183 


275 


183 


122 


H 


.012 


762 


432 


204 


30s 


153 


212 


153 


102 


U 


.013 


654 


371 


175 


262 


131 


196 


131 


87 


I 


.014 


571 


323 


153 


229 


115 


172 


lis 


77 



High Speed Drills 
Practice of the Henry and Wright Mfg. Co. 



Size 
of 
drill 


Feed 
per 
rev. 


Bronze 
brass, 
300 ft. 


C. iron 
ann'ld, 
170 ft. 


C. iron 
hard, 
80 ft. 


Mild 
steel, 
120 ft. 


Drop 
forg., 
60 ft. 


Mai. 
iron, 
90 ft. 


Tool 
steel, 
60 ft. 


Cast 
steel, 
40 ft. 


R.p.m. 


R.p.m. 


R.p.m. 


R.p.m. 


R.p.m. 


R.p.m. 


R.p.m. 


R.p.m. 


Me 


,003 

.004 
.005 
.006 
.007 

.008 
.009 
.010 
.011 
.012 

.013 
.014 






4880 
2440 
1626 
1220 
976 

813 
698 
610 

488 
407 

349 
30s 


3660 
2440 
1830 
1464 

1220 

1046 

915 

732 

610 

523 
458 


3660 

1830 

1210 

915 

732 

610 
522 

458 
366 
305 

261 
229 


2745 
1830 
1375 
1138 

91S 
784 
636 
569 
458 

392 
349 


3660 

1830 

1220 

915 

732 

610 

522 
458 
366 
305 

261 
229 


2440 

1720 
807 
610 
490 

407 
348 
30s 
24s 
203 

174 
153 


H 
Me 
H 
Mo 

Me 

y, 

M 

% 
I 


4575 
3660 

3050 
2614 
2287 
1830 
1525 

1307 
1 143 


5185 
3456 
2593 
2074 

1728 
1482 
1296 
1037 
864 

741 
648 



332 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



In these tests, in cast-iron, the feeds ranged between ifl and lo^^ 
ins. per min., the depth of cut between .14 and i.io ins. and the mater- 
ial removed between 7.39 and 15.23 cu. ins. per min. In steel, the 
feeds ranged between | and lof ins. per min., the depth of cut between 
.10 and I.IO ins. and the material removed between 2.88 and 6.27 cu. 
ins. per min. 



3. 413s sq. ins. of double belt passing over a pulley in i min. will 
remove i cu. in. of mild steel on a miller. 

4. A 4|-in. cutter on a 2-in. arbor running at 70 ft. per min. is 
capable of removing at least 3.63 cu. ins. and possibly as much as 
6.01 cu. ins. of cast-iron, and at least 2.125 cu. ins., and possibly as 
much as 3.03 cu. ins. of mild steel per min. for each inch of width up 
to 8 ins. 



Table 6. — Results of Cutting Tests with an 8-in., i2-bladed, 
High-power Face Mill in Machinery Steel, Cut 5 Ins. 
Wide, Block B 



Depth of 
cut, ins. 


Spindle 

speed, 

r.p.m. 


Feed in 

ins. per 

min. 


H.p. 

delivered to 

machine 


Cu. ins. Cu. 

of metal re- of me 

moved per mov 

min. h.p. 


ins. 
tal re- 
;d per 

min. 


' 


20^ 


12.31 


10.96 


7.69 


702 


J 


20^ 


12.26 


11.52 


7.66 


664 


20| 


12.26 


II. S2 


7.66 


664 


[ 


20^ 


12. 24 


11.52 


7.6s 


664 




20J 


7. SI 


10.96 


7.04 


642 


-5- I 


20 


7-34 


10. 70 


6.88 


643 




20 


7.38 


11.25 


6.92 


615 




20^ 


7.61 


11.52 


7.13 


618 




20 


5-9 


12.62 


7-375 


584 


1 I 


20^ 


5-97 


13- 20 


7.4s 


564 




20 


5-9 


13. 20 


7.375 


558 




20| 


5-97 


13.46 


7.45 


554 




20 


4.S4 


13.20 


7-09 


537 


' -5- I 


20J 


4.66 


13.46 


7.28 


542 




20J 


4.68 


13-20 


7-31 


554 


1, 


I9§ 


4.46 


13.46 


6.97 


518 




20 


3.48 


10.96 


6.53 


596 


1 


20 


3-49 


II. 81 


6-54 


554 


, 


20 


3-54 


12.62 


6.64 


526 



Table 7. — Results of Cutting Tests with an 8-in., i2-bladed. 
High-power Face Mill in Machinery Steel, Cut 5 Ins. 
Wide, Block A 



Depth of 
cut, ins. 


Spindle 
speed, 
r.p.m. 


Feed in 

ins. per 

min. 


H.p. 

delivered to 
machine 


Cu. ins. 
of metal re- 
moved per 
min. 


Cu. ins. 
of metal re- 
moved per 
h.p. min. 






19 


11-34 


8. 113 


7.088 


.873 


i ■■ 




19J 


11.72 


7.818 


7-325 


• 937 




I9i 


11.53 


7.555 


7-21 


.954 






20 


II. 81 


7-555 


7-381 


■ 977 






26 


16.0 


14-843 


10.000 


.674 


* 1 




25 


15.4 


13-473 


9.625 


-714 




25 


lS-4 


13-473 


9.625 


-714 




I 


25 


IS. 4 


13-473 


9.62s 


-714 






20 


7.44 


10.319 


6.975 


.686 


*■ 




20 


7.29 


9.749 


6.834 


.701 




20 


7.32 


10.319 


6.863 


.665 






20 


7.32 


10.319 


6.863 


-665 






I9J 


S.7S 


II. 41 


7.188 


-63 


i < 




20 


5. 81 


II -959 


7.263 


.608 




19h 


5. 69 


1I-138 


7. 113 


.638 




' 


20 


S.84 


II .41 


7-30 


-64 






20i 


4.6 


12.52 


8.568 


-684 


A^ 




20 


4-47 


11.959 


8.32s 


.696 




I 


20 


4-57 


11.959 


8.512 


.712 



More recent tests by Mr. Vernon {Proc. Manchester Asso. of 
Engrs., 1912) have led to the following conclusions: 

1. A s-in. double belt driving a i6-in. pulley at a speed of 400 
r.p.m. (100,531 sq. ins. of belt surface per min.) geared to drive a 
4i-in. cutter at 70 ft. per min., is able to remove as much as 48.1 
cu. ins. of cast-iron and 24.31 cu. ins. of mild steel in a minute. 

2. 2090 sq. ins. of double belt passing over a pulley in i min. will 
remove i cu. in. of cast-iron on a miller. 



Table 8. — Results of Cutting Tests with an 8-in., i2-bladed, 
High-power Face Mill in Machinery Steel, Cut 5 Ins. 
Wide, Block A 



Depth of 
cut, ins. 


Spindle 
speed, 
r.p.m. 


Feed in 

ins. per 

min. 


H.p. 

delivered to 

machine 


Cu. ins. 
of metal re- 
moved per 
min. 


Cu. ins. 
of metal re- 
moved per 
h.p. min. 




20 


II .92 


7-92 


7-45 


•94 


4 < 


20 


11.83 


7-34 


7-39 


1.007 




19^ 


11-73 


7.62 


7-33 


.962 


[ 


19J 


11-73 


7-62 


7-33 


.962 




I9i 


7-25 


7-34 


6.80 


.927 


J- . 


20 


7-29 


7-07 


6.83 


.967 




20 


7-29 


7-07 


6.83 


.967 


, 


I9i 


7-25 


7.07 


6.80 


.964 




19} 


5-7 


8-17 


7.12 


.872 


!. I 


19 


5-58 


7.62 


6.97 


.915 




19 


S-54 


8.17 


6.92 


.847 




19 


5-58 


8.17 


6.97 


.854 


Ar < 


20 


4-53 


8.17 


7.08 


.867 


20 


4-56 


7.34 


7-125 


.972 




20 


4-56 


7.92 


7.125 


.90 




20 


4-53 


8.17 


7.08 


.867 




20 


3-47 


7.07 


6. SI 


.921 


i • 


20 


3-54 


7.62 


6.64 


.873 


20 


3-49 


7-07 


6.54 


.92s 


. 


20 


3-S2 


7-62 


6.60 


.868 



Tests by A. L. DeLeeuw for the Cincinnati Milling Machine Co. 
on knee type machines, gave the results shown in Fig. 7 (Trans. 
A. S. M. E., 1911). The horse-powers given are the net outputs of 
the motors after the motor losses have been deducted from the current 
readings. 

Additional tests by Me. DeLeeuw (Amer. Mack., Aug. 8, 191 2) 
give the results of Tables 6-1 1. As a rule the material is specified by 
reference to a particular block whose physical properties are given 
in Table 12. Where there is no reference of this nature the material 
was machinery steel of a tensile strength of 55,000 lbs. per sq. in., 
.26 per cent, carbon and .5 per cent, manganese. 

In all these tests the efficiency of the motor was taken into con- 
sideration, and the horse-power values given have the motor losses 
eliminated. 

Table 13 gives the results of cuts by Mr. DeLeeuw on German irons 
and steels. The tests of speigeleisen are noteworthy, showing but 
little more power consumed than for cast-iron. 

In a report of some extremely (record) heavy slab milling on 
Niles-Bement-Pond planer type machines, W. H. Taylor (Amer. 
Mack., Jan. 14, 1909) gives data from which the following are ex- 
tracted. The horse-powers are the equivalents of the current read- 
ings, and include the motor losses. 



Slabbing mild steel, average of 15, 2.07 h.p. per cu. in. per min. 

Slabbing mild steel, minimum, 1.52 h.p. per cu. in. per min. 

Slabbing mild steel, maximum, 3.71 h.p. per cu. in. per min. 

Slabbing cast-iron, average of 8, 1.06 h.p. per cu. in. per min. 

{Continued on page 334, second column) 



PERFORMANCE AND POWER REQUIREMENTS OF TOOLS 



333 



6.0 

5,0 

u 
o 
S 4.0 



S3.0 

a 2.0 

1.0 


















^ 














X 


/ 












/ 


^ 






^ 






/ 


/ 










^ 




^ 















__ 




== 


— 













,/ 6.0 
5.0 

H"|i0 
'a LO 


















^^ 














^ 


y 


X 










/ 


/ 






^ 






/ 


^ 




^ 




^ 


■^ 




/ 




^ 


__.^ 


^ 




\Z. 


_ 




' 




" 











2 4 6 8 10 12 14 16 ] 
Feed, In. per Min, 

Spiral Nicked Milling-Cutter 
3'5&" Diameter; 18 Teeth and 
about ^" Pitch. Cutting Cast 
Iron. 



6.0 

5.0 

S 4.0 

3," '^ 3.0 

/l4 CD 

h" ° 2.0 

tq 

'/w LO 



2 4 6 8 10 12 14 16 IS 
Feed, In. per Min. 
Spiral Nicked Milling-Cutter 
3 '/u' Diameter; 14 Teeth and 
about ?4" Pitch. Cutting Cast 
Iron, 






6.0 

,',' 5.0 

S 4,0 

'/la u 
u"m 2.0 

'/':; LO 



2 4 6 8 10 12 14 16 18 
Feed, In. per Min. 
Spiral Nicked Milling-Cutter 
6" Diameter; 16 Teeth and 
about lit" Pitch. Cutting 
Cast Iron. 

















> 


y 












/ 


V 












jH 


/ 




^ 


^ 








y 


y 

^ 




^ 


^ 


-^ 






/. 


^ 






-— 


" 








■ 

















2 4 6 8 10 12 14 IS 18 
Feed, In. per Min, 
End Milling-Cutter 3"Diameter, 
14 Teeth. Cutting Cast Iron. 



6.0 

5.0 

I 10 
o 

c* 

o 3.0 

W2.0 

LO 















1 


n 




6.0 
" 5.0 

W'^3-0 

2 

V" H 2-0 
/« w 

LO 




















6.0 
5.0 

s • 

V" s 

^A'^ ^ 2.0 

'/;1 LO 




















0.0 
5.0 

„ g 4.0 

^ 3.0 

'4.' « 2.0 

"^l 1.0 

'/is 






;/ 




/ 


7 




/^ 












y 


y 














































y 




y 


/ 


/^ 


-^ 










/ 


/ 


^ 




















^ 






















^ 




/ 


/ 


/ 


y 


/ 












/ 


^ 
















^ 


^ 


^ 


^ 


- — 


^ 












^ 


^ 




^^ 




'a 


y 


y 




■^ 


^ 








y. 


^ 


:::^ 








— 








^ 


:-; 


=^ 


■ 


^ 




— ' 






^ 






"^ 


"^ 




—' 




^ 








_- 


-- 




■ 








L 
















=^ 


H 


■ ■ 


— — 












■::ZZ. 



































1 















2 4 6 8 10 12 14 16 18 
Feed, In. per Min. 
End Milling-Cutter 5 Diameter, 
20 Teeth. Cutting Cast Iron. 



6.0 

5.0 
|4.0 



^3.0 
LO 
















/ 


^ 












X 














^ 


y 








^ 


^ 


y 


y 




^ 


^ 


y 










^ 






__^, 






-^ 




^ 


■^ 















2 4 6 8 10 12 14 16 13 
Feed, In. per Min. 
Regular Face Milling-Cutter 9 Vi 
Diameter, 22 Teeth, Cutting 
Cast Iron, 
0.0r 



2 4 6 8 10 12 14 10 18 
Feed, In. per Min. 
High Power Face Milling-Cutter 
8" Diameter, 6 Teeth. Cutting 
Cast Iron 



18 



2 4 6 8 10 12 14 16 
Feed, In. per Min, 
Spiral Nicked Milling-Cutter 31^ 
Diameter, 14 and about ^"Pitch. 
Cutting Machine Steel. 



'/,bW 



5.0 

o 
o 

g3.0 
u 
' 2.0 



LO 



% 4 6 8 10 12 14 16 18 
Feed, In. per Min. 
Regular Face Milling-Cutter 9H 
Diameter, 22 Teeth. Cutting 
Machine Steel, 















V 


/ 














/ 


/ 




y 








/ 


/ 




y 


y 








/ 


/ 




y 


^ 


^ 






/ 


y:. 


y 








-^ 






^ 


— 








— 


~~~' 






! 


1 




8 1 


1 


2 1 


4 1 


6 18 



6.0 

5.0 

I4.O 
o 

o3.0 

u 

W2.0 

LO 















Ay 


/ 
















/ 


/ 




^ 


iA 










/ 




y 


y 


^ 


^« 






/ 


/ 


y 


y 




^ 








/ 


^ 


/ 






-- 


-^ 








^ 


^^ 


— 






r 






>i. 



5.0 




































^ 


g4.U 
iO 

a 

K2.0 
LO 












^ 


^ 


^ 










^ 






- 




^ 




^ 


1 


■^ 






— 




= 











~~~' 











Feed, In, per Min. 
High Power Face Milling-Cutter 
8"Diameter, 12 Teeth, Cutting 
Machine Steel 



2 4 6 8 10 12 14 16 18 
Feed, In, per Min, 
Helical Milling-Cutter ZVz Dia- 
meter, 414' Lead, 2l4' Pitch, 
Cutting Machine Steel. 



2 4 



8 10 12 14 16 18 



Feed, In, per Min, 
Helical Milling-Cutter 3 '4" Dia, 
meter, 414" Lead, '=^/i'o Pitch, 
Cutting Cast Iron 



Radiating lines give depth of cut. The observations are reduced to i in. width of cut. Composition of steel: carbon, . 20; manganese, .50 

Fig. 7. — Power required to drive milling machines in machinery steel. 



Table 9. — Results of Cutting Tests with an 8-in. 
High-power Face Mill in Machinery Steel, 
Wide, Block C 



, 12-BLADED, 

Cut s Ins. 



Depth of 
cut, ins. 


Spindle 

speed, 

r.p.m. 


Feed in 

ins. per 

min. 


H.p. 

delivered to 

machine 


Cu. ins. 
of metal re- 
moved per 
min. 


Cu. ins. 
of metal re- 
moved per 
h.p. min. 




22 


12.96 


6.85 


8.10 


1. 183 


i ■ 


21J 


12.89 


7.14 


8.OS6 


1. 128 


21I 


12.89 


6.8s 


8.056 


1. 177 




22 


13 00 


7.42 


8.I2S 


1.096 




2U 


7.96 


6.8s 


7.462 


1.09 


A \ 


2li 


7.96 


6.8s 


7.462 


1 .09 




2I| 


7.92 


6.8s 


7.42s 


1.084 




21^ 


7.87 


6.8s 


7.378 


1 .078 




21 


6.16 


7.42 


7.70 


1.039 


i ] 


2li 


6.22 


7.14 


7.77s 


1 . 09 


I 


2li 


6.26 


7.14 


7.82s 


1.096 




21I 


4.89 


7.14 


7.64 


1.07 


■h \ 


2U 


4-8s 


7.14 


7.578 


1.062 




21J 


4.88 


7.71 


7.62s 


.99 


, 


22 


4S>2 


7.42 


7.687 


1.037 




21} 


3.80 


7.14 


7.13 


I. 000 


1 


22 


3.86 


6.8s 


7.237 


I. 056 


21} 


3.78 


6.8s 


7.08 


I 034 


\ 


21} 


3 -80 


6.8s 


7.13 


1.042 



Table 10. — Results of Cutting Tests with 4i-iN., io-toothed. 
Spiral-nicked Cutter on Machinery Steel, Cut 5 Ins. Wide 



Depth of 
cut, ins. 


Spindle 
speed, 
r.p.m. 


Feed in 

ins. per 

min. 


H.p. 

delivered to 

machine 


Cu. ins. 
of metal re- 
moved per 
min. 


Cu. ins. 
of metal re- 
moved per 
h.p. min. 


f 


40} 


6.22 


12. 3S 


11.66 


•944 


i \ 


41 


6.25 


12 .60 


11.72 


.93 


, 


AT- 


6.27 


12 .60 


II. 76 


.933 




40 


7.8 


IS. 12 


14-625 


.968 


■' \ 


40 


7.8 


14-55 


14-625 


1. 001 




40 


7.8 


14.82 


14-625 


.987 


40 


6.09 


15-93 


IS- 225 


.955 


} ■■ 


39} 


6.06 


15-93 


15.15 


-95 


39} 


6. OS 


15-93 


15.125 


.948 




40 


6. II 


16 20 


15-275 


-943 




40 


4-71 


17-07 


14.72 


-863 


f \ 


39i 


4.68 


17-90 


14.62 


-816 


39} 


4.68 


18-75 


14.62 


.779 




39 


4.66 


17-07 


14-56 


.853 




39 


3.62 


18. 20 


13-575 


.745 


\ 


39 


3.62 


18.20 


13-575 


• 745 


39} 


3.63 


17-63 


13-61 


.772 


. 


39} 


3.63 


17-38 


13-61 


.782 



334 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table ii. — Results oe Cutting Tests with a io-in., i6-bladed, 
High-power Face Mill in Machinery Steel, Cut 5 Ins. Wide 



Table 12. — Physical Properties of Machinery Steel Blocks 
Used in the Tests 



Depth of 
cut, ins. 


Spindle 
speed, 
r.p.m. 


Feed in 

ins. per 

min. 


H.p. 

delivered to 
machine 


Cu. ins. 
of metal re- 
moved per 
min. 


Cu. ins. 
of metal re- 
moved per 

h.p. min. 






isi 


4.61 


12.64 


11.525 


-912 






iSi 


4.61 


12.94 


II.S2S 


-89 


1 , 




I5i 


4-6s 


12.94 


11.625 


.898 






IS 


S.81 


17. OS 


I4.S2S 


.852 






IS 


S.81 


16,24 


14525 


-893 






IS 


S.81 


16.50 


14525 


.881 






ISI 


7.56 


13-20 


I4-I7S 


1.074 






isi 


7.63 


13.20 


14-30 


1.083 


i ' 




I5h 


7.56 


12 .64 


I4-I7S 


1 . 121 






ISI 


7.66 


12,64 


14-37 


I -137 






isi 


7.63 


12.94 


14-30 


1 . 106 






16 


7.70 


II .00 


12 .02 


1 ,092 


ft - 




16 


7.68 


II , 00 


12.00 


1 ,09 






isi 


7.63 


11.27 


11.92 


1.058 



Block 


Limit of elasticity, 
lbs. per sq. in. 


Elongation per cent, 
of length 


Reduction per cent, 
of section 


A 
B 
C 


36,400 
36,200 
37,400 


36 
36. 5 

36. 5 


66 

59.6 

60 



.79 h.p. per cu. in. per min. 
1.53 h.p. per cu. in. per min. 
2.33 h.p. per cu. in. per min. 
1.69 h.p. per cu. in. per min. 
2,.iZ li-P- per cu. in. per min. 



Slabbing cast-iron, minimum, 
Slabbing cast-iron, maximum, 
Channelling mild steel, average of 6, 
Channelling mild steel, minimum, 
Channelling mild steel, maximum. 

These tests were of extraordinary severity, a feed in steel as high 
as 9I ins. per min. under a cut of f in. depth, consuming 162 h.p. ' 
and removing 82 cu. ins. per min., having been included. In cast- 
iron the extreme duty was the removal of 105 cu. ins. per min. under 
a feed of 7 ins. per min. and a depth of cut of i in., the power consump- 



Table 13. — Power Consumption by Milling Machines Operating on 


German Irons and Steels 




Material 


Cutter 


Speed of 

cutter, 

r.p.m. 


Feed in 

ins. per 

min. 


Depth of 
cut, in. 


Cu. ins. of 

metal removed per 

min. 


Cu. ins. of 

metal removed per 

h.p. min- 


Bar No. i — Cast steel, 56,000-70,000 
lbs. tensile strength; width of mate- 
rial 6 ins. 


Cutter: high-power face mill, 10 ins. 
diameter, high-speed steel. 


14 
14 
14 

14 


4.82 
7.83 
10,3 
6.52 




3-52 
5. 87 
7.74 
9,80 


-837 
1,005 
1, 105 
8,08 




Cutter: spiral mill with nicked teeth, 
-4 ins. diameter, high-speed steel. 


32 
32 
32 


12. 96 

6,18 

. 3.92 


i 

1 

f 


9-73 
9.27 
8.80 


,640 
,608 
.620 


Bar No. 2 — Spiegeleisen, 17,028 lbs. 
tensile strength; width of material, 
4 ins. 


Cutter: spiral mill with nicked teeth, 
4 ins. diameter, carbon steel. 


20 
20 
20 


12.5 

12. 5 

7-S 




6,25 
12.5 
11.25 


1,03 
1 , 02 
.990 


Bar No. 3 — Gray iron, German medium 
hard machine cast-iron, generally 
used; width of material, 4 ins. 


Cutter: high-speed steel, high-power 
face mill, 16 blades, 10 ins. diameter- 


21 
21 
21 
21 


16. 1 
15-7 
IS-S 

14.6 


i 

ft 
i 
1 


8,05 
II . 76 
15-5 

21-9 


2, 12 
2,18 
2,31 
1,63 




Cutter: high-speed steel, spiral mill, 
4? ins. diameter. 


49 
49 
49 
49 


20,82 
20, 70 
16, 24 
12,96 


1 

a 
5 


10, 41 
20.7 
24,4 
25,82 


1,50 
2,67 
2.17 
1.65 


Bar No. 3A — Gray iron, German me- 
dium hard machine cast-iron, gen- 
erally used; width of material, 4 ins. 


Cutter: high-speed steel, face mill, 18 
blades, 8 ins. diameter. 


25 
25 
25 
25 


16.4 

15 -5 

14.6 

9.7 


1 
J 
i 


8. 20 

IS, 5 
21.9 
19-4 


I -412 
I -84 
1-370 
1 ,64 




Cutter: high-speed steel, spiral mill 
with nicked teeth, 4I ins. diameter. 


40 
40 
40 
40 


20.82 

20.70 

20, 52 

7,62 


i 

1 
f 


10.41 
20,70 
30,8 
19.05 


1,14 
1,44 
1-97 
I- 29 


Bar No. 4 — Siemens- Martin steel, 85,- 
000-99.000 lbs. tensile strength; 
width of material, 3 J ins. 


Cutter: high-speed steel, face mill, 8 
ins. diameter. 


17 
17 
17 
17 


0,9 

2.28 
3-04 
4-05 


\ 

1 
1 

\ 


0,786 
1,99 
2-56 
3-52 


-378 
-533 
-540 
.567 




Cutter: high-speed steel, cutter with 
nicked teeth, 4j ins. diameter. 


16 
16 
16 
16 


20, 82 
12,96 
10, I 
5-31 


ft 

1 

i 


9, 10 
8,49 
8,85 
6,97 


-587 
-575 
-548 
.605 


Bar No. 5 — Chrome-nickel steel, 122,- 
000-141,000 lbs. tensile strength; 
width of material, 3f ins. 


Cutter: high-speed steel, high-power 
face mill, 10 ins. diameter. 


14 
14 
14 
14 


4.82 
16, 4 

6,53 
13-3 


J 
ft 


2,l8 

7-42 

4-43 

9, 02 


• 592 
.747 
.720 
-628 




Cutter: high-speed steel, spiral mill 
with nicked teeth, 4 ins. diameter. 


13 
13 
13 
13 


20,86 

20-86 

8-44 

S-3I 


ft 
i 

ft 
J 


4,78 

9-45 
S-70 
4-64 


-467 
.680 
.692 
.493 



PERFORMANCE AND POWER REQUIREMENTS OF TOOLS 



335 



tion being 47 h.p. The h.p. per cu. in. decreased as the amount 
of metal removed increased, as follows, for steel: 

Max. duty, 162 h.p., 82. cu. ins. per min. removed, consumption 
1.99 h.p. per cu. in. 

Min. duty, 29 h.p., 7.82 cu. ins. per min. removed, consumption 
3.71 h.p. per cu. in. 
and as follows for cast-iron: 

Max. duty, 89. h.p., 105 cu. ins. removed per min., consumption 
.85 h.p. per cu. in. 

Min. duty, 40 h.p., 26 cu. ins. removed per min., consumption 1.53 
h.p. per cu. in. 

Sizes of Motors for Machine Tools 

The sizes of motors for machine tools, according to the practice of the 



Westinghouse Electric and Mfg. Co., are given in Table 14 in which 
the horse-power recommended is based on average practice; it may be 
decreased for very light work and must often be increased for heavy 
work. The class of motor is indicated by the symbols A, B, C, ex- 
plained below. The meaning of these symbols is sometimes modi- 
fied by notes under the tables. 

(A) Adjustable-speed shunt- wound direct-current motors wherever 
a number of speeds are essential. 

(B) Constant-speed shunt-wound direct-current motors where 
the speeds are obtainable by a gear-box or cone-pulley arrangement 
or where only one speed is required. 

(C) Squirrel-cage induction motor where direct current is not avail- 
able. A gear-box or cone-pulley arrangement must be used to obtain 
different speeds. 



Table 14.- 
BoLT AND Nut Machinery 

BOLT CUTTERS 



-Sizes of Motors for Machine Tools 

Bulldozers or Forming or Bending Machines 
Motor— .gi or C^ 





Motor— ^, B, or C 




With, ins. Head movement 
29 14 


ins. 


H.p. 








s 




Size, ins. 


H.p. 


34 16 




7h 


Single 


I, li, li 


I to 2 


39 16 




10 




if, 2 


2 to 3 


45 18 




IS 




2i,3i 


3 to s 


63 ■ 20 




20 




4, 6 


S to 7i 


1 Compound-wound motor. 






Double 


I, li 


2 to 3 


2 Wound secondary or squirrel-cage motor with approximately lo per cent. 




2, 2| 

I, li 2 


3 to s 
3 to 7I 


slip. 






Triple 










bolt pointers 




Buffing Lathes 








Motor— 5 or C 




Motor— 5 or C 








li. 2i 


I to 2 










■*■ Z J " z 




Wheels 








NUT TAPPERS 




No. Diam., ins. 




H.p. 




Motor ^,£, or C 




2 6 




itoj 


Four-spindle 


I, 2 


3 


2 10 




I to 2 


Six-spindle 


2 


3 


2 12 




2 to 3 

3 to 5 


Ten-spindle 


2 


S 


2 14 






NUT facing 




For brass tubing and other special work use about doubl 


e the above horse- 




Motor— 5 or C 




power. 








I, 2 


2 to 3 


















Bolt Heading, Upsetting and Forging 
Motor— yl,i£,2or C3 
Size, ins. H.p. 

f to ij S to 7i 

15 to 2 10 to 15 

2j to 3 20 to 25 

4 to 6 30 to 40 

1 Speed variation is sometimes desired when different sizes of bolts are headed 
on the same machine. 

2 Compound-wound direct-current motor. 

3 Wound secondary or squirrel-cage motor with approximately lo per cent, 
slip. 



Drilling and Boring Machines 
Motor— ^, B, or C 



B 


oring 


and Turning 


Mills 






Motor — A, B, or 


C 










Horse- 


power 


Size 




Average 






Heavy 


37 to 42 ins. 




S to 7J 






75 to 10 


SO ins. 




7l 






75 to 10 


60 to 84 ins. 




72 to 10 






10 to IS 


7 to 9 ft. 




10 to IS 








10 to 12 ft. 




10 to IS 






30 to 40 


14 to 16 ft. 




15 to 20 








16 to 25 ft. 




20 to 25 









Sensitive drills up to § in. 
Upright drills, 12 to 20 ins. 
Upright drills, 24 to 28 ins. 
Upright drills, 30 to 32 ins. 
Upright drills, 36 to 40 ins. 
Upright drills, 50 to 60 ins. 



Radial drills, 3-ft. arm. 

Radial drills, 4-ft. arm. 

Radial drills, s to 6 and 7-ft. arm. 

Radial drills, 8 to 9 and lo-ft. arm. 



H.p. 
ito i 
I 
2 
3 
S 
S to 7i 
Horse-power 
Heavy Average 

3 1 to 2 

S • to 7i 2 to 3 

S to 7i 3 to 5 

7J to ro s to 7i 



Cylinder Boring Machines 
Motor — A, B, or C 

Diam. of Max. boring H.p. 

spindle, ins. diam., ins. 

4 10 72 

6 30 10 

8 40 IS 



336 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 14. — Sizes of Motors 

Pipe Threading and Cutting-off Machines 
Motor — A, B, or C 

Size pipe, ins. H.p. 

J to 2 2 

i to 3 3 

I to 4 3 

I? to 6 3 to s 

i to 8 3 to 5 

3 to 10 S 

4 to 12 S 
8 to 18 7i 

24 10 

Planers ' 
Motor— ^,15, lor C 
Distance 

Width, ins. under rail, ins. H.p. 

22 22 3 

24 24 3 to 5 

27 27 3 to s 

30 30 S to 7i 

36 36 10 to 15 

42 42 IS to 20 

48 48 IS to 20 

54 54 20 to 25 

60 60 I 20 to 25 

72 72 25 to 30 

84 84 30 

100 100 40 
Normal length of bed in feet is about J the width in inches. 
See also second table below. 
' Compound-wound motor. 

ROTARY PLANERS 

Motor— ^, B,otC 

Dia. of cutter, ins. H.p. 

24 5 

30 74 

36 to 42 10 

48 to 54 IS 

60 20 

72 25 

84 30 

96 to 100 40 

Hydrostatic Wheel Presses 

Motor— B or C 

Size, tons H.p. 

100 S 

200 75 

300 7i 

400 10 

600 15 

Rolls — Bending and Straightening 



Width, ft. 

4 

6 

6 

6 

8 
10 
10 
24 



Motor— fii or C^ 
Thickness, ins. 



H.p. 

5 
5 

72 

IS 

25 

35 
50 
50 



FOR Machine Tools — {Continued) 

Punching and Shearing Machines 
Presses for notching sheet iron, motor — A, B, or C, 5 to 3 h.p. 

punches 
Motor— £> or C^ 
Dia., ins. Thickness, ins. 



H.p. 






I 




2 


to 


3 


2 


to 


3 


3 


to 

S 
5 

7l 


S 


7^ 


to 


10 


10 


to 


IS 


10 


to 


IS 


IS 


to 


25 



1 Compound-wound motor. 

2 Wound secondary or squirrel-cage motor with approximately lo per cent, 
slip on the larger sizes. 

SHEARS 

Motor— 51 or C 

Horse-power 
Width, ins. Cut |- n. iron Cut j in. iron 

30 to 42 3 5 

50 to 60 4 7J 

72 to 96 5 10 

Bolt shears 7I h.p. 

Double-angle shears 10 h.p. 

' Compound-wound motor. 

2 Wound secondary or squirrel-cage induction motor with lO per cent. slip. 

LEVER SHEARS 

Motor— £1 or C^ 

Size, ins. H.p. 

1 Xi 5 
I2XI2 72 

2 X2 10 
6 Xi 

24X2J IS 

I X7 

2|X2f 20 

1^X8 

35X35 30 

4I round 

1 Compound-wound motor. 

2 Wound secondary or squirrel-cage motor with approximately 10 per cent. slip. 

PLATE SHEARS 

Motor— 51 or C 

Length of 



Size of metal ^ . ^^..e,-.. ^. -.. 

, . Cut per mm. , . H.p. 

cut, ms. stroke, ins. 

|X 24 35 3 10 

1 X 24 20 3 IS 

2 X 14 IS 4i 30 
I X 42 20 4 20 
15X 42 15 45 60 
iiX 54 18 6 75 
15X 72 20 si 10 
iJXioo 10 to 12 7I 75 

' Compound-wound motor. 

2 Wound secondary or squirrel-cage motor with approximately 10 per cent. slip. 
TIN PLATE SQUARING SHEARS 

Motor- £ or C 
Cuts per min. H.p. 

30 7 5 



> Standard bending roll motor. 

' Wound secondary induction motor. 



Size of plates, ins. 
54X54 
Ts packs 
72X 72 
A packs 



30 



PERFORMANCE AND POWER REQUIREMENTS OF TOOLS 



337 







Table 


14.- 


—Sizes of I 




SHAPERS 










Motor- 


-A,B, 


or 


C 




Stroke, ins. 








H.p. 


single head 


12 to i6 










2 


i8 










2 to 3 


20 to 24 










3 tos 


30 










5 to 7I 


TRAVERSE HEAD SHAPER 




20 










7^ 


24 










10 



Sizes of Motors for Machine Tools — {Continued) 



Saws — Cold and Cut Off 
Motor — A, B,ox C 
Size of saw, ins. 
20 
26 



Miscellaneous Grinders 

Motor— 5 or C H.p. 

Wet tool grinder 2 to 3 

Flexible swinging, grinding, and polishing machine 3 

Angle cock grinder 3 

Piston rod grinder 3 

Twist-drill grinder 2 

Automatic tool grinder 3 to 5 

Grinding Machines (Grinding Shafts, Etc.) 
Motor— .4, B,otC 
Length work, Horse-power 



32 
36 
42 



H.p. 

3 

S 

7i 

10 to 15 

20 

25 



Dia. wheel 
ins. 
10 
10 
10 
10 
14 



Slotting and Key Seating 
Motor — A, B, OT C 
Stroke, ins. H.p. 

6 3 

8 3 to 5 
10 5 

12 5 

14 S to 7i 
16 77 

18 7I to 10 

20 10 to 15 

24 10 to 15 

30 10 to IS 

Horizontal Boeing, Drilling and Milling Machines 

Motor— ^, B,OT C 

Size of spindle, ins. H.p. for single spindle 

32 to 4J S to 7^ 

4i to si 7i to 10 

55 to 65 10 to 15 

For machines with double spindles use motors of double the horse-power 



ins. 

SO 

72 

96 
120 

72 
120 
144 
168 



Average 


work 


Heavy work 


S 




7i 


S 




7i 


S 




7i 


S 




7l 


10 




15 


10 




IS 


10 




IS 


10 




IS 





Gear Cutters 






Motor— ^, B, or C 




Size, ins. 




H.p. 


36X 9 




2 to 3 


48X10 




3 to 5 


30X12 




S to yi 


60X12 




5 to 7I 


72X14 




72 to 10 


64X20 




10 to IS 



Hammers 
Motor— Bi or C- 
Size, lbs. 
IS to 7s 
100 to 200 

Bliss drop hammers require approximately 



H.p. 

itos 

S to 7I 

h.p for every loo-lb. weight 



of hammer head 

' Compound- wound motor. 

2 Wound secondary squirrel-cage motor with approximately 10 per cent, 
slip. 





Multiple Spindle Drill 








Motor— ^, B,oT C 






Size of drill. 


, ins. 


Up to 




H.p. 


^toi 




6 to 10 spindle 




3 


Atof 




10 




S 


Atoi 




10 




7i 


i tof 




10 




10 


1 to I 




10 




10 to IS 


2 




4 




7h 


2 




6 




10 


2 




8 




IS 




Emery Wheels, Grinders, 


Etc. 








Motor— B or C 








Wheels 








No. 




Size, ins. 




H.p. 


2 




6 




i to I 


2 




10 




2 


2 




12 




3 


2 




18 . 




S to 7^ 


2 




24 




75 to 10 


2 




26 




7? to 10 



Swing, ins. 

12 

14 

16 

18 
20 to 22 
24 to 27 

30 
32 to 36 
38 to 42 
48 to 54 
60 to 84 



Size, ins. 

48 

SI to 60 

79 to 84 

90 

100 

' Standard machine-tool traverse motor. 



Lathes 






Motor — A, B, or C 






ENGINE lathes 






Horse-power 


Average 

1 




Heavy 


f to I 




2 to 3 


I to 2 




2 to 3 


2 to 3 




3 to 5 


3 




72 to 10 


5 




7I to 10 


5 to 7I 




7I to 10 


72 to 10 




10 to 15 


10 to IS 




IS to 20 


15 to 20 




20 to 25 


20 to 25 




2S to 30 


WHEEL lathes 






H.p. 


Tail stock motor^ h.p. 


15 to 20 




S 


IS to 20 




S 


25 to 30 




s 


30 to 40 




S to 7* 


40 to so 




S to 7f 



22 



338 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 14. — Sizes of Motors foe Machine Tools — {Continued) 

AXLE LATHES 

H.p. 

Single 5, 75> 10 

Double 10,15, 20 



Milling Machines 
^ Motor — A, B, or C 
vertical slabbing machines 
Width of work, ins. H.p. 

24 73 

32 to 36 10 

42 IS 



horizontal slab millers 



Width between hous- 
ings, ins. 
24 
30 
36 
60 
72 





Horse-power 




Average 




Heavy 


72 to 10 




10 to 15 


75 to 10 




10 to IS 


10 to 15 




20 to 2S 


25 




50 to 60 


25 




75 



VERTICAL MILLING MACHINES 

Height under work, ins. H.p. 

12 5 

14 7i 

18 10 

20 IS 

24 20 

PLAIN MILLING MACHINES 



Table feed, 


Cross feed, 


Vertical feed 


ins. 


ins. 


ins. 


34 


10 


20 


42 


12 


20 


SO 


12 


21 



H.p. 

7i 
10 

IS 



UNIVERSAL MILLING MACHINES 

Machine No. H.p. 

1 I to 2 
i§ I to 2 

2 3 to 5 

3 S to 7I 

4 75 to 10 

5 10 to 15 



Sizes of motors for various metal and wood-working machines, accord- 
ing to L. R. PoMEROY {General Electric Review, 1907), are given in 
Table 16. 

The sizes of motors suitable for various commercial presses forms the 
subject of an article by F. C. Fladd {Amer. Mach., May 28, 1903). 
The list is made up of presses and motors in satisfactory use, includes 
direct-belt and chain-driven presses (belt-driven preferred by Mr. 
Fladd) and is given in Table 1 7. 

Methods of making more accurate determinations of motor capacity 
for machine tools, especially under heavy and fluctuating loads, have 
been explained by A. G. Popcke, Industrial Elect. Engr., Westing- 
house Electric and Mfg. Co. {Amer. Mach., Sept. 26, 1912) as follows: 

The preliminary data required are: On direct current: Horse- 
power, speed and voltage. On alternating current: Horse-power, 
speed, voltage, frequency and phase. 

The voltage, frequency and phase are determined by the electric 
circuit in the shop. The horse-power depends upon the work done. 
Whether to use a high- or a low-speed motor depends on the gears 
that must be used. A comparatively low speed is usually necessary. 

The power required to drive a machine tool depends upon the 
following: The tools used. Amount of metal removed in a given 
time. The metal cut. 

The tools used are of three general types: Lathe type tools, 
used on lathes, boring mills, planers and shapers. Drills. Milling 
cutters. 

The amount of metal is usually expressed in cubic inches re- 
moved per minute. The rate of removing metal for a given job 
depends upon the tools used, the strength of the machine tool, the 
strength of the work, the accuracy desired, and the nature of the 
metal cut. 

In roughing work, the question of horse-power must be carefully 
considered so that the most economical motor is applied. The 
tendency is to select a motor to suit the maximum capacity of the 
machine tool. This is very rarely reached for any length of time. 
Modern motors will operate successfully, at loads 100-125 P^r cent, 
above the rated loads. The following information must be obtained 
to determine the horse-power of the motor to be used for any tool- 
cutting metal: 

Type of tool. Average cut taken: Depth (all tools considered) 
in inches. Feed per revolution in inches. Cutting speed in feet 
per minute. Duration. Maximum cut taken: Depth in inches. 



Feed per revolution. Cutting speed in feet. Duration of peak 
maximum load. Number of peaks per hour. 

With this information it is possible to estimate the average and 
maximum horse-power required from the cubic inches of metal re- 
moved per minute for the cuts taken for average and maximum. 

In all cases the area of the cut is taken as equal to the depth 
multiplied by the feed. 

For revolving work or tools, the cutting speed may be quickly 
determined from Fig. 8, the use of which is explained below it. The 
chart may obviously be used in the reverse direction with equal 
facility. The cubic inches of metal removed per minute may be 
quickly determined from Fig. 9, the use of which is explained below 
it. To determine the horse-power required the cubic inches of 
metal removed per minute are multiplied by a constant. 

For estimating purposes the constants of Table 15. based on aver- 
age shop conditions are useful in figuring horse-power at the cutting 
tool when round-nose tools are used : 

Table 15. — Relation between Power and Volume of Metal 

Removed 

Cast-iron 3 to .5 h.p. per cu. in. per min. 

Wrought iron 1 , , 

, , , 7 , ^ . 6 h.p. per cu. m. per mm. 

Machinery steel J 

Steel, 50 carbon and harder. . i .00 to 1.25 h.p. per cu. in. per min. 

Brass and similar alloys 2 to . 25 h.p. per cu. in. per min. 

For twist drills the consumption of power per cu. in. of metal 
removed is approximately double the above. 

The constants will vary with the angle and sharpness of the tool, 
but are close enough to determine the size of motors in the majority 
of cases. A few tests in any shop using motor-driven tools can 
easily be made to determine the constants for their particular tools. 
The tendency is to use tools in accordance with the conditions ar- 
rived at from tests made by F. W. Taylor, and others, which see. 
The above constants hold under these conditions. 

The friction load of the machine tool should be added to obtain 
the total horse-power. However, where the horse-power to remove 
metal is large, the friction is a small percentage and can be neglected. 

In selecting the size of a motor it must be remembered that the 
load is intermittent in the majority of cases. The heating of a 
motor in supplying power for work of an intermittent nature is de- 

{Conlinued on next page, second column) 



PERFORMANCE AND POWER REQUIREMENTS OF TOOLS 



339 



Table i6. — Sizes of Motors for Various Metal and Wood- 
working MACmNES 
Bolt and Nut Machinery, Helve Hammers 

Motor h p. 
required 
to drive 

One and one-half inch single-head bolt cutter i ^ 

Pratt & Whitney No. 4 turret bolt cutter 2 

Two-spindle stay bolt cutter 2 

One and one-half inch Acme double-head bolt cutter 25 

One and one-half to two and one-half Acme nut facer 25 

Six-spindle nut tapper 3 

One and one-half inch triple-head bolt cutter 3 

Three-fourths to two and a half inch double-head bolt cutter ... 3 

Two-inch triple-head bolt cutter S 

Four-spindle stay bolt cutter S 

Bradley hammer 75 

Grinders 

Air-cock grinder i 

Link grinder 3 

Sellers universal grinder for tools 5 

Norton iSXgfi-in. piston-rod grinder 5 

Saws for Wood 

Band saw, 36-in. wheel 3 

Band saw, 42-in. wheel 5 

Swing cut-off saw 5 

Band saw, 48-in. wheel 75 

Greenlee No. i J self-feed rip saw 10 

Greenlee vertical automatic cut-off saw 15 

Forty to forty-six inch saws 15 

Automatic band resaw 20 

Greenlee No. 6 automatic cut-off saw 20 

Greenlee No. 3 rip saw 20 

Woods No. 4 rip saw 20 

Extra heavy automatic rip saw 25 

Wood-working Tools 

Fay-Egan single-spindle vertical boring machine 3 

Fay-Egan three-spindle vertical boring machine 4 

Fay-Egan No. 6 vertical mortiser and borer 6 

Fay-Egan No. 7 tenoner or gainer yf 

Fay-Egan universal wood worker 7^ 

Fay-Egan four-spindle vertical borer 7 1 

Fay-Egan five-spindle vertical borer 10 

Fourteen-inch inside molder 12 

Fay-Egan universal tenoner and gainer 12 

Fay-Egan vertical tenoner 12 

Greenlee automatic vertical tenoner 15 

Fay-Egan No. 3 gainer, also Greenlee 15I 

Greenlee extra-range five-spindle borer and mortiser 15 

Greenlee vertical mortiser 15 

Fay-Egan automatic gainer, also combination gainer and mor- 
tiser 20 

Fay-Egan No. 8 vertical saw and gainer 20 J 

Vertical hollow chisel mortiser and borer 20 

Fay-Egan i4j-in. double-cylinder surfacer 2o| 

Heavy outside molder 20 

Six-roll direct-connected planer and matcher 25 

Double-cylinder fast flooring machine 30 

Double-cylinder planer and matcher 30 

Fay-Egan No. 8 automatic tenoner 30J 

Woods No. 2 7 matcher 35 

Four-side timber planer, heavy 60 



Table 17. — Horse-power of Motors for Various Presses 

Name and number H.p. of 

of press. motor required. 

Bliss, No. 21 2 

Bliss, No. 20 I 

Bliss, No. 19 I 

Bliss, No. 18 I 

Bliss, No. 52 2 

Bliss, No. 30A 3 

Bliss-Stiles punch, fly-wheel pattern, No. o 3 

Bliss-Stiles punch, fly-wheel pattern. No. i i 

Bliss-Stiles punch, fly-wheel pattern. No. 2 i 

Bliss-Stiles punch, fly-wheel pattern. No. 3 2 

Bliss-Stiles punch, fly-wheel pattern. No. 4 3 

Bliss-Stiles punch, fly-wheel pattern, No. 5 3 

Bliss-Stiles punch, geared pattern. No. 5 doing heavy work .... 5 

Bliss, geared for heavy work. No. 36 5 

Bliss, double crank geared, No. 35 5 

Bliss, double crank geared. No. 5 75 

Bliss circular shear. No. 105 | 

Bliss double action, fitted with double roll feeds. No. 68N 2 

Stiles special five-slide gang press, geared. No. 102 3 

Bliss automatic feed armature disk press. No. 16A i 

Bliss toggle. No. 35 10 

Bliss-Stiles 200-lb. automatic board lift drop, improved 3 

Bliss-Stiles 400-lb. automatic board lift drop, improved 4 

Bliss-Stiles 800-lb. automatic board lift drop, improved 75 

Hilles & Jones combined punch and shear. No. 2 5 

Ferracute, No. SG86 5 

Ferracute, direct geared. No. Cs 3 

Ferracute, 18-inch throat. No. P21 2 

Bliss, geared with side cut-off attachment. No. 745 3 

Bliss, deep throat for light punching. No. 475 i 

Hilles & Jones, combined punch and shear. No. 3, 36-in. throat, 

capable of punching i f-in. hole through i-in. stock 10 

Hilles & Jones, combined punch and shear. No. 3, 12-in. gap, 

punching i-in. hole through i-in. stock 7I 

Hilles & Jones, single punch, 36-in. throat, punching i|-in. holes 

through I i-in. stock 10 

Hilles & Jones, horizontal punch. No. 3, 20-in. throat punching 

i-in. holes through f-in. stock 7§ 

Hilles & Jones angle shear. No. 3 cutting i-in. stock 10 

Williams & White, bulldozer. No. 6, 20-in. stock, 38-in. die face. 7 J 

Bliss geared power shear, 36-in. cut, cutting sheet steel J-in. thick. 5 
Heavy alligator geared cut-off shear, capable of shearing 5-in. by 

I-in. bar iron 5 

Small armature disk notching press i 

Large coining presses at U. S. Mint, striking up silver dollars, 

r.p.m., 80; pressure, 160 tons 75 

Smaller coining presses, striking up quarter dollars, r.p.m., 100; 

pressure, 60 tons ■ 3 

Planchet presses at Mint, double roll feed 3 

Double cut-off shear at Mint 3 

termined by the square root of the mean square of the power required 
to perform the various operations taking place in a complete cycle. 
This value will be termed the root mean square value, or r.m.s. 
value. The method of figuring the r.m.s. value of any intermittent 

load is best explained by working out an example. Suppose the 
power fluctuates as follows during a cycle of operations: 

Power required Duration 

10 h.p. 10 seconds 

S h.p. 30 seconds 

3.5 h.p. 25 seconds 

I h.p. 20 seconds 

o h.p. 20 seconds 



340 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



■The r.m.s. value is figured as follows : 

1. Multiply the square of each value of power by its duration. 

2. Add the products thus obtained. 

3. Divide the sum by the time to complete the cycle (the sum of 
the times of the various components). 

4. Take the square root of the quotient. 

The result will be the r.m.s. value. The time can be expressed in 
seconds or minutes and the power in horse-powers, kilowatts or — 



current induction motor or commutating-pole, direct-current motor 
will carry this successfully under the given conditions. A S-h.p. 
motor would, therefore, be used in this case. 

The limits above rated loads to be considered when selecting 
alternating-current motors to carry widely fluctuating intermittent 
loads are: 

1. Pull at the starting torque. 

2. Speed regulation. 
For direct-current motors: 

1. Commutation. 

2. Speed regulation. 

3. Stability. 
The pull at the starting torque of an induction motor is from 2.5 to 
3.5 times the full-load torque. 

The speed regulation of induction motors is the percentage drop 
in speed between no load and full load based on the maximum speed; 
it is usually called the slip. The slip at full load is usually about 
S to 7 per cent. At other loads it is approximately proportional to 
the load, therefore, at twice full load the drop in speed will be 
approximately 10 to 15 per cent. 

fore commutating-pole motors were built, commutation 

limited the overloads on direct-current motors. At present 

to-date commutating-pole motor will carry 100 

to 125 per cent, overload, that is, 2 to 2.25 times 

the full load without sparking. 




Spindle Speed.R.P.M. 

Trace vertically from the r. p. m. to the intersection with the horizonal through the diameter where read cutting speed, 
piece of work of 3 in. diameter at 60 r. p m has 47 ft. per min. surface speed. 

Fig. 8. — Relation of spindle speed, cutting speed and diameter of work. 



Thus a 



the voltage being constant- 
throughout a given problem 
is determined as follows: 
(i) 



-in amperes, the same units being used 
In the above example the r.m.s. value 



10^X10 = 


1000 




5^X30= 


750 




3.S'X2S = 


306. 


25 


1^X20 = 


20 




X20 = 









2076 


25 


10+2O+2S + 


20 4-2 


= 1 


2076.25 

- —10.8 

105 







(2) 



(4) \/ig.8 = 4.4S h.p. =r.m.s. value 

Thus 4.4s is the r.m.s. value of this cycle and the heating of the 
motor will be the same as if it were run at a constant load of 4.45 
h.p. The maximum load on a S-h.p. motor would be twice the full 
load, or 100 per cent, overload for 10 sec. An up-to-date alternating- 



It is customary to express the speed regulation of direct-current 
motors in terms of the full-load speed, because the full-load speed 
is the rated speed of the motor. At full load the speed regulation 
is 10 to 15 per cent., depending on the rating of the motor. At over- 
loads the effect on noncommutating-pole motors is a decrease in 
speed proportional to the load; but on commutating-pole motors 
the speed in many cases tends to increase between full load and 100 
per cent, overload. 

This type of motor will, therefore, have approximately the same 
speed at twice full load as it has at full load. If the effect of the 
interpoles is too strong the tendency is to make a commutating-pole 
motor oscillate in speed. This speed oscillation will cause a similar 
variation on armature current of gradually increasing intensity, 
until something gives way; a fuse will blow, a circuit breaker open 
or the motor will be injured by "bucking over," that is, flashing 
across brushes, or burning out the armature. 

There is a relation existing between speed regulation and stability. 
A commutating-pole motor can be designed to be stable at over- 
loads. This will increase the drop in speed. Better speed regula- 



PERFORMANCE AND POWER REQUIREMENTS OF TOOLS 



341 



tion makes stability less certain. Reliable designers of this type of 
motor strike a happy medium between these two, and the commer- 
cial result is that in most cases these motors can be safely operated 
on intermittent loads where the maximum load is twice the rated 
load. 

A large reduction in speed giving a stable motor, is an advantage 
in machine-tool work. It often occurs when long, continuous cuts 
are taken, that on one part of a casting the depth of cut is greater 
than on another due to irregularities in casting. When cutting 




a graphic recording ammeter will be used for this purpose. The 
record was taken on a lathe driven with a direct-current, adjustable- 
speed motor. The cycle of operations when turning the shaft. 
Fig. II, is as follows: 

Calculated r.m.s. 
(Amp.) 2 X time 

1089 X 180 = 196,020 

900X140 = 126,000 

oXri2= o 

900X171 = 153,900 

81X260= 21,060 

0X190= o 

496,980 



= 470 
^053 
V4 70 =21.7 amperes 

which is the r.m.s. value of current. At 220 volts, the voltage of 
the circuit, the r.m.s. h.p. input to the motor is 
21, 7X 220 





Amp. 


Time 


Cut ab 


33 


180 sec. 


Cut be 


30 


140 sec. 


Idle 





112 sec. 


Cut de 


30 


171 sec. 


Cut ec 


9 


260 sec. 


Idle 





190 sec. 






I0S3 


A similar cycle 


is then 


repeated. 
496,980 



looc X 746 



= 6.4 h.p. input 



30 



40 50 60 70 80 90 100 ICO 
Cutting Speed, Ft. per Mln, 



Trace vertically from the cutting speed to the intersection with the hori- 
zontal through the area of cut (produced by feed and depth of cut) where 
read cu. ins. of metal removed per min. Thus for 1^ in. feed and 5 in. 
depth of cut (.015 sq. in, area of cut) and 60 ft, per min, cutting speed, 
ii-i-cu, in, per min, are removed. 
Fig. 9 — Relation beween area of cut, cutting speed and volume of metal removed. 



through the heavy part the speed should be reduced, thus protecting 
the cutting tools and machine tool as well as the work. For this 
reason adjustable-speed motors with a speed reduction as high as 
25 per cent., can be used to advantage. 

Let us apply these principles in determining the horse-power of a 
motor in actual machine-tool service. A record. Fig. 10, taken with 



,_,, , . amperesX volts , 

(The h.p. mput to motor= — — z — for any direct-current 

1000X746 -^ 

motor.) 

The efficiency of the motor being 86 per cent., the r.m.s. h.p. 
output is 5.5 h.p. 

The maximum load occurs when the cut ab is taken, which requires 
33X220 

1000X746 = 9-" ^'■P- 
input or, at 85 per cent, efficiency, an output of 8.3 h.p. 

If a 5-h.p. motor is used, 8.3 h.p. will be 66 per cent, 
overload, and considering what has been said above, a 
modern S-h.p. motor will pull this satisfactorily. The 
r.m.s. value, or that upon which heating depends, is 10 
per cent, above the rated load. All S-h.p. motors made 
by reliable manufacturers will carry this load without 
overheating. 

The cycle just discussed represents maximum work 
done in this lathe, the average load being less severe. 
Hence a 5-h.p. motor is the proper motor to drive it. 
Usually a test cannot be conveniently made. In these 
cases the power cycles can be figured from the rate of 
removing metal and the time required for each cut. The 
method of estimating the power has already been ex- 
plained. The time of a cut is estimated as follows when 
knowing the length of the cut, the feed per revolution 
and the spindle speed while taking the cut: 

The product obtained by multiplying the feed per 
revolution and the r.p.m. of the spindle will give the 
advance of the cutting tool per minute. Dividing this 
into the length of the cut will give the time to complete 
the cut in minutes. With this information and the 
time to make adjustments when the motor is shut down 
or running idle, the r.m.s. value can be figured with the 
rules already given. 

The practice in the past, still largely used, is to select 
the motor with reference to the size of a machine, as the 
swing of a lathe. The strength of a lathe and, therefore, 
the horse-power it can transniit naturally increases with 
the size of the lathe; but the quantities which determine 
the horse-power are those just discussed and their application will 
avoid misapplications. In many cases, heavier cuts are taken on 18- 
than on 24-in. lathes, the smaller-swing lathe, therefore, requiring 
the larger motor. 

On machine tools where light cuts are taken, it is not necessary 
to figure the horse-power for cutting because 2-h.p. is required to 



400 &00 



342 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



start the tool and run it idle and will do the light cutting success- 
fully. 

The rule for figuring horse-power just described is applicable in 
determining the power to cut metal wherever the round-nose type 
of tool is used, as in vertical boring mills, shapers, slotters, and 
planers. On planers the peak load for reversing must be considered 
in determining the size of motor. 

On planers the general tendency is to use motors that are too 
large. This tendency originated when non-interpole, direct-current 
motors, only, were available, and a peak load caused considerable 



4U 
















1 
















b ,^ 




lb _ ■ 






1 


* 


"*1 














. *' 


*- 






_i!jni 










<'~a"< 






\. rf ' 








30 














T 


"b 




* 








m 




















































































p, 20 




















o 








H 




















u 








<! 


















d 






rt 




















^e 






(3 








(V^, 
















[VH. 






^ 


C 






' 1 




1 




u 
















f^lV 






[ 


>l 


, 














ll 1 1 


ri 


(— J 


L_f Li 1 l_ 




IL 







u 


1_ 





10:00 



9:30 

Time 

Fig. 10. 



9:00 



):00 



w/^/////m 






i-li i ~i 



Fig. II. 

Figs, io and ii. — A piece of lathe work. and a graphic record of 
current readings. 

sparking when the planer was reversed. A large motor was, there- 
fore, necessary on account of the reversal. When using alternating- 
current induction motors, or direct-current, commutating-pole mo- 
tors this precaution need not be taken. 

Table i8 shows the results of tests made on various sizes of planers 
with a graphic meter. Note the difference between the motors 
usually specified and those recommended. The recomrhended 
motors are alternating current and are operating their planers 
successfully. 

Fly-wheels can be used to advantage on the countershaft from 
which the forward and reverse belts are driven. The fly-wheel wUl 
reduce the peak load on the motor occurring when the planer is 



reversed. In this way the horse-power of the driving motor can 
often be reduced. 

It is evident that making an investigation as outlined results in 
the selection of the most economical size of motor for the work 
done, and in a smaller motor than that usually specified, because 
the tendency is to select motors to suit the maximum capacity of 
machine tools and no advantage is taken of the fact that motors will 
stand heavy overloads for short intervals. 

The selection of electric motors for machine driving includes other 
questions than that of horse-power. These have been explained 
by A. G. PopCKE, Indust. Elect. Engr. Westinghouse Electric & 
Mfg. Co., {Amer. Mack., Oct. 3, 1912) as follows: 

The speed of the shaft on the machine where power is applied is 
the principal factor which determines the speed of the motor to be 
connected. On forging machines using large fly-wheels these speeds 
are as low as 50 to 60 r.p.m. ; on machine tools, such as lathes, drills, 
millers, etc., they average between 200 and 300 r.p.m. Speeds as 
high as 1000 to 2000 r.p.m. occur on grinders and wood-working 
machines. 

Modern practice is to standardize the speeds of motors This 
practice has been brought about by the extensive use of alternating 
current. Since 60 cycles are used in the majority of alternating- 
current systems, the standard speeds of direct-current motors are 
approximately the same as the speeds of 60-cycle, alternating-current 
motors. 

The speeds obtainable with the 60-cycle motors mostly used are 
1700 to 1800; iioo to 1200; 850 to 900; 650 to 720, and 550 to 600 
r.p.m. The higher speed given in each case is the synchronous speed 
at which the motor runs when not loaded. The speed decreases 
from 5 to 7 per cent, as the motor is loaded. 

On 25-cycle circuits the speeds of motors most frequently used 
are 700 to 750; 550 to 600, and 350 to 375 r.p.m. The speeds of 
direct-current motors are given in the second column of Table 19. 
A reference thereto wiU show the relation to the speeds of the alter- 
nating-current motors just given. 

Whe7i a belt drive is to be used the quantities to be considered are : 
Speed reduction; pulley sizes; belt speeds; motor speed; distance 
between pulley centers; arc of contact; size of belt; use of idle pulleys; 
mounting of the motor. 

Obtaining the required speed reduction involves the size of the 
motor pulley, machine pulley and belt speed. The sizes of the pul- 
leys used on motors have been standardized according to ratings. 



Table 18. — Motors for Planers 



Manufacturer 



Size 



Motor 

used for 

test 



Kw. 

cut 
stroke 



Kw. 

return 
stroke 



Kw. 

reversal 

to cut 



Kw. 
reversal 
to return 



Remarks 



Motor 
recom- 
mended 
based on 
test, h.p. 



Motor 

usually 

specified, 

h.p. 



Gray 

Gray 

Gray 

Gray 

Gray 

Bement-Miles 

Chandler 

Detrick & Harvey 

Bement-Miles 

Bement-Miles 

Pratt & Whitney. 
Gray 



ins. ft. 
56X15 
56X15 
56X15 
54X16 

54X16 
48X12 
24X10 

42X12 
open side 
48X12 
37X 8 
36X 8 
36X 8 



3 

5 

S 
30 
30 

5 

5 

7l 



30 
5 
5 
5 



1-3 
1.8 

2.S 

4 

4 

1.8 

2 



5 
1.8 

i-S 
1.8 



3-5 
2.8 



6 

7 

2-3 
7 
4-S 

2-S 

10 

3 
2 
2 



4.0 

3-5 
6 
8 
10 

3-5 
8 

4-3 

5 

14 
4 

2.5 
3 



5-3 

5-3 

6 
10.5 
12 

5-5 

9 

5-5 



19 

6 



Average work, 
S tons on tab. 
Short stroke 
Aver, stroke 
Short stroke 
Aver, stroke 
Aver, work 
Motor geared 
balance wheel 
Aver, work 

No. bal. wheel 
Aver, work 
Aver, work 
Aver, work 



7^ 
5 



7i 
5 
3 
3 



15 



IS 

15 

7i 

IS 

IS 
10 

5 
S 



PERFORMANCE AND POWER REQUIREMENTS OF TOOLS 



343 



Table 19. — Standard Motor Ratings Standard and Minimum 
Pulleys and Belt Speed with Standard Pulley 



I 


2 


3 1 4 


5 


6 


7 


8 






Standard 


Minimum 


Belt speed 


Leather 
belt 


H.p. 


R.p.m. 


pulley 


pul 


ey 


standard pul- 
ley, ft. per min. 




Dia. 


Face 


Dia. 


Face 


I 


1700 


3J 


2} 


3 


ij 


1560 


Single 


2 


1700 


3J 


3 


3 


3 


1560 


Single 




1200 


4 


3 


3 


3 


1250 


Single 




850 


4 


4 


3} 


4 


890 


Single 


3 


1800 


4 


3 


3 


3 


1890 


Single 




1150 


4 


3 


3i 


4 


1200 


Single 




850 


5 


4h 


4 


4i 


mo 


Single 


5 


1800 


4 


4 


3i 


4 


1890 


Single 




1200 


5 


4i 


4 


4i 


IS70 


Single 




850 


6 


5 


4i 


5 


1340 


Single 


7i 


1700 


5 


4i 


4 


4i 


2220 


Single 




IISO 


6 


5 


4i 


5 


1800 


Single 




975 


7 


6 


5 


6 


1790 


Single 




850 


7 


6 


5 


6 


1560 


Single 




650 


8 


7 


6 


7 


1360 


Single 


10 


1700 


6 


5 


4i 


5 


2670 


Single 




1300 


7 


6 


5 


6 


2380 


Single 




1150 


7 


6 


5 


6 


2100 


Single 




850 


8 


7 


6 


7 


1780 


Single 




730 


8 


7 


6 


7i 


1530 


Single 




600 


9 


8 


6i 


9 


1410 


Single 


IS 


1700 


7 


6 


5 


6 


3100 


Single 




1250 


8 


7 


6 


7 • 


2620 


Single 




1 100 


8 


7 


6 


7i 


2300 


Single 




825 


9 


8 


6i 


9 


1940 


Single 




675 


10 


9 


7 


8 


1770 


Single 




600 


II 


10 


7j 


9i 


1730 


Single 


20 


1700 


8 


7 


6 


7 


3560 


Single 




IIOO 


9 


8 


6^ 


9 


2600 


Single 




900 


10 


9 


7 


8 


2360 


Single 




750 


II 


10 


7i 


9i 


2160 


Single 




650 


II 


10 


8 


9i 


1870 


Single 


25 


1400 


9 


8 


6i 


9 


3300 


Single 




IIOO 


10 


9 


7 


8 


2880 


Single 




950 


II 


10 


7J 


9i 


2730 


Single 




825 


II 


10 


8 


9h 


2370 


Single 




600 


12 


12 


9 


10^ 


1880 


Double 


30 


1700 


9 


8 


6} 


9 


4000 


Single 




II50 


II 


10 


7^ 


9i 


3300 


Single 




975 


II 


10 


8 


9i 


2800 


Single 




725 


12 


12 


9 


lOj 


2280 


Double 




600 


13 


12 


10 


II 


2040 


Double 


35 


1700 


10 


9 


7 


8 


4450 


Single 




1150 


II 


10 


8 


9i 


3330 


Single 




850 


12 


12 


9 


loj 


2670 


Double 




675 


13 


12 


10 


II 


2300 


Double 


40 


1700 


II 


10 


7i 


9i 


4900 


Double 




950 


12 


12 


9 


lOj 


3000 


Single 




775 


13 


12 


10 


II 


2640 


Double 




600 


14 


12 


12 


13 


2200 


Double 


56 


1700 


II 


10 


8 


9l 


4900 


Double 




975 


13 


12 


10 


II 


3320 


Double 




750 


14 


12 


12 


13 


2750 


Double 




565 


16 


13 


12^ 


IS 


2360 


Double 



i.e., horse-power, and speed of motor. These are given in Table 19, 
column 3. This fixes standard practice for belt speeds (see Table 19, 
column 7). 

As the size of motor pulley is reduced on any motor, the strains 
on the motor bearings and shaft are increased. A minimum pulley 
is, therefore, specified by motor manufacturers for each motor rating 
(see Table 19, column 5). The maximum size of the pulley on a 
motor is required only where speeds higher than the motor speed 
are required. This is, in nearly all cases, limited by the belt speed, 
which should not exceed 5000 ft. per min. 

In some cases, with small motors especially, the size and location 
of the motor are such that the diameter of the motor limits the largest 
pulley. 

The success of a belted motor application depends largely upon 
the arc of contact. The distance between centers of motor pulley 



and machine pulley, as well as the speed reduction, determines the 
arc of contact on the smallest puUey, usually the motor pulley. 

Motors can be furnished with idler pulley attachments, and these 
are applied to advantage where it is necessary to overcome a small 
arc of contact. When necessary to obtain extremely low speeds 
back-geared motors should be used. 

A good standard for the back-geared type of motor is one having 
a speed reduction of 6 to i between its armature and countershaft 
speeds. Usually, if the required reduction in speed exceeds 6 to i, 
a back-geared motor should be used. For instance, if the reduction 
is 12 to I between the motor speed and the machine speed, a back- 
geared motor with a 6 to i speed reduction should be used, and the 
further reduction 2 to i obtained by means of a pulley on the counter- 
shaft of the back-geared motor. 

The pulleys furnished with motors make provision for the proper 
width of the belt. Table 19 shows whether a single or double belt 
should be used. The width of the belt should be i in. narrower 
than the pulley face on pulleys up to 12-in. face; above that it should 
be 2 ins. narrower than the pulley face. 

The cost of a motor of given horse-power increases as the rated 
speed decreases. For instance, the cost of a lo-h.p. motor at 1200 
r.p.m. is approximately the same as a 5-h.p. motor at 600 r.p.m. 
The cost increases in the same proportion as the square root of the 
torque figured at i-ft. radius. From a cost point of view, therefore, 
as high a speed motor as possible should be used, but the diameter of 
minimum pulley specified should not be gone below. 

When the machine pulley is fixed, as when belting to a fly-wheel, 
the motor pulley must suit the requirements of the machine. Care 
must be taken not to go below the minimum motor pulley and the 
arc of contact must also be carefully considered, for in these cases 
the reduction is usually large. 

When the machine pulley can be chosen to suit, the standard 
motor pulley, Tables 19 and 20, will assist in selecting the proper 
speed of motor and size of pulleys. Table 20 gives the machine 
speed at the left column and the motor speeds at the top of the table. 
The figures in the body of the table are the speed reductions for any 
combination of machine and motor speed indicated. 

The letter B indicates that the motor is to be belted directly, and 
the symbol Bbg indicates that a back-geared motor be belted. The 
figure after Bbg indicates the reduction between the motor counter- 
shaft and the driven machine, if a back-geared motor with a 6 to i 
reduction is used. 

The heavy-faced type indicate the method recommended in the 
majority of cases for the combination where it occurs. Thus, for 
machine speeds between 600 and 1500, use i8oo-r.p.m. motors; 
between 350 and 600, use 1200-r.p.m. motors; between 250 and 350 
use 900-r.p.m. motors; between 150 and 250, use 720-r.p.m. motors. 
For the smaller power requirements, and between 150 to 200 for 
the large power requirements, use 600-r.p.m. motors. Below 100 
and 150 r.p.m. it is best to use back-geared motors. 

It is poor practice to use back-geared motors whose initial speed 
is 1 700-1800 r.p.m. in the majority of cases. In applications re- 
quiring from 10 to 20 h.p., 1200-r.p.m. back-geared motors should 
be used; above this 900-r.p.m. or 720-r.p.m. back-geared motors 
should be used. 

Before deciding upon any belt drive the arc of contact should be 
carefully checked. In machine-tool work, for applications where 
belts are used, the distance between centers is usually between 3 and 
S ft. Motor pulleys range from 3 to 12 ins., and the arc of contact 
is usually considered when the ratio of reduction is between 3 and 6. 

Table 21 shows the arc of contact, knowing the size of the motor 
pulley, ratio of reduction and the distance between pulley centers. 
Table 22 shows the effect of the arc of contact on the transmitting 
power of the belt. The decrease with decreased arc of contact is 
expressed by a percentage which the power transmitted at a given 
arc of contact is of the power transmitted at 180 deg. 



344 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



To transmit the required power the pulley and belt width must 
be increased or an idler pulley must be used to increase the arc of 
contact. An example will best illustrate the application of Tables 
14, 15 and 16. The speed of the machine is 185 r.p.m.; the h.p. re- 
quired is 7i; the distance between centers is 5 ft. What motor 
speed and what pulleys should be used for the belt drive? 

Refer to Table 20. This shows that for 150 to 200 r.p.m. a 720- 
r.p.m. motor should be used. 

Refer to Table 19. A 75-h.p. 650-r.p.m. motor has an 8X7-in. 
standard pulley and a 6 X7-in. minimum pulley. 

The speed reduction with this motor is 
650 



185 



= 3-5 



Refer to Table 21. The arc of contact for a ratio of reduction 
of 3.5 (average of 3 to 4), the distance between centers of s-ft. and 
8-in. motor pulley is 160 deg. (average of 164 and 157), and with 
a 6-in. motor pulley is 165 deg. (average of 162 and 168). Either 
will give successful service. The machine pulley would be with an 
8-in motor pulley 3.5X8 = 28 ins. and with a 6-in. motor pulley, 
3.5X6 = 21 ins. 

The face in either case will be 7 ins. and a single 6-in. leather belt 



should be used. The combination of 8-in. motor puUey and 28-in. 
machine pulley is preferred because the motor pulley is standard. 

The above example covers a case where the machine pulley can 
be seclected at will. In cases where a motor is to be belted to a 
fly-wheel or to a pulley on a machine which cannot be easily changed, 
the procedure is as explained in the following example: The size 
of the machine pulley (fly-wheel) is 72 ins.; the speed of the puUey is 
100 r.p.m.; the h.p. required is 15, and the distance between centers 
is 6 ft. What motor speed and motor pulley should be used? 

Consider a reduction of 6 : i belted directly. The motor speed 
must be 600. The size of the motor pulley 

machine pulley 72 
ratio of reduction 6 

Table 19 shows that a 12-in. pulley can be used with a is-h.p., 
600-r.p.m. motor. It is i-in. above the standard pulley diameter. 

Table 21 shows that for a 12-in. motor pulley, a ratio of reduction 
of 6, and 6 ft. distance between centers, the arc of contact is outside 
the limits of the table and the arc of contact very small (below 120 
deg.) 

A successful drive can be obtained by using a i2Xio-in pulley 
on the motor and employing an idler pulley. It is not customary 







Table 20. — Relation of 


Machine and 


Motor Speeds. Recom 


MENDATIONS FOR BeLT DeIVE 










Approximate motor speed 




1800 


1200 


900 


720 


600 




1500 


1.2 B 






















1000 


1.8 B 


i-S 


B 


















800 


2.2 B 


i-S 


B 




1. 12 B 














600 


3 B 


2 


B 




i.S B 


1.2 B 




I. 2 


B 




_c 


500 


3-6 B 


2.4 


B 




1.8 B 


1,44 B 




I. 2 


B 




u 


400 


4.5 B 


3 


B 




2.25 B 


1.8 B 




I-S 


B 




6 


350 


S.13 B 


3-4 


B 




2.52 B 


2.06 B 




1-7 


B 




a 


300 


6 B 


4 


B 




3 B 


2.4 B 




2 


B 




'u 


250 


7.2 


4.8 


B 




3.6 B 


2.9 B 




2.4 


B 




TD 


200 


9 


6 


B 




4-5 B 


3.6 B 




3 


B 




-a 


ISO 


12 


8 


Bbg 


1-33 


6 B 


4.8 B 




4 


B 




0) 


100 


18 


12 


Bbg 


2 


9 Bbg 1 . 5 


7.2 Bbg 


I. 2 


6 


B 




90 


20 


13-4 


Bbg 


2.23 


10 Bbg 1.67 


8 Bbg 


1-33 


6.7 


Bbg 


I. II 




80 


22. s 


IS 


Bbg 


2-5 


II. 3 Bbg 1.88 


9 Bbg 


i-S 


7-S 


Bbg 


1-25 




70 


25-3 


17. 1 


Bbg 


2.85 


12.9 Bbg 2.15 


10.2 Bbg 


1-7 


8.6 


Bbg 


1-43 




60 


30 


20 


Bbg 


3-33 


15 Bbg 2.5 


12 Bbg 


2 


10 


Bbg 


1.67 




SO 


36 


24 


Bbg 


4 


18 Bbg 3 


14.4 Bbg 


2.4 


12 


Bbg 


2 



B= Motor belted direct. Bbg. = Back-geared motor belted. Bbg = 1.33, etc., the number indicates reduction from countershaft speed. 
The heavy-faced type indicates the motor speed recommended in most cases. 



Table 21. — Relation Between Motor Pulley, Distance Be- 
tween Centers of Pulleys, Ratio of Reduction and Arc 
OF Belt Contact. 



Table 22. — Relation of Arc of Contact to Power 

Transmitted 



Ratio of 


Distance 

between 

centers, ft. 


Diameter of motor pulley, ins. 


• reduction 


3 1 4 


5 


6. 


7 


8 


9 


10 


II 


12 




3 


170 


166 


163 


160 


157 


153 


150 


147 


145 


141 


3 


4 


173 


170 


167 


165 


163 


161 


158 


156 


155 


151 




S 


17s 


172 


170 


168 


167 


164 


162 


161 


160 


157 




3 


i6s 


160 


iSS 


150 


145 


142 


156 


132 


126 


122 


4 


4 


168 


i6s 


162 


158 


154 


152 


148 


144 


140 


137 




S 


172 


168 


166 


162 


159 


157 


155 


151 


148 


146 




3 


160 


153 


148 


142 


134 


128 


122 








S 


4 


165 


161 


157 


IS2 


146 


142 


138 










5 


168 


164 


162 


157 


153 


ISO 


146 










3 


I S3 


147 


139 


131 


122 












6 


4 


161 


iS6 


ISO 


144 


138 














5 


164 


161 


156 


152 


146 















Arc of contact 


Per cent, of poWer 
transmitted 






180 


100 






170 


94 






160 


89 






150 


83 






140 


78 






130 


72 






120 


67 





for motor manufacturers to supply idler attachments on such large 
motors. 

When a geared drive is to be used the points to be considered are the 
following: Speed reduction; pitch of the gears, number of teeth 
on the gears (pinion and gear); face of the gear; pitch-line speed; dis- 
tance between centers; use of idler gears and mounting of the motor. 
{Continued on next page, first column) 



PERFORMANCE AND POWER REQUIREMENTS OF TOOLS 



345 



Table 23. — Standard Motor Ratings and Data for Geared 
Connections 



Table 24. — Adjustable Speed Motor Ratings and Data for 
Geared Connections 



Number of teeth 



Stan- 
dard 
pin- 
ion 



7i 



IS 



1700 

1200 

1700 

1200 
850 
1800 
IISO 

8so 
1800 
1200 

850 
1700 

IISO 

97S 

850 

6so 

1700 

1300 

IISO 

8so 

730 

600 

1700 

1250 

IISO 

82s 
67s 
600 
1700 
1100 
900 
7S0 
6so 
1400 

IIOO 

950 
825 
600 
1700 

IISO 



6 
6 
6 
6 
5 
S 
5 
6 
6 
5 
S 
5 
5 
S 
5 
S 
5 

4i 

4l 

5 

S 

4§ 

4i 

4 

S 

4i 

4J 
4 
4 
5 

4l 



22 
18 
22 
18 
21 
18 
21 
19 
19 
20 
21 
22 
19 
20 
21 
21 
19 
20 
21 
21 
22 
22 
20 
21 



Min. 
raw- 
hide 
pin- 
ion 



Min. 
pin- 
ion 
steel 



IS 
IS 
IS 
20 
21 
20 
21 
18 
21 
18 
19 
18 
19 
18 
18 
18 
19 
19 
18 
18 
iS 

19 
18 
18 
18 
19 
18 
19 
18 
19 



40 



SO 



97S 


4 


72s 


4 


600 


3i 


1700 


4§ 


IISO 


4 


8so 


4 


67s 


3i 


1700 


4i 


9S0 


4 


775 


3i 


600 


3 


1700 


4 


975 


3i 


750 


3 


S6S 


3 



22 


18 


22 


19 


21 


18 


21 


19 


22 


18 


22 


19 


21 


18 


22 


19 


21 


19 


22 


19 


21 


18 


22 


19 


20 




22 


18 


21 


18 


22 


19 


20 




22 


19 


22 


19 


20 




18 




21 


18 


20 




18 




20 





13 
13 
13 
19 
19 
19 
19 
18 
19 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
19 
18 
18 
18 
19 

iS 

19 
18 
19 
18 
19 
18 
19 
18 
19 
18 

l8 
19 
19 
18 
18 
iS 
18 
18 
18 
18 
19 
18 
18 
IS 
18 
18 
IS 
18 



Face 



Raw- 
hide 
and 
cloth 



Std. 
pitch 
line 
speed 



Min. 
dia. 



Max. No. of 

teeth for 
p.l. speed of 



1000 ft. 
per min. 



2000 ft. 
per min. 



I5 

li 

li 
li 
If 
li 

li 

2f 

2| 

2i 

2i 

3 

3 

3 

2i 

2i 

3 

3 

3i 

3i 

3 

3 

3§ 

3^ 

4 

4 

3 

3i 

4 

4 

4i 

3i 

4 

4 

4i 

4i 

3i 
4 

4i 

4i 

4l 

4 

4i 

4i 

4i 

4 

4i 

4§ 

4J 

4i 

4l 

4i 

44 



li 



2i 
2i 

2i 
2i 
3f 

2i 

at 
3i 
3i 

3l 

3i 
3i 

3l 
3l 

3i 

01 
3l 

Ai 

4i 

3l 

3i 

Ai 

Ai 

5 

5 

3i 

Ai 

5 

5 

5i 

Ai 

5 

S 

Si 

Si 

4i 

5 

Si 

Si 



5 

Si 



Si 



940 
665 
940 
870 

6IS 

1300 
830 
670 

1300 
940 
990 

1400 

loso 
970 
850 

68s 

1420 

1250 

1150 

890 

80s 

665 

1700 

1300 

1210 

910 

870 

770 

1780 

1220 

1150 

960 

890 

1550 

1400 

1220 

1130 

860 

1880 

1470 

1330 

lOSO 

970 
2180 
1580 

1220 
1080 
2180 
1370 
1250 

940 
2340 
1580 
II70 

990 



1.63 
1.63 
1.63 
2.38 
2.38 
2.38 
2.38 

3 

2.38 
3 
3 
3 

3 

3.6 

3.6 

3.6 

3 

3 

3.6 

3.6 

3.6 

3-8 

3.6 

3.6 

3.6 

3.8 

4 

4. 22 

3.6 

3.8 

4 

4.22 

4-5 

3.8 

4 

4. 22 

4.5 

4-5 

3.8 

4. 22 

4. 

4. 

S- 

4 

4 

4 

S 



25 
18 
25 
36 



26 
27 



19 
27 



20 

19 



22 
26 
31 



S 
S 
53 

S 

5 

53 
4.22 
45 
5-53 
S 
4- 
S- 
5 
6 



23 
25 
29 



19 
23 
23 



18 
2S 



18 
18 



19 



IS 
20 



36 
50 
36 

SO 

72 

34 

S2 

54 

34 

38 

54 

26 

40 

38 

44 

58 

27 

35 

33 

44 

52 

62 

22 

30 

35 

46 

50 

S8 

22 

35 

38 

46 

46 

27 

31 

36 

36 

50 

22 

30 

31 

42 

40 

20 

27 

36 

36 

20 

32 

32 

38 

i3 
25 
30 

40 



Here also, each motor rating has a minimum pinion for the same 
reason, limiting stresses. The pitch, number of teeth and face for 
motor pinions have been standardized for back-gear motors and the 
best practice when gearing a motor directly to machines is to use 
these motor pinions as far as possible. 

The pitch-line speed is limited by noise when steel pinions are used. 
A speed of 1000 ft. per min. should not be exceeded if quiet operation 
is desired. Between 1000 and 2000 r.p.m., rawhide or cloth pinions 
should be used; 2000 ft. per min. should not be exceeded if it can 
possibly be avoided. 

Table 23 gives the standard motor ratings and additional data 
for geared drives all of which are useful when working out geared 
motor applications. 





R.p.m. 1 


Smallest 
pulley 


Gear data | 


Pitch-line 


thnot 

2000 

m. at 

2d 






01 


6 
a 

Q 


Pace 1 


J3 


S 
a 

d 
S 


speed, at 
min. diam. 




(5 







p. 




1 

ClJ 

Pi 


S 1 6 " 


w 


d-g 




Max. t 
to exc 
ft. per 
max. s 


I 


740 


2200 


3 


3 


8 


ij 


2i 


19 


2.38 


460 


1380 


27 




600 


1800 


3 


3 


8 


If 


2i 


19 


2.38 


375 


82s 


46 




450 


1800 


3 


4 


8 


If 


2i 


19 


2.38 


280 


II20 


3 


2 


IIOO 


2200 


3 


3 


8 


If 


2i 


19 


2.38 


690 


1380 


2 




740 


2200 


3 


4 


8 


li 


2i 


19 


2.38 


460 


1380 


27 




450 


1800 


4 


Ai 


6 


2i 


3f 


18 


3.0 


35S 


1420 


2S 


3 


1000 


2000 


3 


A 


8 


li 


2i 


19 


2.38 


62s 


I2S0 


30 




660 


2000 


4 


Ai 


6 


2i 


3f 


18 


3.0 


520 


1560 


23 




450 


1800 


Ai 


5 


6 


2i 


3f 


18 


3-0 


355 


1422 


2S 




375 


1500 


s 


6 


6 


2i 


3f 


18 


3.0 


294 


II76 


30 


S 


1000 


2000 


4 


Ai 


6 


2i 


3l 


18 


3.0 


790 


1580 


23 




750 


ISOO 


Ai 


S 


6 


2i 


3l 


18 


3.0 


590 


I18O 


30 




600 


1800 


s 


6 


6 


2i 


3l 


18 


30 


470 


I4IO 


25 




4S0 


1800 


6 


7 


S 


3 


3i 


18 


3.6 


425 


1700 


21 




375 


1500 


6 


7i 


S 


3i 


4i 


18 


3.6 


355 


1420 


25 


n 


900 


1800 


S 


6 


6 


2i 


3i 


18 


3.0 


70s 


141O 


25 




800 


1600 


5 


6 


5 


3 


3i 


18 


3.6 


755 


iSio 


24 




600 


1800 


6 


7 


5 


3 


3i 


18 


3.6 


S70 


1710 


21 




500 


ISOO 


6 


7i 


5 


3J 


4i 


18 


3.6 


475 


142s 


25 




450 


1800 


61 


9 


S 


3i 


4i 


19 


3.8 


450 


1800 


21 




350 


1400 


6J 


9 


5 


3j 


4i 


19 


3.8 


350 


1400 


27 


10 


850 


1700 


6 


7 


5 


3 


3i 


18 


3.6 


800 


1600 


22 




750 


1500 


6 


7 


S 


3 


3i 


18 


3.6 


710 


1420 


25 




600 


1800 


6 


li 


5 


3i 


4i 


18 


3.6 


570 


1710 


21 




500 


ISOO 


6i 


9 


S 


3l 


4i 


19 


3.8 


500 


ISOO 


2S 




450 


1800 


6i 


9 


5 


3^ 


4i 


19 


3.8 


450 


1800 


21 




375 


ISS>0 


7 


8 


4j 


4 


5 


18 


4.0 


390 


1560 


23 


15 


780 


1560 


6i 


9 


S 


3l 


4i 


19 


3.8 


780 


iS6o 


24 




600 


1200 


7 


8 


Ai 


4 


5 


18 


4.0 


630 


1260 


28 




500 


1500 


7J 


9i 


4J 


4 


5 


19 


4.22 


555 


i66s 


23 




400 


1200 


8 


9j 


4 


4i 


Si 


18 


4-5 


470 


1410 


2S 




375 


ISOO 


9 


lOi 


4 


4i 


Si 


18 


4-5 


440 


1760 


20 


20 


6so 


1300 


75 


9i 


Ai 


4 


5 


19 


4.22 


720 


1440 


26 




550 


IIOO 


8 


9i 


A 


4i 


5i 


18 


4-5 


645 


1290 


28 




500 


ISOO 


9 


loj 


A 


4i 


si 


18 


4-5 


590 


1770 


20 




400 


1200 


10 


II 


3i 


44 




18 


5. S3 


580 


1740 


21 




300 


1200 


12 


13 


3 


Ai 




IS 


SO 


390 


1560 


19 


25 


550 


IIOO 


9 


loi 


4 


4i 


Si 


18 


4-5 


64s 


1290 


28 




400 


1200 


12 


13 


3 


44 




IS 


5-0 


52s 


IS7S 


19 




300 


1200 


I2I 


15 


3 


4i 




18 


6.0 


470 


1880 


19 


30 


550 


IIOO 


10 


II 


3i 


Ai 




18 


5 53 


800 


1600 


23 




350 


lOSO 


I2J 


IS 


3 


Ai 




18 


6.0 


SSO 


1650 


22 




250 


1000 


14 


18 


3 


Ai 




18 


6.0 


390 


1560 


23 


40 


550 


IIOO 


12 


13 


3 


4l 




15 


5-0 


720 


1440 


21 




350 


1050 


12| 


IS 


3 


Ai 




18 


6.0 


550 


1650 


22 




2S0 


1000 


16 


21 


3 


Ai 




19 


6.33 


415 


1660 


23 


SO 


500 


1000 


I2i 


15 


3 


Ai 




18 


6.0 


790 


1580 


23 




32s 


975 


16 


21 


3 


4i 




19 


6.33 


540 


1620 


23 



If reductions greater than 7 to i are required, it is usually necessary 
to obtain the reduction by the use of two sets of gears. Back- 
geared motors can be used to furnish one set of gears in these cases. 
Thus if a reduction of 10 to i is desired, a back-geared motor with a 
standard 6 to i reduction, with a further reduction from the counter- 
shaft of the motor to the machine of -r to i or 1.66 to i will fulfill 



the requirements. 

An example will explain how to proceed in a motor application 
where gears are to be used: The speed of the driven shaft of the 
machine is 210 r.p.m.; the h.p. is 10; the motor is mounted on the 
machine and the limiting distance between centers is 12 ins. What 
are the sizes of gear and pinion to be used? The machine is a punch 
and shear. 

In this case a pitch-line speed of approximately 1000 ft. per min. 

will be employed. Table 23 shows that a lo-h.p. motor at 850 r.p.m. 

is the highest speed motor that can be used for this pitch-line speed. 

The ratio of reduction is then 

850 

= 4.05 (use 4 to i) 

210 ^ -^ ^ 

(Continued on page 347, fint column) 



346 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 25. — Power Requirements of Machine Tools in Groups 



Kind 



Size 



Observed 
horse- 
power, 

maximum 



Observed 
horse- 
power, 
average 



Remarks 



Kind 



Size 



Observed 
horse- 
power, 

maximum 



Observed 
horse- 
power, 
average 



Boring Machines 

BuUard, single head... . 

BuUard, double head.. . 
Cam Cutters 

Brainard 

Brainard 

Brainard 

Lathe type, single head. 

Lathe type, double head 

CUTTING-OFF MACHINES. 

Hurlbut-Rogers 

Hurlbut-Rogers 

Hurlbut-Rogers 

Drilling Machines. 

Prentice Bros, radial.. . 

Prentice Bros, radial.. . 
t 

Woodward & Rogers \ 



D wight-Slate 

Woodward & Rogers | 

Woodward & Rogers. . 

Woodward & Rogers.. 

Prentice upright 

Prentice upright 

Prentice upright 

Prentice upright 

Blaisdell upright 

Blaisdell upright 

Blaisdell upright 

Blaisdell upright 

Blaisdell upright 

Blaisdell upright 

Blaisdell upright 

Blaisdell upright 

Gear Cutters 

Brainard 

Gould & Eberhardt... , 

Brown & Sharpe 

Grinders 

Brown & Sharpe cutter 
and reamer grinder 

C. H. Besly &Co. gard- 
ner grinder. 

Brown & Sharpe plain. 

Brown & Sharpe surface 

Brown & Sharpe surface 

Brown & Sharpe uni- 
versal. 

Brown & Sharpe uni- 
versal. 

Diamond wet tool 
grinder. 

Leland & Faulconer 
wet grinder. 
Drop Hammers 

Blondell 

Pratt & Whitney 

Pratt & Whitney 

Pratt & Whitney.... 

Pratt & Whitney. . . . 

Pratt & Whitney. . . . 

Billings & Spencer. . 
Power Hammers 

Bradley 

Bradley 

Keyseater 

Baker Bros 

Lathes 

Reed boring lathe. . . 

Reed boring lathe. . . 

Reed engine lathe. . . 

Reed lathe 

Prentice 



36 ins. 
42 ins. 

No. 2 
No. 4 
No. s 



1 if ins. 

2 ins. 

3 ins. 

No. o 

No. I 
Sensitive 

single- 
spindle 
2-spindle 
Sensitive 
3-spindle 
4-spindle 
6-spindle 

16 ins. 

18 ins. 

20 ins. 

22 ins. 

24 ins. 

26 ins. 

28 ins. 

30 ins. 

34 ins. 

36 ins. 

46 ins. 

so ins. 

No. 4J 
No. 3 
No. 3 

No. 3 

No. 4 

No. II 
No. 2 
No. 3 

No. I 

No. 2 



40 lbs, 
250 lbs. 
400 lbs, 
600 lbs, 
800 lbs, 
1000 lbs, 
1500 lbs. 

100 lbs, 
ISO lbs. 

No. 4 

20 ins. 
30 ins. 
12 ins. 
14 ins. 
16 ins. 



.78 
1 . 72 



.48 



.28 
• 34 



.S2 
1.08 

.67 
.32 
.32 
.32 
.SO 

. 12 

. 14 to . 18 

. 20 to . 22 

.72 
I . 12 



.64 



.48 



.32 

.35 

.48 
.71 
.25 

• 33 
.42 
.S9 
.47 
.22 
■ 25 
.30 

• 45 
.53 
.63 
.83 

. 15 to .32 
.20 



.32 

.33 

.80 
.40 

■ SO 
.60 

.76 

■ 97 

.41 to .82 



2 . 00 
2.50 
3.00 
3.50 

4. 00 
5-00 

I. SO 

1.75 

. 28 to .32 

.35 
.41 
.24 
.26 

■34 



Carrying 
one 20-in. 
wheel 
Carrying 
two 24-in. 
wheels 



Lathes — {Contin ued) 

Reed 

Blaisdell 

Blaisdell 

Reed 

Reed 

Blaisdell 

Prentice. 

Draper 

Reed speed lathe. . . . 

Reed speed lathe. . . . 

Putnam squaring - up 
lathe. 

Gisholt turret lathe.. . 

Potter & Johnston 
semi-automatic. 

Jones & Lamson fiat 
turret. 

Wood turning lathe.. . , 

Wood turning lathe.. . , 

Wood turning lathe 
(Putnam gap). 
Milling Machines 

Brainard 

Brainard 

Brainard 

Brainard 

Brainard 

Brainard 

Brainard 

Brainard 

Becker vertical 

Becker vertical 

Becker-Brainard 

Brown & Sharpe 

Brown & Sharpe 

Brown & Sharpe 

Reed 

Pratt & Whitney hand 
Planers 

Whitcomb 

Whitcomb 

Putnam 

Putnam 

Putnam 

Putnam 

Putnam ■. . . . 

Powell 

Pond 

Wood panel planer. . . 

Wood surface 

Polishing Stands 

Brown & Sharpe 

Diamond 

Punch Presses 

Bliss 

Profiling Machines 

Garvin 

Pratt & Whitney 

Band Saw 

Fay & Co 

Circular Saws 

Kimball Bros 

Whitney 

White 

Hack Saw 

Screw Machines 

Brown & Sharpe auto- 
matic. 

Pratt & Whitney auto- 
matic. 

Pratt & Whitney 



16 ins. 
18 ins. 
20 ins. 
22 ins. 
24 ins. 
24 ins. 
28 ins. 
38 ins. 
10 ins. 

14 ins. 

15 ins. 

Size H 
No. I 

2X 24 ins. 

14 ins. 
16 ins. 
36 ins. 



No. I 
No. 3 
No. 4 
No. 4i 
No. 6 
No. 7 
No. 14 
No. IS 
No. 3 



.48 



No. 
No. 
No. 
No. 
No. 
No. 



No. ij 

17 ins. 
22 ins. X S ft. 
22 ins. X S ft. 
24 ins. X 6 ft. 
26 ins. X 5 ft. 
30 ins. X6 ft. 
30 ins. X 8 ft. 
36ins. X loft. 
so ins. X9 ft. 

34 ins- 

24 ins. 

No. 3 

No. s 

No. 3 

No. I 
No. 6 

36-in. wheels 

9-in. b.ade 
9-in. blade 
13-in. blade 

12 to 14 ins. 

No. I 

No. 2 

No. 2 



.37 



1.63 
1.97 



■47 
.64 



2 . 01 

2.34 
1.44 



1.59 
4.91 
5.46 
4.00 
2.94 
7.7s 
3-40 



3-00 

3.77 
3.75 
5^82 



.36 
.39 

.44 
.32 

.25 

.31 
.31 

.58 
.10 
. 12 

■ 23 

.70 
.33 to .63 

1.20 to 1.80 

■ 31 

■36 

1.30 



■ 30 
.26 

. 19 to .29 
.13 to .19 

.26 

• 83 

• 25 

• 25 

.26 

.55 

17 to . 2S 
•15 

• 25 

• 30 
.83 



1 . 00 to .43 
1. 16 to .S3 

.70 

.84 

.81 
1. 31 
I.S6 
1 , 60 
I, 14 
3-70 



I . 00 
1, 19 

1 . 26 

.50 
.40 

.87 

1 .05 
1.04 
I. 21 

.06 
.60 
• 37 
.72 



PERFORMANCE AND POWER REQUIREMENTS OF TOOLS 



347 





Table 26. — Power Requirements of 


Machine Tools in Groups — (Continued) 






Kind 


Size 


Observed 
horse- 
power, 

maximum 


Observed 
horse- 
power, 
average 


Remarks 


Kind 


Size 


Obseived 
horse- 
power, 

maximum 


Observed 
horse- 
power, 

average 


Remarks 


Screw Machines— (Co» 

Pratt & Whitney 

Brown & Sharpe auto- 


tinned) 
No. 3 
No. 3 

No. 3-0 

No. 3-B 
No. 00 

Jin. 

2 ins. 

2f ins. 




.80 
.80 

.90 

.90 
.36 
.40 
.87 
.90 




Screw Machines— (Cok 

Pratt & Whitney hand. 

Pratt & Whitney hand. 

Pratt & Whitney hand. 
Shapers 

Lodge & Davis 

Hendey 


tinned.) 
No. 2 
No. 2j 
No. 3 

14 ins. 
20 ins. 
24 ins. 
28 ins. 

No. 2 




• 43 
•47 

• SO 

• 35 

• 50 
.52 to .70 
.52 to .70 

. 10 










1 . 04 
1 . 04 






Pratt & Whitney auto- 






Pratt & Whitney 

Brown & Sharpe 




















Cleveland . . 




Tapping Machine 
Pratt & Whitney 






Cleveland 


1 . 04 













The distance between centers for any set of gears is determined 

by the formula: 

b 

2P 
where a is the distance between centers in ins., b is the sum of 
the number of teeth in both gears and P is the diametral pitch. In 
this case 

b 



2X5 



or b =120 



The number of teeth in the pinion is, 

b 120 

-=; — : ; ; ; ; =■ = 24 = number of teeth 

Ratio of reduction plus i 5 

in the motor pinion. The number of teeth in the gear is 4X24 = 96. 

Table 18 shows that the pitch-line speed for this motor with 20 

teeth is 890 ft. per min. The pitch-line speed with 24 teeth is 

24 

— X 890 = 1070 ft. per mm. 

If quiet operation is desired a cloth or rawhide pinion should be 
used with a sJ-in. face. Thus the gears are specified as follows: 
Motor pinion (rawhide) P = s, face 3! ins., 24 teeth. Machine 
gear (steel) P — S, face 3 ins., 96 teeth. 

Applications of adjustable speed-motors are dealt with in a similar 
way. The belt speeds and pitch-line speeds must be carefully con- 
sidered on the maximum speeds of these motors. The minimum 



pulleys and pinions are determined by the minimum speeds of the 
motors. 

Table 24 contains the ratings mostly used and pulley and gear 
information for this type of motor. 

Power Reqtiirements of Machine Tools in Groups 

Data relating to the power required to drive machine tools in 
groups are much more difficult to obtain and are correspondingly 

Table 26. — Friction Load Due to Line and Countershafts 

Per cent. 
Department of friction load to 

the total load 

Cam-cutting department 26 

Cutting-off department 43 

Cuttermaking department 27 

Chucking department 26 

Light drilling department 23 

Heavy drilling department 34 

Grinding department 21 

Lathe department 25 

MUling department 25 

Planing department 26 

Patternmaking department 17 

Jig and fixture making department 37 



Table 27. — Power Requirements of Machine Tools in Groups 



Kind of machine 



Kind of work 



Per cent, 
of ma- 
chines 

running 



Floor area 
for ma- 
chine and 
operator 
in sq. ft. 



Total aver- 
age power 
per machine 
in watts' 



Total aver- 
age power 
per ma- 
chine in 
watts^ 



Friction 
and line- 
shaft load 

per ma- 
chine in 

watts^ 



Average 
power used 
in doing 
actual 
work, in 
watts^ 



Total 

power per 

sq. ft. of 

floor area, 

in watts 



No. 2 horizontal Rockford boring mills. 

No. 4 Cincinnati millers 

i6-in. Lodge & Shipley lathes 

Double disk grinders; double buffers; 
two- wheel emery stands. 

24-in. BuUard vertical lathes 

24-in. Gould & Eberhardt gear cutters. . 
Four-head IngersoU milling machines. . 

Baker single and Bausch multi-spindle 

drills. 
Heald grinders, No. 60 internal grinders. ■ 

No. 6 Whitney hand millers 

Landis No. 2 grinders 

Norton 10 by 50-in. grinders 

Jones & Lamson flat turret lathes 

Eight spindle Cincinnati gang drills. .. 

Potter & Johnston automatics 

ij-in. Gridley automatics 

No. 4 Warner & Swasey turret lathe.. . 
24-in. Cincinnati drill presses 

1 Deducting idle machines. 



Boring bearings in aluminum cases. 

Light milling on aluminum 

Turning small forgings 

Grinding and polishing 



Heavy cuts on cast-iron fly-wheels. 

Cutting small cast-iron gears 

Making four cuts on cast-iron cyl- 
inders. 
Drilling and tapping cast-iron 



Cylinder grinding 

Key sea ting small cast-iron gears. . . 

Grinding cam shafts 

Grinding pistons and small forgings 

Machining small forgings 

Drilling and reaming connecting 
rods (8 holes). 

Turning small cast-iron gears 

Machining cast-iron pistons 

Machining small forgings 

Small drilling on forgings 



100 
60 
SS 

100 
100 
100 

40 

85 
60 
80 
70 
8S 
100 

100 
100 
65 
90 



150 

120 

S5 

55 

100 

65 
300 

70 

70 
40 
90 

100 
65 

no 

75 

200 

55 

40 



1620 
995 
900 

1800 

1350 

333 

3550 

1530 

2830 
36s 

1875 

2000 
675 

2840 

690 

1520 

560 

520 



1320 
995 
555 

1000 

1350 

333 
3550 

637 

2430 
220 

1500 

1400 
560 

2840 

690 
1520 
360 

474 



IIOO 

830 
500 
300 

350 

250 

2300 

SSO 

i860 
120 

1000 

1100 
200 

2000 

440 
1250 
310 
345 



300 

500 

87 

830 

1000 

83 

1250 
217 



8.3 
10. 1 
18.2 

13. S 

5.1 

II. 8 

9.1 



500 


34.7 


165 


5. 5 


62s 


16.7 


450 


14 


375 


8.6 


840 


25.8 


250 


9.2 


270 


7.6 


70 


6.5 


100 


11.8 



* Including idle machines. 



^ Exhaust fan not considered. 



348 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



less numerous than those relating to tools fitted with individual 
motors. The individual motor must be proportional to the maxi- 
mum requirements of the tool, while the group motor has to meet 
only the average requirement, this average taking into account the 
fact that some of the tools are normally idle at any one moment. 
Group driving therefore calls for much smaller motor capacity than 
individual driving. 

An excellent determination of the requirements for group driving, 
by L. P. Alford {Amer. Mach., Oct. 31, 1907), is given in Table 
25, which includes the results of many thousand observations ex- 
tending over a period of about six months in a plant comprising over 
2000 machine tools. The experiments were made prior to the intro- 
duction of high-speed steel and on machine tools, generally speak- 
ing, of the Ught or medium class used in making light automatic 
machinery. The rough parts were made with a small surplus of 
material to be removed, making the work of the cutting tools Ught. 
Since the chief use of high-speed steel is in removing large quantities 
of stock, the tests have permanent value for the conditions under 
which they were made. 

The tests were made possible by the arrangement of the works in 
departments, each department being devoted to tools of a given 
kind, not usually differing much in size. 

In the case of departments containing a variety of sizes, the re- 
sults were arrived at by a process of elimination. The tests were 
made under strictly working conditions. 

The motor capacity for a department is subject to correction from 
the total obtained from Table 25 to cover the factor of departmental 
slip due to that lessening of the average horse-power values of 
machine tools due to working conditions in the department. During 
the progress of the tests the horse-power actually used by all of the 
machine tools in the plant was checked against the value obtained 
by computation after applying the individual horse-power values 
to the entire machine-tool equipment. Certain classes of machine 
tools were eliminated, such as speed lathes, grindstones, tool grinders, 
snagging grinders and others which are intermittent in their use. 
A comparison of the gross horse-power value so obtained showed 
that the sum of the individual power values was 20 per cent, higher 
than the actual horse-power used in the factories. Therefore in 
using these data for the purpose for which they were collected, the 
values obtained for the various departments by using the individual 
machine-tool values as given in Table 25 were reduced 20 per cent, 
before being used to determine the size of motor required. Two 
other exceptions were also made in its use. The power values were 
reduced one-half when applied to the machine tools of the jig- and 
fixture-making department and to the experimental department. 
The reason is obvious, as the tools are there used intermittently and 
with light cuts and fine feeds. 

To the machine tool load as thus determined, the load due to the 
line- and countershafts is to be added. Table 26 gives these friction 
loads for the plant at which the tests were made. For additional 
information on the friction of shafting see Index. 

Additional data of the same character are given in Table 27 by 
H. C. Spillman (Mchy., June, 1913). The tests embodied in this 
table were made in an automobile engine factory after it had been in 
operation about nine months. 

The transmission equipment consists of a 20-h.p. motor for each 
department, driving two or three lengths of line-shafting, each about 
70 ft. long. The shafting is 2^ ins. diameter, running at 240 r.p.m., 
and supported on Hyatt roller bearings at intervals of 10 ft. (For 
additional information on the friction of shafting with plain and 
Hyatt bearings, see Friction of Shafting.) The shafting was care- 
fully lined up when installed and re-checked with a surveyor's level. 
The equipment was in excellent condition. 

In making the power tests, every precaution was taken to avoid 
errors. The motors were all tested for their efiiiciency at different 
loads and the electrical instruments were carefully caUbrated. Read- 



ings were taken when the motor was driving the line-and counter- 
shafts only, no machines being in operation. The electrical losses 
were deducted, which gave the net shafting and countershaf ting loss, 
and this was carefully proportioned among the different machines. 
The second reading was taken with the machines running light, 
which gave the total friction loss. After these tests were completed, 
the machines were put into operation and readings were taken every 
fifteen minutes. A record was also made of the number of machines 
in operation and the kind of work the machines were doing. Each 
motor in this factory drives only one kind of machine tools, which 
greatly assisted in obtaining accurate results for the amount of 
power consumed by each size and kind of machine. The data were 
carefully tabulated and the floor area occupied by each machine, 
and space allotted for the operator were noted; also the total length 
of hne-shafting. The trucking aisles and all other space not used for 
manufacturing was deducted, so that the unit values per sq. ft. of 
floor area include the machine and sufficient space for the operator 
and material. The results show that the line-shafting and counter- 
shafting consume 30 per cent, of the total power, and the total 
friction losses absorb 72 per cent, of the total power. This makes a 
42 per cent, loss of power from the countershaf ting to the machine 
tools, and only 20 per cent, of the total power is utilized in doing work. 
The electrical loss shows 8 per cent, of the total power. In the table 
there are two items mentioned as follows: Total average power per 
machine, deducting idle machines; total average power per machine, 
including idle machines. These items include all the mechanical 
power of that department, such as line-shafting, countershafting, 
machine friction and power consumed in doing work on the machines. 
In the first case this total power is equally divided among all the 
machines which are in operation. In the second case it is divided 
equally among all the machines, both running and idle. The elec- 
trical losses are omitted in all cases. 

Power Constants for Punching and Shearing 

The power required for punching and shearing formed the subject 
of experiments by Prof. G. C. Anthony {Amer. Mach., May 22, 
19 13). The apparatus employed consisted of a hydraulic bolster 
below the die and connected to an ordnance indicator by which indi- 
cator diagrams of the pressures were obtained {Trans. A. S. M. E., 
Vol., 33). Examples of these diagrams to a reduced scale are given 
in Fig. 12. The steel plate tested was from the Lukins Iron and Steel 
Co., and was of 5, j^, f , ^, 5 and | in. thickness, having an aver- 




FiG. 12. — Indicator diagrams from punching and shearing 
experiments. 

age tensile strength of 59,000 lbs. per sq. in. with elongation of 27 
per cent, and reduction of area of 55 per cent. Flat, bevel and spiral 
punches of 3 deg. of clearance were included in the tests. The cards 
were interpreted for both maximum pressure and ft.-lbs. of work 
required. 

Fig. 13 gives the work and maximum pressures developed when 
using flat punches having .06 in. clearance. Figs. 14 and 15 give 
the effects of clearance and shape of the punch on the work and 
maximum pressure required for punching. The character of the 
punch and amount of clearance are given at the top of the charts; 
the ft.-lbs. of work and maximum pressure are at the left, and the 



PERFORMANCE AND POWER REQUIREMENTS OF TOOLS 



349 



thickness of the plate is indicated on the charts. Save in two or 
three cases the rainimum values for work and pressure were obtained 
by the use of the flat punch, and in one of these cases, the spiral punch 
in f-in. plate, the value is questionable by reason of insufficient data. 
While efficiency in the use of bevel and spiral punches in thick 



Table 28. — Shearing Values of Hot Steel Blooms of .20 
Carbon and 70,000 Lbs. Ultimate Strength 



Plate Thickness, Ins. 



1000 

900 

800 

700 

600 

500 

■0 400 

I 300 

« 200 

O 

fe 100 



Plate Thickness, Ins. 
» 5 3 .7_ 1 

3 16 S 16 S 

















k 






-Jj— 




^s 


'^i / 




'*' 


/ /-«■ 






^J/ 




/ 


% 






// 






<; 


'1 






r 

























































EOOOO 

48000 

46000 

44000 

„• 42000 

■^ 40000 

3 38000 

t 36000 

a 34000 

a 32000 

M 

I 30000 
28000 
26000 
24000 
22000 
20000 









y 








/ 








/ 






/ 








/ 






^ / 






^ 


if 






.,s 














1 


/ 








f 






/ 






^V 


f 




















/ 






■— 


f 







Size of 
bloom, ins. 


Temperature 
Fahr. about 


Max. pressure, 
lbs. per sq. in. 


Energy, ft.-lbs. 
per sq. in. 


9X9 
6X6 
4X6I 


2500 
2500 
2500 


5,000 

9,000 

1 1 ,000 


540 
800 



Table 29. — Shearing Values of Cold Steel Bars of 70,000 
Lbs. Ultimate Strength 



Thickness 

of bars, 

ins. 


Angle of 

knives, 

deg. 


Max. pressure, 
lbs. per 
sq. in. 


Energy, in.- 

Ibs. per inch 

of width 


I 


Flat 


48,000 


1200 


I 


4 


36,000 


1000 


I 


8 
Flat 


22,000 
48,000 


700 
2500 




4- 
8 


45, 000 
32,000 


2000 
1600 



Fig. 13. — Work and pressure of punching steel plates with flat 
punches having .06 in. clearance. 

plates has been frequently questioned, it has been believed that a 
decrease in pressure was general when they were used in punching 
thin plates, but the results of these experiments do not confirm this. 
The bevel and spiral punches crowd the metal to the walls of the die, 
thus producing unnecessary friction, while 
the real cutting edge, which is on the die, 
does not produce the eJBfect of a bevel 
shear. 

Fig. 16 gives the work and maximum 
pressures required per sq. in. for punching 



y [^ Spiral 
Clearance 

015 "N^ 



_ Flat (Z^ 
C<^ Bevel Y r^ 




Clearance 




single shear; that the ultimate strength of the plate in double shear 
is 1.95 greater than in single shear. 

Experiments with similar apparatus were made and reported by 
H. V. Loss (Journal of the Franklin Institute, Dec. 1899). Mr. 
Loss's experiments covered the shearing of hot blooms from 4X4 ins. 
to 10X10 Ins., and of cold bars from f to 2 ^ ins. thick and 4 to 8 ins. 
wide. His results are summarized in Tables 28 and 29. The appar- 
ent anomaly of greater energy consumed when cutting hot metal is 
apparent only. With cold metal the bar breaks after a compara- 
tively small depth of penetration, while, with hot metal, the shear 
blade plows through the entire thickness before the parts separate. 
At a temperature of about 1800 deg. Fahr. 
the maximum pressure increases about 50 per 
cent, for the larger and 100 per cent, for the 



P4 



u in 



•B, ja ^_ 3 



IBOO 
I 1400 
A 1300 

[C 

a 1200 
I 1100 
'° lOGO 



^. 900 
c 

"2 800 

°l 700 

3 600 

£ 500 



P 



. 62000 

". 60000 

^ 68000 

a '2 66000 

3 i 54000 

a ■^ 

•3 2 52000 

I I 50000 

4; "g 48000 

3 46000 

a 

■a 44000 
S 42000 



>3 P4 



04 a< 



■ 








T< 


dsIg 


Q-V-i 


iluc 




















































V 








\ 








\ 








s 








1 










\ 








\ 








\ 








\ 










^ 











-Effects of clearance and of form 
of punch on work of punching. 



Fig. 15. — Effects of clearance and of form of 
punch on maximum pressure of punching. 



Fig. 16. — Punching and shearing values of 
steel plate. 



and for single and double shear of plate and rivet. The tension 
value has been added for purpose of comparison. 

It will be observed that the work required for punching is approx- 
imately double that for shearing; that the ultimate shearing strength 
of the plate is about 75 per cent, of the tensile strength; that the ulti- 
mate strength of the rivet in double shear is 1.82 greater than in 



smaller sizes. At the same temperature the energy increases about 
40 per cent, for the larger and 75 to 80 per cent, for the smaller 
sizes. 

The pressure required to drive rivets may be obtained from Fig. 17 
(Amer. Mack., July 13, 1911), which is based on formulas by 
Wilfred Lewis. 



350 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Pressure Eequired, in Tons 
BO 75 100 125 150 



175 



200 



1?4 



•gl!. 
u 

5 

































































'^C 




























w'' 


\\>- 






















- •^6'' 


.-^a- 












* 














#■? 


k^^- 
























^- 


•^s. 












„, 














<(^ 


^•''^ 






nn><^ 


-frv^ 


u*^ 
















<' 


<o^ 


■iV 




c'^'^" 


^!f- 


-ol' 


frVve 


■ 














y 


:>^ 








f;'>T 
















/ 


i^ 




^; 






















.<v 




^'' 


'^■bo 
























• 




''' 


















1 







50,000 100,000 160.000 200,000 250,000 300.000 350,000 4(J3,000 
Pressure Required, Lbs, 

Fig. 17. — Pressure required to drive rivets. 



Power Constants for Centrifugal Fans 

The power required to drive Sturtevant centrijiigal fans is given in 
Tables 30 and 3 1 from tests by the Interior Conduit and Insulation 
Co. {Amer. Mach., Dec. 31, 1896). 

Centrifugal fans consume an amount of power which is dependent 
upon the opening of the outlet and the amount of air which the fan 
is allowed to pass — an obstruction in the outlet operating to decrease 
the power consumed, which is at a maximum when the outlet is 
entirely free. As fans are actually used for blowing fires and many 
other purposes, the resistance of the fire operates as an obstruction 

Table 30. — Power Required to Drive Sturtevant Steel 
Pressure Blowers 

The Upper Figures of Each Set Give the Power Consumed with the Outlet 

One-third Open; the Middle Figure, Two-thirds Open, 

and the Lower Figure Fully Open 



Pressure of 
blast 


4 


oz. 


S oz. 


6 


oz. 


7 


oz. 


8 oz. 


Size No. of 
blower 


Rev. 


H.p. 


Rev. 


H.p. 


Rev. 


H.p. 


Rev. 


H.p. 


Rev. 


H.p. 






• 7 




I. 




1-3 










2 


3103 


1.4 

2. I 
I. 


3445 


2.0 
3.0 

1.4 


3756 


2.6 

3-9 

1.8 










3 


2456 


2. 
30 

1.4 


2753 


2.8 

4.2 

1-9 


3006 

■ 


3.6 

5-4 

2.5 










4 


2224 


2.8 

4.2 

2.0 


2470 


3-8 

5-7 

2.8 


2692 


5.0 
7-5 

3.6 




4.6 






5 


1814 


4.0 

6.0 
2.6 


2026 


5.6 

8.4 

3.6 


2215 


7.2 
10.8 

4-7 


2387 


9.2 
13.8 

6.0 




7.3 


6 


1619 


S-2 

7.8 
3.6 


1797 


7.2 
10.8 

5.0 


i960 


9.4 
14. 1 

6.5 


2009 


12. 
18. 

8.3 


2258 


14.6 
21.9 

10.4 


7 


1344 


7.2 
10.8 

45 


IS07 


10. 
15-0 

6.4 


164 1 


13.0 
19.5 

8.4 


1768 


16.6 
24.9 

10.6 


1898 


20.8 
31-2 

13.0 


8 


1200 


9.0 

13.5 

S.9 


1330 


12.8 
19.2 

8.3 


1445 


16.8 
25.2 

10.9 


1565 


21.2 
31.8 

13.8 


1675 


26. 
39-0 

16.9 


9 


1035 


II. 8 
17.7 

7.9 


1145 


16.6 
24.9 

II. 2 


1250 


21.8 
32.7 

14.5 


1350 


27.6 
41.4 

18.4 


1446 


33.8 
50.7 

22.5 


10 


902 


15.8 
23.7 


995 


22.4 
33.6 


1085 


29.0 

43-5 


1168 


36.8 
55.2 


1253 


45.0 
67. 5 



and the power consumed is, during normal conditions, reduced from 
the maximum. Nevertheless, at various times this resistance is 
reduced or may be absent, when the power consumed at once mounts 
up to the maxirrium. 

With fan driven by a belt or special engine, this increase is a matter 
of little moment so long as the belt or engine is able to drive the fan, 
and on this account the figures for power given in the catalog of fan 
makers show what is supposed to be the average or normal require- 
ments. When fans are driven by electric motors the conditions are 
changed, since an electric motor has no limit of capacity beyond 
which it slows down or stalls, but, on the other hand, takes more and 
more current in the endeavor to drive the load, until a burn-out 
results. Consequently electric motors for fans should be propor- 
tioned with reference to the maximum requirements, and not, as 
with steam engines, to the mean. 

The figures of the tables are no doubt the equivalents of the current 
readings which necessarily exceed the actual power consumed by 
the fans. 

A pressure of 4 oz. is amply sufiicient for ordinary forge fires. 
There is a tendency toward specifications for higher pressures than 
this, even up to 8 oz., but it is doubtful if such pressirres ever reach 
the fire, the convenient blast gate cutting the pressure down to lower 
figures. 

The horse-power required to drive centrifugal Jans has been investi- 
gated by A. E. Guy, and the results are given below (Amer. Mack., 
June 29, 1911). 

When the fan takes the air from the atmosphere and delivers into 
a duct, and particularly when that duct or pipe is circular, it is 
comparatively easy to measure the approximate capacity of the 
apparatus when the air handled is at a moderate temperature. The 
instrument needed for the operation is very simple and can be easily 
made. Fig. 18 represents a conbination of Pitot and pressure tubes 
connected to a glass U-tube containing water. The end of the assem- 
bled tubes should be inserted into the delivery pipe as shown. A 
straight part of the pipe should be selected where the flow is not 
likely to be disturbed by the influence of bends, valves, etc. The 
gage should be inserted into the pipe for about one-sixth the diameter 
and turned so that the open end of the Pitot tube is against the cur- 
rent. If the tube is not so placed the readings will not be correct. 



To Pitot Tube 



Pressure Tube~ 




Fig. 18.- — Apparatus for finding the pressure and flow of air in blast 

pipes. 

with the two rubber tubes in place the difference in the heights 
of the columns of water in the U-tube shows the velocity head causing 
the flow in the duct. Disconnecting the Pitot tube from the glass 
gage and measuring the height between the two levels, will indicate 
the pressure head against which the air is delivered. Again connect- 
ing the Pitot tube and disconnecting the pressure tube, wiU show, by 
the difference in the hights of the water columns, the total head pro- 
duced by the fan. This total head is composed of the static head 
measured by the pressure tube, plus the velocity head shown when 
the two tubes are used together. 



PERFORMANCE AND POWER REQUIREMENTS OF TOOLS 



351 



Table 31. — Power Required to Drive Sturtevant Monogram Blowers 
The Upper Figures of Each Pair Give the Power Consumed with the Outlet One-half Open, and the Lower Figure with the Outlet Fully Open 



Pressure of blast 


I oz. 


ll 


oz. 


2 OZ. 


2i 


OZ 


3 oz. 


3i 


oz. 


4 


oz. 


5 oz. 


Size No, of blower 


Rev. 


H.p. 


Rev. 


H.p. 


Rev. 


H.p. 


Rev. 


H.p. 


Rev. 


H.p. 


Rev. 


H.p. 


Rev. 


H.p. 


Rev. 


H.p. 





1863 


.15 
.30 


2274 


.28 
.56 


261S 


.44 
.88 


2912 


.6 
I. 2 


3177 


.8 
1.6 














I 


1632 


.21 
.42 


1992 


.38 
• 76 


2291 


.60 
I. 20 


2550 


•8 
1-7 


2782 


1. 1 
2. 2 


2992 


1-4 
2.8 










2 


1373 


.28 
.56 


1677 


•5 

I. 


1928 


■ .79 
1.6 


2147 


1. I 

2. 2 


2343 


1-5 
2.9 


2520 


1.8 
3-7 










3 


1 167 


.40 
.80 


1425 


■ 7 
1.4 


1638 


1. 1 
2. 2 


1824 


1.6 
3-1 


1900 


2.0 

4.1 


2140 


2.6 
5-2 


2279 


3.2 
6.3 


2527 


4-4 
8.8 


4 


loso 


• 5 
1. 1 


1227 


i.o 
2.0 


1410 


1-5 
3.0 


1570 


2. 1 
4.2 


1713 


2.8 

5.6 


1842 


3.5 
7.0 


1961 


4-3 
8.6 


2176 


6.0 
12.0 


5 


852 


• 7 
I-S 


1038 


1-3 

2.7 


1 194 


2. I 
4.2 


1330 


2.9 
5-9 


1450 


3.9 

7.8 


1560 


4-9 
9.8 


1662 


6.0 
II. 9 


1843 


8.4 
16.7 


6 


726 


1. 

2. I 


886 


1.8 
3.7 


1018 


2.9 

S-7 


1134 


4.0 
8.0 


1237 


5-3 
10. 5 


1331 


6.7 
13-4 


1417 


8.1 
16.2 


1571 


11. 4 
22.7 


7 


632 


1-3 

2.7 


772 


2.4 
4-9 


878 


3.8 

7-5 


988 


5.3 
10.6 


1078 


6.9 
13-9 


1159 


8.7 
17-3 


1234 


10.7 
21.4 


1368 


150 
30. 


8 


545 


1-7 
3-4 


665 


3.1 
6.2 


766 


4.8 
9-5 


852 


6.7 
13.4 


930 


8.8 
17.6 


1000 


II. I 
22.3 


1065 


13.5 
27.0 


1180 


18.2 
36.5 


9 


477 


2. 2 
4-5 


583 


4-1 
8.2 


671 


6.3 

12.7 


748 


8.9 
17.8 


815 


II. 7 
23-4 


876 


14.8 
29.7 


932 


18. 
36.1 


1034 


25-3 
50.6 


10 


426 


2.9 
5.9 


519 


5.4 
10.8 


598 


8.3 

16.7 


667 


II. 7 
23-3 


726 


15-4 
30.7 


78X 


18.0 
36.1 


831 


23.7 
47-3 


922 


33.2 
66.4 


36 


362 


4.0 
8.1 


443 


7-4 
14.8 


Sii 


II. 4 
22.9 


567 


16. 

32.0 


611 


21. I 

42.1 


66s 


26.7 
53-4 


712 


32.5 
64.9 


78s 


45-5 
91 . 


37 


316 


5-3 
10.7 


345 


9.8 
19.7 


413 


15-2 
30.3 


493 


21. 2 
42. 5 


538 


27.9 
55-9 


579 


35.4 
70.9 


615 


43.0 
86.1 


683 


60.3 

120. 7 



Calling the velocity of flow v ft. per sec, the velocity head /j ins. 
of water, and the static pressure head H ins. of water. 



Substituting for the volume and velocity their respective values: 
D^XvXHXUo6.7+H) 



^=^\n-r\ 406.7+ff ='32iY 



air h.p. = - 



6350X1242 



4o6.7+fl- ^■'^ \(4o6.74-ff) 
in which, 

p = pressure in lbs. per sq. ft., 

(f = weight in lbs. of i cu. ft. of free air at 50 deg. Fahr. 
= .077884, 
406.7 =ins. of water, corresponding to atmospheric pressure. 
Knowing the inside diameter D, in ins., of the delivery pipe, the 
volume discharged in cu. ft. per sec. is 

—— Xv 

4X144 

But this air is at a pressure H and the corresponding volume of free 

air per min. would be 

7iD^XvX6oXUo6.7+H) D^XvX {406.7+H) 

— rz — z = cu. ft. per mm. 

4X144X406.7 1242 

The horse-power in air delivered would be 

volume per min. X pressure per sq. ft. 

33,000 

One cu. ft. of water weighs 62.35 lbs.; i in. of water equals 

62-35 
12 

Hence, 

vol. per min. X5 .i96Xg 
33,000 
or 

cu.i ft.ipermin. Xfl^ 



= 5.196 lbs. per sq. ft. 

r 
=air h.p. 

^^11 1 ft 1 rtt^r mr 

air h.p. = - 



As the efficiency of ordinary blowers is about 50 per cent., multiply- 
ing the air horse-power as just obtained by 2 gives approximately 
the shaft horse-power necessary to run the blower. While reading 
the gages the speed should be kept constant, and the time selected 
when the flow of air is uniform. 

The gage readings and particularly that of the velocity head 
should be very close, for which reason it is preferable to use a U-tube 
of rather small diameter. 

The formulas given are intended for approximate work only. The 
density of the air depends so much upon the temperature that the 
method would not apply to hot-blast work, for instance. Corrections 
should also be made for altitude and humidity. 

Power Constants for Moving Heavy Loads 

The power required to move heavy loads on wheels may be obtained 
from Fig. 19, by A. D. Harrison {Amer. Mack., June 18, 1908). 
The chart was originally designed for hoists and cranes but is appli- 
cable to analogous conditions. It represents the forrnula: 



VWS 1 

Brake h.p. = .oog7\-jr-{d+.7) 



6350 



in which W = total weight of structure, tons, 
5 = speed, ft. per min., 
£> = diameter of wheels, ins,, 
d = diameter of axles, ins. 



352 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



The coefficient of rolling friction is taken at .035, the coefficient Table 32.— The Compression of iJXii-iN. Lead Plugs under 
of sliding friction at .1 and the efficiency of the gearing at .70. 
The use of the chart is shown by an example below it. 



Falling Weights 
Note. — Where division lines include two or more plug numbers, the weight 
was dropped upon as many plugs at one time as there indicated. 



120 
110- 
100- 

90- 

,„ 80- 
to 

(3 

^70- 

'O 

a 60- 
o 
1-1 

« 
o 
H 40- 

30-1 

20 
Brake horse power required 

at Motor ^''Hl 

40 35 30 25 20 15 10 5 

_1 1_1 I I I I L 




Traveling Speed, Ft» per Min 




Diameter of Axle, Ins. Diameter of Traveling Wheels, Ins. 

Starting with a load of 80 tons, trace to the right to the speed 
— 60 ft. per min. — then down to the wheel diameter — 27 ins. — 
then to the left to the axle diameter — 8 ins. — and then up to the 
horse-power — 13. 

Fig. 19. — Power reqiiired to move heavy loads on wheels. 

Measuring the Energy of Hammer Blows 

The measurement of the energy of hammer blows by the compression 
of lead plugs, formed the subject of experiments by the Niles-Bement- 
Pond Co., which were reported by W. T. Sears, Mech. Engr. of the 
company {Amer. Mach., Mar. 10, 1910). 

Previous experiments of this kind have, usually, been made by 
comparing the compressions obtained under hammers with the meas- 
ured compressions obtained under testing machines. The speed of 
the compression is, however, known to affect the results and, hence, 
these tests were made under falling weights at speeds equal to those 
obtained in actual practice in a steam hammer, in order to get final 
results which could be depended upon in steam-hammer work. 
These results are, finally, compared with those obtained from slow- 
speed or static tests. 

The velocity of a hammer ram at the instant before impact with 
the anvil, depends on friction, the total mean effective pressure on 
the piston and the distance it has fallen through. 

For a Niles-Bement-Pond iioo-lb. steam hammer of 28-in. stroke, 
the maximum velocity, assuming a constant pressure of 100 lbs. per 





1 

.5 .5? 


.1 

ft 

2 
Q 


h- 1 


Plug dimensions, ins. 






Plug 
No. 


c 

1-1 




ft 

t3 


a 

|3 


TO (U 
a b. 
> 3 

< 


Striking 

velocity, 

ft. per 

sec. 


13 


20 


240 


4,800 


1.489 


■ 979 


• 51 




35-9 




14 


20 


120 


2,400 


1.494 


1. 16s 


.329 




25-4 




IS 


20 


240 


4,800 


1.492 


.983 


.509 




35.9 




16 


20 


360 


7,200 


1.492 


.848 


.644 




43-9 




17 


50 


120 


6,000 


1.492 


.911 


.S8i 




2S-4 




18 


50 


240 


12,000 


1.492 


.656 


.836 




35-9 




19 


so 


360 


iS.ooo 


1.492 


.50s 


.987 




43-9 




20 


100 


360 


36,000 


1. 491 


-307 


1. 184 




43-9 




21 


100 


240 


24,000 


1. 491 


.409 


1.Q82 




35 9 




22 


100 


120 


12,000 


1.489 


.652 


.837 




25 -4 




23 


150 


360 


54,000 


1-491 


.194 


1-297 




43.9 




24 


150 


240 


36,000 


1-495 


.290 


1.205 




35-9 


. 


2S 


200 


120 


24,000 


1-492 


.401 


1.091 




25-4 


c 

3 

ft 






26 


200 


240 


48,000 


1-492 


.219 


1.274 




35-9 


27 


200 


120 


24,000 


1.489 


.40s 


1.084 




25-4 


.00 

J. 


28 


150 


120 


18,000 


1.492 


.501 


.991 




2S.4 


CIS 


29 


150 


240 


36,000 


1.502 


.275 


1 . 227 




35-9 


-4J 


30 


ISO 


240 


36,000 


I.S03 


.275 


1.228 




35-9 




31 
32 


ISO 


240 


36,000 


1.498 
1.498 


.502 
.507 


.996 
.991 


.993 


35-9 




33 
34 
35 


150 


240 


36,000 


l-S 

1.498 

1-495 


.670 
.681 

.677 


.830 
.817 
.818 


.822 


35-9 




36 
37 
38 
39 


ISO 


240 


36,000 


1-5 
1.5 
1.498 
1-5 


.769 

-778 
.778 
-771 


.731 
.722 
.720 
-729 


-72s 


35-9 




40 


ISO 


240 


36,000 


1.498 


.279 


1.219 




35-9 




41 
42 
43 
44 


200 


240 


48,000 


1-497 
I -502 
I. 502 

1-502 


.663 
.660 
.660 
.650 


.834 
.842 
.842 
.852 


.842 


35-9 




45 
46 
47 
48 


2 00 


356. s 


71.300 


1-5 
i-SOi 
I-S 
1.502 


-SIS 
-525 
-525 
-SIS 


• 985 

• 976 
-975 
-987 


.981 


43-7 





sq. in. on the piston on its downward stroke, and neglecting friction, 
would be in the neighborhood of 35 ft. per sec, and this corresponds 
to the speed due to gravity alone, acting through a distance of about 
19 ft. 

The plugs, which were 1 5 ins. diameter by i ^ ins. long, were tested, 
in most cases, one at a time by placing them on an anvil, having 
a weight of over 8000 lbs. and striking them with different size cylin- 
drical weights, weighing from 20 to 200 lbs. dropping from different 
heights up to 360 ins. In addition to a drop on a single plug, the 



PERFORMANCE AND POWER REQUIREMENTS OF TOOLS 



353 



150-lb. weights were tried with two, three and foiir plugs, and the 
200-lb. weight with four plugs. 

The falling weights were guided by two lengths of piano wire 
stretched tight vertically. The weights were tripped without giving 
any initial velocity, and there is not much question but that the 
actual and theoretical velocities at instant of impact were, very 
closely, the same, the friction loss due to the guides and air being, 
undoubtedly, very slight. 

There was certainly some loss, even if small, and therefore the 
compressions obtained were perhaps a trifle less than they should 
have been. 

Table 32 gives the results of these tests. 

In plotting the energy curves. Fig. 20, which are the values that 
were wanted, it was found that there was not so much difference 
in the higher speeds, as was perhaps to be expected from the consider- 
able difference that occurred at the low speeds. 

In other words, the higher the velocity, up to the maximum of 
speeds tested, the less the energy curves varied. 



Somewhat similar, though less complete, tests using copper cylin- 
ders, which have been often referred to, were made by Prof. R. H. 
Thurston {Amer. Mack., Dec. 24, 1903). The original object of 
these tests was to determine the comparative efSciencies of crank 
and friction roll (board drop) presses. Two hammers of each type 
were tested, the falling weights being about 300 and 900 lbs. respec- 
tively. They were adjusted to fall 28 ins., that being the maximum 
lift of the crank drop hammer. The effect attainable by utilizing 
the full 60-in. lift of' the friction roll hammer was not deter- 
mined experimentally, but it is easily calculable from the data 
obtained. 

The gages used in measuring the work done by the hammers were 
cylinders of pure merchant copper, prepared for the purpose. They 
measured: Size No. i, 25 ins. long, ij ins. diameter; size No. 2, 
2 ins. long, i in. diameter; size No. 3, i\ ins. long, | in. diameter. 

Of these, a considerable number were prepared and divided into 
three sets: one for use with each kind of hammer, and one for testing 
and standardizing in testing machines. The work done by crushing 



Table 33. — Work Done by Drop Hammers as Measured by 


THE Compression of 


Copper Cylinders 






Friction roll drop hammer 


Crank lift drop hammer 


Weight of drop 


903 lbs. 


319 lbs. 


925 lbs. 


290 lbs. 


Size of copper 


iiX2r 


1X2" 


1X2" 


fXii" 


liX2j" 


1X2" 


1X2" 


IX ir 


cylinders 


No. I 


No. 2 


No. 2 


No. 3 


No. I 


No. 2 


No. 2 


No. 3 


Area in sq. ins. under compression curves 


AD E AHI 


AN OARS 


ABC AFG 


ALM APQ 


(see chart). 


45.22 45.26 


I3-7S 1376 


35.10 36.25 


10.75 10-50 




Average 45.34 


Average 13-755 


Average 35.67 


Average 10.67J 


Reduced to work done or in.-lbs. 


22,715 22,630 


6,87s 6,880 


17,550 18,07s 


5,875 5,250 




Average 22,672 


Average 6,877 


Average 17,812 


Average 5,312 


Reduced to work done or ft.-lbs. 


Average 1,884 


Average 576 


Average 1,484 


Average 443 


Work done per lb. of drop in ft.-lbs. 


Average 25.10 


Average 21.56 


Average 19.14 


Average 18.30 


Work done per lb. of drop in ft.-lbs. 


Average 2 .09 


Average 1.8 


Average 1.6 


Average 1.52 



This is quite clearly illustrated in the chart. Fig. 20, which gives the 
energy ciu^ves worked up from the Niles-Bement-Pond tests and from 
tests made at Purdue University. It would seem as if, after a speed 
of say 10 ft. per sec. was obtained, that a furtier increase in compres- 
sion speed makes very little change. 

The speed of 10 ft. is simply a guess, and it may be 5 ft. or i, or 
even less. 

This was a point which was not important to the company, but 
it would seem to be vitally important in measuring energy of blows 
where the speed is low, for all tests so far made show that energy 
calculations of a slow-moving blow cannot be even closely estimated 
unless the speed is known. 

Curve A is worked up from slow-moving or static-pressure tests 
at Purdue University. 

Curves B and C are the energy curves, resulting from Niles- 
Bement-Pond slow-moving or static tests of which the speeds are 
given. 

Curve D is the result of the low velocity drop tests made by the 
Niles-Bement-Pond Co., which are not shown in the table, but 
which were made roughly in a hammer, having a falUng weight of 
1330 lbs., an anvil weight of 16,400 lbs. and a maximum drop of 38 ins. 

Curve E is plotted from the tabulated results given in Table 27, 
and is the curve that is used for hammer calculations. 

Curve F is plotted from the published results of Purdue drop tests, 
in which the maximum velocity is igi^o ft. per sec. 

In order to check up new lots of plugs from time to time, static 
or slow-moving tests are obtained, and if these agree with previous 
ones, it is assumed that the action at the high speeds will also be 
practically the same, thus giving fairly dependable results. 

No appreciable difference has been noted in new lead obtained 
from time to time, or in lead that has been used for tests and remelted. 

The lead should be reasonably pure, though small amounts of 
impurities do not appear to affect the accuracy of the results. 
23 



the standards in the testing machine to the same extent that com- 
panion specimens were crushed under the hammers, gave a measure 
of the action of the latter, and permitted a fair comparison to be made. 
The amount of work done in the slowly acting testing machine 
in producing a given compression is somewhat less than where the 
same effect is suddenly produced, as by a falling weight; but this 
difference effects the two hammers nearly alike, and, if the difference 
were measurable, it would be found to tell against the drop which 
falls most rapidly — the friction roll hammer, in this case. 

The results of the experiments thus made are exhibited in Table 
i:^ and Fig. 21. The final results of the table are given in ft.-lbs. 
of work per lb. of hammer, and the unavoidable differences in size 
are thus eliminated. 

The chart. Fig. 21, was made thus: The compression of each set 
of gage cylinders was averaged for each of the two styles of hammer. 
These average compressions were laid off, on a convenient scale, 
horizontally from the left toward the right. Erecting ordinates at 
the extremities of the abscissas thus measured off, proportional to 
the loads required to produce the same compression as determined 
by the testing machine, and shown on the chart by the curve laid 
down by plotting the loads and compressions obtained by test, a 
measure of the work done by the hammer is obtained. 

This was done for each hammer, and a set of measures is thus 
given of the work done by each machine, and the effects produced 
by the hammers are rendered easily comparable. 

Comparing the tabulated figures, it is seen that the friction roll 
drop hammers performed, respectively, 25.1 and 21.56 in.-lbs., or 
2.1 and 1.8 ft.-lbs. of work per lb. of weight of drop or hammer, 
while the crank lift hammer gives 19.14 and 18.3 in.-lbs., or 1.5 ft.-lbs. 
per lb. of hammer falling 275 ins. The theoretical effect would 
be 275 in.-lbs, or 2.25 ft.-lbs. The "efficiencies" of the two are, 
therefore, 90 per cent, for the friction roll hammer, and less than 
70 per cent, for the crank lift hammer. 



354 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



1.5 



to 3 

►1 Pi 



rl.0-, 



"0.5 1, 



1 































































































































































































































































































































































































































































































































































































































































































































N.:^.p 


Worked frorA Bressurej 


Curve 


Standard 
















































u 




























Veloci'ty [= 


.00007 


Ft. ^er 


S^c. 
















:^.B 


P. 


Higli Velocity Drop Tests 






























tur 


aU< 




[vVorked 


from 


Pressure Curye 


^ 








-^ 
























25 


to 


44 Ft. perlSe'c. 








-^ 














































._ 


— 


— 




:- 


— 








__ 




-s 


















._ 


_is 


P= 


^ 


— 


— "^ 




= 


r^ 


^ 


=1 


E^ 






— 


: — 






















^ 


•^ 


^- 


-- 


""* 






' 







— 


^^ 














K 


.^ 




LiS= 


-^ 


















N.B.P. 


Low Vel 


jcity 


Drop 


Tests 




















-^ 


-^ 


















_ 






^ 


s^ 


s= 




]sf.B 


fe 


Worked from High SpeU Pressur^C^ 


urye 




6 to 14.2 Ft. 


pfjr Sec 


_ 






5 












y 


^ 


-d 


?^ 




[.^ 


J 


"U- 




^ 


3?= 


=^ 






















Velocity 


= 


.OOOBFt. perSeb. 














































> 




^ 


6 




::h 


l-i 


'■^ 




























































































/ 
/ 


/ 


/ 


/ 


.-• 


..^^ 


£:iiz 
































































































/ 


y 


/^ 




a^T 




Purdue Dr 


op 


Te'stE 























































































// 


/. 


<^ 


^ 




•^^ 




Velocitj 4 1' 


).7 


Ft 


per 


3ec 


.Max. 






































































y 




^ 












































































































/^ 




^ 










































































































i 


r 










































































































- 














































































































R 


r 






































_ 








































_j 
























_ 


_ 




HH 



4,000 8,000 12,000 16,000 20,000 24,000 28,000 32,000 36,000 40,000 44,000 48.000 52,000 

Energy, In. Lbs. 



Fig. 2o. — Energy curves from compression tests of lead plugs 



lia ' r ■ \- 




_ .. . ^ . _ ._ . . ,. — . . 


" ;:: :::: ::: ::::_:.;""_ " " " : : i- ::: ^ -i ^_ : ::: :-: :_:-;_ :: : : ::: 








___ - ^'Z- . _. 






"" " ■ " " _ : ,<= :~_"." _ _ 






-rnP 


fiOOflO — A^ ' \ 








'- ' '- ' - - ---^i SS< '- I , = £ - Jj_ - 


— :::::;: — X""' "" "" "" -;-::::;::- - - ^-voJi.' - -- - U -- -^ - -- -- -- - - --- 


\^^ -M IW-tTl 


■ " - <<\.'i.' t-- ,a-; 


"_.:;:"" ::.::::::. ;.:::. ..:.. "" " :::.; ::'.:.::-;: ; :: ,\-K' : :: :.::;: : ; ::: i,-" iF--- :::_::: :: i- 






50.000 " " " SS'' :;' J "" 






- --_ -- ^ .?«' = ' " .Vlit.^'' 1 ---- 




cs^-Pn ro^^ VtTIi 


,\J\' — -- r,\a.v>.'' r i 


^^eV;-- .V'%- " - - - J- - 


OJO^y , _ _ -^ 1.^., - __ _.L ... . J U 


fin npn ? *-• -rfi tvv '• -^t 1 i 






-.j-r'^n aei-ifl-rT i 


^v, jj ^^^jv ,= L . ^_ .J _ 




-:;.-_:::::..::;:.;;::_:.::: : -..ig.^. : :: "^S^S^w ------ :_-::_. :;i-:_:;_:.. -_--\ :::::: 


Je>it<\ , f "V'^r-rTT 1 


T;?>' — „< '-'" ^ - ---- — J — 


„„ ««« --- :::::::::::: : ::: ^i^: :: : : :±^s2;-r -::.:.--::: -:::::--;:-:-:: :;[: = — ~-- -:::-;-::" Ij--::. -" - - :_:::: 


30 t'oo Civ-f-f ^vit^' Vn 1 11 






LkT ■c«'=j-rrTT 1' 1^ 


,^: -Ifo^-f -^ vpr^/a-^'rr"i4--'-1 


__< ouv, , ij etei.-|.4J,--4== -1 


A\ -rT . 1 ' \"- ^ -M — nlTl^' i 


_ _i., ,' ;;' — . st\ J-i-LTX-i li: J 






''".000 J^^ i-Ki x-T ■'^"u rr-rt^rT rM" Ti ' 


_ ,' 2 ,rK\\'^iL.it'^ + f J 


::::::::;:<•;::: ::::: ..':::: ::::::::: : ; :,(i:^BS^" ' i" " "1""":"""-::::::::: : 1: : :::: -i -i-i: : 


r^ -Hi f- -iiwei^ij-ifrTT .1 


-rl J-nl cio«*-'>ffrrT ' 1 ' 


yfl 'fT -vesS^H; — MTT i 


^ 2 ■".aC'-i "" "il" " i t" ■ J - 


? -- \^--'-- - - -h - ^- --- ---\ r- -- --- --J--- 


A yri 14-HtT - ! 


10 000 4 J-^r 1-h'T^ ' \ ,- 1 . 




" ;':_ ^' ;--"' - . . .-ij. . - - — j-- p. - j . - _-_ 










/i'D-nil 111 11 




MIfflILIIii||illi|l111l|illlllllllllllll|i|||||ll|i[l[^^l4M"rir[[fi|i|^^ 



.100" .200" .300" .400" .600" 

Fig. 21. — Work done and pressures obtained by drop hammers as measured by the compression 



.600" .TOO" 

of copper cylinders. 



Cutting Capacity of Power Presses 

The cutting capacity of power presses has been analyzed by E. W. 
Zeh {Amer.Mach.,Oct. 12, 1905), the result being the chart, Fig 22. 
The chart is based on the principle that the cutting length increases 
inversely as the square of the thickness of the material. That is, 

A 

in which I = cutting length, ins. . 

A = energy required to shear a flat bar using parallel cutting 
edges, in. -lbs., 
/ = thickness of material, ins., 
5 = ultimate resistance to shearing. 
The fact that the material is severed before the upper knife has 
descended the full thickness of the bar brings in another influence 



which has to be taken into account. This depth of penetration 
as it may be called, varies greatly. It is influenced by the ductility 
and thickness of the material and it increases as the thickness de- 
creases, but not in simple proportion. Table 34 gives the results of 
some experiments with soft steel, in which t stands again for the 
thickness of the material, and p for the depth of penetration. 

Table 34. — The Depth of Penetration in Soft Steel 



t = 

P = 

i = 

P = 



I 


f 


f 


i 


f 


■25 


■ 31 


■34 


• 37 


• 44 


i 


^ 


i 


^ 


^ 


• s 


• 56 


.62 


• 67 


■75 



16 

•47 

1 

32 
.87 



In Fig. 21 the curve C shows in a graphical manner how the depth 
of penetration varies. 



PERFORMANCE AND POWER REQUIREMENTS OF TOOLS 



355 



Taking into consideration the depth of penetration, formula (a) 
for the cutting length will now have the following form: 

^ (6) 



t = 



t^ps 



When the cutting length of a power press for.a certain thickness t 
and resistance 5 is given, and it is desired to know the cutting length 
h for another thickness ti of the resistance Si, the following formula, 
which is derived from (b), will give the answer: 

iHps 
ilpisi 

in which pi stands for the depth of penetration for the new thickness 
introduced. 



h=: 



(c) 



To find the angle which the upper knife must have to keep the 
maximum pressure within the limit of the presS; formula (d) may be 
inverted thus: 

P 



cot « = 



■Si's 



ie) 



When closed cutting dies are used, for instance, a round blanking 
die, it is customary for practical reasons to give the die (or punch) 
several high points, and the question arises: How does the number 
of cutting points affect the pressure which is required to penetrate 
the material? 

In answer to this question refer to Fig. 24, in which c is the devel- 
oped circumference or cutting length of a round die. According to 




Fig. 22. — Cutting capacity of power presses. 



While the cutting length of which a press is capable may be 
obtained from these formulas, another factor has also to be taken 
into consideration. The pressure required to force the knife or die 
through the material must not exceed a certain maximum which is 
fixed for each press. To keep the pressure within this limit, it is 
often necessary to incline the edge of one of the dies. 

When the cutting edge AB, Fig. 23, descends, it finds a resistance 
P = .St^ cot. as, (d) 

in which < = thickness of the material, 

5 = ultimate resistance to shearing, 
a = angle of the knife, 
This formula is limited in one direction to which attention should 
be called, viz., the width of the bar divided by its thickness must be 
greater than cot a. 



formula (d) the pressure necessary for one inclined side of the punch 
is equal to .5^^ cot. <xs, consequently for both sides of one cutting 
point the pressure is twice this amount or t^ cot. «5. If n be the num- 
ber of cutting points, the total pressure necessary is 



p=/2 cot. asn 

d 

cot a = r and d = 



(/) 



c 

2» 



consequently 



2» 



This value, substituted in formula (/), gives: 

I'^cs 



P = 



26 



(g) 



In this formula the number of high points does not appear at all, 



356 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



showing that the pressure is not influenced by the number of such 
points and merely depends upon the amount of shearing b. To 
ascertain the amount of shearing b which the die must have to keep 
the pressure within the limit P, formula (h) may be used: 

Although developed from a plain round cutting die, these formulas 
will hold good for irregular dies. In this case it is only necessary to 
observe that all sections of the cutting edge have the same inclination. 




Section through Line A-B 
Showing Greatest, or True 
Slope of Lip Surface 



Horizontal Line 




Fig. 25. — Definitions of tool elements. 



Following are two practical applications of the formulas: 
(i) A die has a cutting length of 14 ins. in |-in. soft steel. Which 
size press is required? 

The chart shows that the No. 5 press cuts l-in. steel about 14 ins. 
long. Consequently this press will do the work. This press, exert- 
ing a maximum pressure of 25 tons, we have to shear the die sufli- 
ciently to keep the pressure within this limit. According to formula 
(h) this shear must be at least 

1X14X60,000 
64X2X50,000 
This being the extreme limit, it would be well to increase the shear 
somewhat, say to .2 or .25 in. 

Taylor's Tool Forms 

The shape and duty of roughing tools formed the subject of 
exhaustive experimental investigation by F. W. Taylor and his 
associates (Trans. A. S. M. E., Vol. 28). Mr. Taylor defines the 
various elements of cutting tools by means of the outline sketches, 
Fig. 25. Regarding the values of the various angles for roughing 
tools he makes the following recommendations: 

Contrary to the opinion of almost all novices in the art of cutting 
metals, the clearance angle and the back-slope and side-slope angles 
of a tool are by no means among the most important elements in the 
design of cutting tools, their effect for good or evil upon the cutting 
speed and even upon the pressure required to remove the chip being 
much less than is ordinarily attributed to them. 

The clearance angle should have the following values: 

(A) For standard shop tools to be ground by a trained grinder or 



Sbbp« o£ Standard 
K"Too1, correB- 
poDdlng to those 
glTon In Fig.27 




8hapo of }i Rumid 
Nosed Standard Tool, 
Two-Thirds Ground 
away 




Fig. 26. — Taylor's standard f-in. rough- 
ing tool. 



Fig. 27. — Blunt tool for cutting hard 
steel and cast-iron. 



Fig. 28. — Sharp tool for cutting medium 
and soft steel. 



The cutting capacity of a given power press for a certain thickness 
of a certain material being known, it is possible to draw a character- 
istic curve which shows the cutting lengths for any other thickness 
of the same material. Such curves have been drawn in Fig. 22 for 
sizes Nos. 3 to 9 of the Zeh & Hahnemann power presses. They 
axe based upon steel of an ultimate strength of 60,000 lbs. per sq. in. 

These presses are graded in such a manner that they exert a max- 
imum pressure in tons equal to the square of their number. The 
No. 5 press, for instance, exerts a maximum pressure of 5X5 or 25 
tons; the No. 6 of 6X6, or 36 tons, etc. These pressures must be 
known to a die- maker in order to enable him to determine the proper 
amount of shear by the formulas given to obtain a safe result. 



on an automatic grinding machine, a clearance angle of 6 deg. 
should be used for all classes of roughing work. 

(B) In shops in which each machinist grinds his own tools a clear- 
ance angle of from 9 deg. to 12 deg. should be used. 

The latter recommendation is based on the fact that when the 
workmen grind their own tools they usually grind the clearance 
and lip angles without gages, merely by looking at the tool and guess- 
ing at the proper angles; and much less harm will be done by grinding 
clearance angles considerably larger than 6 deg. than by getting them 
considerably smaller. 

(C) For standard tools to be used in a machine shop for cutting 
metals of average quality: Tools for cutting cast-iron and the harder 



PERFORMANCE AND POWER REQUIREMENTS OF TOOLS 



357 



steels, beginning with a low limit of hardness, of about carbon .45 
per cent., say, with 100,000 lbs. tensile strength and 18 per cent, 
stretch, should be ground with a clearance angle of 6 deg., back 
slope 8 deg., and side slope 14 deg., giving a lip angle of 68 deg. 

(£>) For cutting steels softer than, say, carbon .45 per cent, having 
about 100,000 lbs. tensile strength and 18 per cent, stretch, tools 
should be ground with a clearance angle of 6 deg., back slope of 8 
deg., side slope of 22 deg., giving a lip angle of 61 deg. 

(E) For shops in which chilled iron is cut a lip angle of iron 86 
deg. to 90 deg. should be used. 

(F) In shops where work is mainly upon steel as hard or harder 
than tire steel, tools should be ground with a clearance angle of 6 
deg., back slope 5 deg., side slope 9 deg., giving a lip angle of 74 deg. 

(G) In shops working mainly upon extremely soft steels, say, 
carbon .10 per cent, to .15 per cent., it is probably economical to use 
tools with lip angles keener than 61 deg. 

(H) The most important consideration in choosing the lip angle 
is to make it sufficiently blunt to avoid the danger of crumbling or 
spalling at the cutting edge. 

(/) Tools ground with a lip angle of about 54 deg. cut softer 
qualities of steel, and also cast-iron, with the least pressure of the 
chip upon the tool. The pressure upon the tool, however, is not the 
most important consideration in selecting the lip angle. 

(/) In choosing between side slope and back slope in order to grind 
a sufficiently acute lip angle, the following considerations, given in 
the order of their importance, call for a steep side slope and are op- 
posed to a steep back slope: (a) With side slope the tool can be ground 
many more times without weakening it; (b) the chip runs off sideways 
and does not strike the tool posts or clamps; (c) the pressure of the 



chip tends to deflect the tool to one side, and a steep side slope 
tends to correct this by bringing the resultant line of pressure 
within the base of the tool; (d) easier to feed. 

(K) The following consideration calls for at least a certain amount 
of back slope: An absence of back slope tends to push the tool and 
the work apart, and therefore to cause a slightly irregular finish 
and a slight variation in the size of the work. 

Fig. 26 shows Mr. Taylor's standard f-in. tool and Figs. 27 and 28 
show the dimensions of other sizes. 

Feed and Depth of Cut 

Following are Mr. Taylor's conclusions regarding the relation 
between feed and depth of cut : 

(A) With any given depth of cut metal can be removed faster, 
i.e., more work can be done, by using the combination of a coarse 
feed with its accompanying slower speed than by using a fine feed 
with its accompanying higher speed. In most cases it is not practic- 
able for the operator to take the coarsest feeds, owing either to the 
lack of pulling power of the machine or the elasticity of the work. 
Therefore, the above rule is only, of course, a broad general 
statement. 

(B) The cutting speed is affected more by the thickness of the 
shaving than by the depth of the cut. A change in the thickness 
of the shaving has about three times as much effect on the cutting 
speed as a similar or proportional change in the depth of the cut has 
upon the cutting speed. Dividing the thickness of the shaving by 3 
increases the cutting speed 1.8 times, while dividing the length that 
the shaving bears on the cutting edge by 3 increases the cutting 
speed 1.27 times. 















Table 


35.— Taylob 


's Cutting Speeds in Steel 
















Peed, 
ins. 


Standard iH-in. tool 


1 Standard l-in. tool 


1 Standard J^-in. tool || Standard %-in. tool || Standard %-in. tool || Standard J-a-in. tool 


Depth of 




Grades of steel and cutting speec 


s, ft. per min., for tools to last i hr. 30 


min. before regrinding 






cut, ins. 


Soft 


Medi- 
um 


Hard 


Soft 


Medi- 
um 


Hard 


Soft 


Medi- 
um 


Hard 


Soft 


Medi- 
um 


Hard 


Soft 


Medi- 
um 


Hard 


Soft 


Medi- 
um 


Hard 




Mi 










I 














536 


268.0 


122.0 


500 


250.0 


114,0 
72,2 


He 


H2 


























350 


175 .0 


79. 6 


316 


158,0 


Me 


























229 


1 15.0 


52.1 


199 


100,0 


45 - 3 
































Hi 


533. 


266.0 


121.0 


490.0 


245.0 


III.O 


477 


238,0 


108.0 


465 


233.0 


106,0 


4S6 


228,0 


104.0 


425 


218.0 


99-0 


0/ 


Hi 


37S.O 


188.0 


85.2 


340-0 


170.0 


77-2 


32s 


163.0 


73.9 


302 


156.0 


70,9 


299 


149.0 


67.9 


275 


137.0 


62,4 


/3 2 


Me 


264,0 


132.0 


60.0 


2350 


118. 


53-5 


222 


III.O 


50.4 


209 


105.0 


47.6 


195 


97.7 


44-4 


173 


86,6 


39.4 




Hi 


215.0 


108.0 


48.9 


189.0 


94-6 


43-0 


177 


88.3 


40.2 


165 


82,8 


37.2 


152 


76.2 


34.6 


132 


66,1 


30 -0 




Hi 


461.0 


231.0 


105. 


427.0 


214.0 


97.2 


420 


210.0 


95-5 


413 


207,0 


93.9 


410 


205.0 


93.1 


396 


198,0 


90.0 




H2 


325.0 


163.0 


73.9 


296.0 


148.0 


67.3 


286 


143.0 


65.1 


277 


139-0 


62.9 


268 


134.0 


60.9 


250 


125.0 


S6.8 


M 


Me 


229.0 


115.0 


52. I 


205.0 


102.0 


46.6 


195 


97.8 


44-4 


186 


92-9 


42.2 


175 


87.6 


39.8 


158 


78.8 


35-8 




^2 


186.0 


93-2 


42.4 


165.0 


82.5 


37.5 


156 


77-8 


35-4 


147 


73-5 


33-4 


137 


68.4 


3I-I 






' 




H 


161.0 


80.6 


36.7 


142.0 


71.0 


32.3 


133 


66.7 


30.3 


123 


61.6 


28.0 
















Hi 


377.0 


189.0 


85.8 


357-0 


179.0 


81.2 


352 


176.0 


80. 1 


350 


175-0 


79-6 


351 


176.0 


79-9 


350 


175.0 


79-6 




Ht. 


265.0 


133-0 


60.4 


247.0 


124.0 


56.2 


240 


120.0 


54-6 


235 


118. 


53-4 


230 


1 15.0 


52.2 


22 1 


110,0 


50.2 


^Xn 


Me 


187.0 


93-6 


42.6 


17 1.0 


85.6 


38.9 


164 


82.0 


37.3 


157 


78,8 


35-8 


151 


75.7 


34-4 








/I 6 


H2 


152.0 


76.2 


34-7 


138.0 


68.9 


31.3 


130 


65.2 


29.7 


125 


62.4 


28.3 
















H 


132.0 


65.9 


30.0 


1 19.0 


59.3 


26.9 


112 


55.9 


25.4 






















Me 


107.0 


53.7 


24.4 


95-4 


47-7 


21.7 




























Hi 


328.0 


164.0 


74-6 


314-0 


1570 


71-4 


312 


156.0 


70.9 


313 


IS7.0 


71.2 


319 


160.0 


72,6 


322 


16 1,0 


73-3 




H-i 


231.0 


116.0 


52. s 


218.0 


109.0 


49.4 


213 


106.0 


48-4 


2 10 


105. 


47.8 


209 


105,0 


47.5 








M 


Me 


163.0 


81.3 


37-0 


lSI-0 


75-3 


34-2 


14s 


72.6 


33-0 


141 


70.5 


32.0 














/4 


?^2 

H 
Me 


132.0 
114. 
93-2 


66.2 
57-2 
46.6 


30. I 
26.0 
21.2 


121. 

104-0 


60.6 
52.1 


27-5 
23-7 


116 


57.8 


26.3 






















Hi 


270.0 


135.0 


61.3 


265.0 


133 


60.3 


26 s 


132.0 


60. I 


269 


135,0 


61.3 


280 


140.0 


63.6 










H2 


190.0 


95. I 


43.2 


183.0 


91.9 


41.8 


18 I 


90.3 


41.0 


181 


90,4 


41. I 














% 


Me 
H2 
H 


134.0 
109.0 
94-2 


66.9 

54. 5 
47. I 


30.4 
24.8 
21.4 


127.0 

102.0 


63.6 
51.2 


28.9 
23.3 


123 


61.6 


28.0 






















Hi 


236.0 


118.0 


53.6 


2340 


117.0 


53-2 


237 


118. 


53.8 




















\4, 


H2 


166.0 


83.0 


37.7 


162 ,0 


80.9 


36.8 


162 


80.8 


36.7 




















/2 


Me 

?^2 


117.0 
95-2 


58.5 
47-6 


26.6 
21.6 


112. 


55-9 


25.4 



























358 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 













Table 36. — Taylor's 


Cutting Speeds in 


Cast Iron 
















Feed, 
ins. 


Standard iH-in. tool || Standard l-in. tool || Standard %-in. tool |j Standard %-in. tool || Standard 5^-in. tool || Standard !.2-in. tool 


Depth of 


Grades of iron and cutting speeds, ft. per min., for tools to last i hr. 30 min. before regrinding 


cut, ins. 


Soft 


Medi- 
um 


Hard 


Soft 


Medi- 
um 


Hard 


Soft 


Medi- 
um 


Hard 


Soft 


Medi- 
um 


Hard 


Soft 


Medi- 
um 


Hard 


Soft 


Medi- 
um 


Hard 


%2 


H2 

Me 

H2 

H 
He 


252.0 
200.0 
150.0 
124.0 
109.0 
89. S 


126.0 
100. 

74-9 
62.2 
54-4 
44-7 


73-5 
S8.4 
43.7 
36.3 
31-7 
26. I 


231.0 
180.0 
133.0 
109.0 
94-5 
76.9 


116.0 
90.2 
66.4 
54-6 
47-2 

38. S 


67.2 
52. 7 

38.7 
31.8 
27.6 

22.5 


223.0 
172.0 
122.0 
102.0 
87.7 
70.9 


112.0 
86.0 
61.0 

SO. 9 

43.9 

3S.4 


65.2 
50.2 
35-6 
29.7 
25.6 
20.7 


218.0 
165.0 
118. 
9S.0 
81.3 
65.0 


109.0 
82. 5 
58.9 

47.5 
40.7 

32. S 


63-6 

48. I 
34.4 
27. I 
23-7 
19.0 


2 14.0 
158.0 

IIO.O 

87-0 
74-0 


107.0 
78.9 
54. 9 
43-7 
37.0 


62.3 
46 -0 
32-0 
25-5 
21.6 


209 .0 
148 -O 
99-2 

77. I 
64-4 


104.0 
73.8 
49.6 

38. S 

32.2 


60,9 
43-0 
28-9 
21,9 
18.8 


Vs 


Hi 

H2 

Vie 

H2 

H 

?16 


224.0 
178.0 
133-0 
lll.o 
96.6 
79.6 


112. 
88.9 
66.6 
55-4 
48.3 
39.8 


65.4 
SI. 9 
38.8 
32.3 
28.2 
23.2 


207 .0 
162.0 
120.0 
98.2 
85.1 
69 -3 


103.0 
81.2 
59.8 
49. I 
42. 5 
34-6 


60.4 

47-4 
34-9 
28.6 
24.8 
20.2 


204.0 
157.0 
112. 
93.0 
80.2 
64.8 


102.0 

78.6 

55.8 
46.5 
40. 1 
32.4 


59. 6 

45-8 
32-5 
27- I 
23.4 
18-9 


200.0 
15 I. 
108 .0 
87.0 

74-4 
59.4 


99-8 
7S-5 
539 
43.5 
37-2 
29-8 


58.2 
44.0 
31.4 
25.4 
21.7 
17.4 


198.0 
146.0 
102.0 
81-8 
68.4 


99-0 

73-0 
50-8 
40.4 
34-2 


57 -7 
42-6 
29.7 
23.6 
20.0 


197-0 
139-0 
93-4 
72-S 
61 2 


98.4 
69.4 
46,7 
36.3 
30.6 


57.4 
40.5 
27-3 
21.2 
17-7 


He 


Hi 

H2 

Me 

H2 

H 

?16 


19 1. 

152.0 
114. 
94-4 
82.4 
67.8 


9S.5 
75.9 
S6.8 
47.2 
41.2 
33.9 


55.7 
44-3 
33- I 
27-5 
24.0 
19.8 


iSl.o 
14 1. 

104.0 

, 85.2 
73.8 
60. I 


90.3 
70.5 
51.8 
42.6 
36.9 
30 


52.7 
41. I 
30.2 
24.9 
21.5 

17 S 


178.0 
137.0 
100. 
91.0 
69.8 
56.4 


88.9 
68.4 
SO. I 
40. S 
34-9 
28.2 


SI-9 
39.9 
29.2 
23.6 
20.4 
16. 5 


177.0 
134.0 
95-7 
77.2 
66.1 
52. 9 


88.7 
67.1 
47.9 
38.6 
33-0 
26.4 


SI. 7 
39. I 
27.9 
22-5 
19-3 
15.4 


179 -0 
132 -0 
91-8 
73.0 
61-8 


89-4 
6S-9 
45-9 
36.5 
30-9 


52- 1 
38.4 
26.8 
21.3 
18.0 


I8I-O 
128.0 
86-0 
66.9 


90.7 
64. I 
43.0 
33-5 


52.9 
37-4 
25. I 
19-5 


K 


M64 
H2 

Me 

H2 

H 
?1e 


171. 
136.0 

102.0 
84.6 

73-9 
60.8 


85.6 
68.0 
51.0 
42-3 
37-0 
30.4 


50.0 
39-7 
29.7 
24.7 
21.6 

17.8 


164,0 
128.0 
94-2 
77.4 
67.1 
54-6 


82. I 
64.0 
47. I 
38.7 
33 5 
27.8 


47-9 
37.3 
27 -S 
22.6 
19-6 
IS. 9 


163.0 
125.0 
90.9 

74- I 
63.9 
SI. 6 


81-3 
62.6 
45-5 
37- I 
31.9 
25-8 


47.4 
36. 5 
26.5 
21.6 
18.6 
IS- I 


164.0 
124.0 
88.4 
71.3 
61. I 


81.9 
61.9 
44.2 
3S.7 
30. 5 


47-8 
36-1 
25.8 

20-8 

17-8 


167 .0 
123.0 
85.6 
68. I 


83-4 
61-5 
42.8 
34- I 


48-6 
3S-9 
25.0 
19-9 


172.0 
121,0 
81.6 


86.0 
60,5 
40,8 


50.2 
3S-3 

23.8 


Vs 


Hi 

H2 

Hi 

H2 

H 
He 


147.0 
1 16.0 
87.9 
73 
63.8 
52. 5 


73.9 
58.2 
44.0 
36. 5 
31-9 
26.2 


43. I 
33-9 
25-7 
21.3 
18.6 
15-3 


144.0 
112.0 
82.7 
68.0 
58.9 
47.9 


72, I 
56.2 
41.4 
34 
29.4 
24.0 


42. I 
32.8 
24. 1 
19.8 
17.2 
14.0 


144-0 
III.O 

80.6 
6S-7 
56.7 
4S.8 


72.2 
55-6 
40.3 
32.9 
28.3 
22.9 


42. I 
32-4 
23 5 
19.2 
16.5 
13-4 


147.0 
III.O 

79.3 
64.0 


73. 5 
S5.6 
39.6 
32 .0 


42.9 
32. 5 
23. I 
18.7 


152 .0 
112 .0 
78.0 


76.0 
56.1 
39.0 


44-3 
32-7 
22-7 








K 


Hi 

H2 

Me 

H2 

H 
Me 


133.0 
106.0 
79-4 
66.0 
57.6 
47.4 


66.7 
53 
39.7 
33 
28.8 
23.7 


38.9 
30.9 
23.2 
19.3 
16.8 
13.8 


132.0 
103.0 
75.6 
62. I 
53.8 
43-8 


65.9 
SI. 4 
37.8 
31. I 
26.9 
21.9 


38.4 
30.0 
22. I 
18. 1 

15-7 
12.8 


133-0 
102.0 
74-2 
60.5 
52.2 


66.4 
51. I 

37- I 
30.3 
26. I 


38-7 
29.8 
21.7 
17.7 
14-9 




















H 


Hi 

Me 

H2 

H 
Vie 


1 16.0 
92.2 
69. I 

57-4 
50. I 
41.2 


58.1 
46. I 
34-6 
28.7 
25. I 
20.6 


33-9 
26.9 
20.2 
16.8 
14.6 
12.0 

































(C) Expressed in mathematical terms, the cutting speed varies 
with the standard round-nosed tool approximately in inverse propor- 
tion to the square root of the thicliness of the shaving or of the feed. 

(D) With the best modern high-speed tools, varying the feed and 
the depth of the cut causes the cutting speed to vary in practically 
the same ratio whether soft or hard metals are being cut. 

(£) The same general formula expresses the laws for the effect 
of depth of cut and feed upon the speed, the constants only requiring 
to be changed: 

{F) The same general type of formula expresses the laws governing 
the effect of the feed and depth of cut upon the cutting speed when 
using the different sized standard tools. 

Tables 35 and 36 give Mr. Taylor's determinations regarding 
depth of cut, feed and speed for high-speed steel tools: 

A study of Mr. Taylor's and Prof. J. T. Nicholson's experiments, 
the latter made at the Manchester School of Technology, has led 
E. C. Herbert {Amer. Mach., June 24, 1909) to the discovery of a 
law by which apparent discrepancies between the experiments are 
reconciled. Mr. Herbert expresses his law, which he calls the cube 
law, thus: 



Since the maximum thickness of the chip is generally proportional 
to the feed or traverse, we will call 
t = traverse, 
c = depth of cut, 
c = area of the cut, and 
5 = cutting speed. 
From what has been said above it follows that if h, Ci, ai, si repre- 
sent the values of these factors for any given working conditions, and 
h, C2, 02, S2 represent their value for another set of working conditions, 
then the heating of the cutting edge and, by assumption, the dura- 
bility of the tool will remain unaltered so long as the relation 

holds good. From which it follows that for constant durabihty of 
the cutting tool 

\<2ffl2 

or, since a=tc, 



Si = Si' 



PERFORMANCE AND POWER REQUIREMENTS OF TOOLS 



359 



The cube law may be most conveniently stated thus: 

The cutting speed varies inversely as the cube root of the product of 
traverse by area of cut; or alternatively, the cutting speed varies itiversely 
as the cube root of the product of depth of cut by traverse squared. 

In cutting cast-iron, the cube law as stated above is only applicable 
in the case of coarse feeds. When the feed is less than xs in. the 
thickness of the chip has very little influence on the speed, which 
varies inversely as the cube root of the area of cut approximately. 




100 90 80 70 60 
Feeds in Turns per In, 



40 30 20 10 



140 130 120 110 100 SO 80 70 60 50 
Feeds in Turns rer In, 

A piece of cast-iron 2 ins. diameter is to be turned down to if ins. diameter. Follow the line marked turning of the chart for 
cast-iron to its intersection with the J-in. depth of cut line whence trace vertically to the bottom where read the feed 34.2 turns 
per in. and from the same point trace horizontally to the right where read the speed 52 ft. per min. 

Fig. 29. — Feeds and speeds in average practice. 



An examination of a large collection of data led Stanley K. 
Moore to construct Fig. 29 {Amer. Mach., Dec. 25, 1902) for the best 
feed and speed values for various depths of cut, the term "best feed 
and speed" being understood to mean that combination that will 
remove a maximum amount of material when due consideration is 
given to economy and the time required for changing and grinding 
the tools. 

The use of the charts is explained by an example below them. 

Speeds for Tapping and Threading 

Cutting speeds for tapping and threading, as followed in the shops 
named, are as follows {Amer. Mach., Aug. 3, 1911): 
By the F. E. Wells Co., for tapping cast-iron: 

inch holes 
r. p. m. 



382 



25s 



^53 



using an oil or soda compound. 
For soft steel and iron: 



91 



127 



76 



inch holes 
r. p. m. 



299 153 115 
using oil as a lubricant. 

The National Machine Company uses 233 r.p.m. up to i in. 
diameter and 140 r.p.m. for sizes between i and i in., using a screw- 
cutting oil as a lubricant. 

They tap holes as deep as four tap diameters by power. 

By the Landis Machine Co., for threading cast-iron in machines 
of the bolt-cutter type: 



200 150 125 ic 

with petroleum as a lubricant. 
For soft steel and iron: 



4 
280 



I 
85 



I 
"5 



SS 



15 

7S 



2 
4S 



ms. 
r. p. 



ins. 
r. p. 



m. 



20 175 140 

with compound or screw-cutting oil. 

The speeds are for high-speed steel dies. Some users of the 
machines run at a much higher rate, the figures given being conserva- 
tive and easily attained. 

The BignaU & Keeler Mfg. Co., aims to have its pipe-threading 
machines run at a cutting speed of 15 ft. per min. They advise 
nothing but lard oil on the dies. 

The Standard Engineering Co. also recommends a cutting speed 
of 15 ft. per min. 

The number of teeth in milling cutters may be determined from 
Fig- 30, by W. G. Groocock, which gives the practice of the Woolwich 
arsenal {Amer. Mach., Aug. 17, 1911). The chart contains also lines 
for the lead of the spiral. Mr. Groocock's practice is to use a 14-deg. 
spiral on end and finishing mills and 20 to 25 deg. on roughing end and 
slab mills, with an occasional slab mill of 30 deg. spiral and fewer teeth. 

Milling machine cutters of greatly increased pilch of teeth formed 
the subject of extended tests by the Cincinnati Milling Machine Co., 
which were reported on by A. L. DeLeeuw {Trans. A. S. M. E. 
Vol. 33). The dimensions found most advantageous, as regards 
capacity and power consumption, are shown in Fig. 31. For the 
power consumption obtained in these tests, see Power Requirements 
of Milling Machines. 



360 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



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Fig. 30. — Number of teeth in milling cutters. 



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90 



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m 



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30 S 
h4 



10 




Teeth 
DiuD. 



Side Mills 



Spiral Shell Cutters 



Fig. 31. — Coarse pitch milling cutters. 



f 



CAST-IRON 



The following particulars and tables relating to the properties 
and uses of cast-iron are extracted from the report of Dr. J. J. Porter, 
chairman of a committee of the American Foundrymen's Association 
(Trans. A.F. A., Vol. 19). 

^ Cast-iron is a complex alloy of six or more elements. The common 
elements are: Iron, carbon, silicon, sulphur, phosphorus, manganese; 
and the other elements sometimes present are: Copper, nickel, 
oxygen, nitrogen, aluminum, titanium and vanadium. 

Carbon is the most important element in cast-iron. It exists in 
many forms, all of which are included under the two heads of graph- 
ite and combined carbon. The total carbon is dependent upon 
the temperature in the blast-furnace, the conditions of melting and 
the percentage of other metalloids. Graphite weakens iron. The 
amount depends upon the per cent, of total carbon, the rate of cool- 
ing, the per cent, of silicon, the per cent, of sulphur, and the per cent. 
of manganese. Combined carbon hardens iron and may increase 
or decrease the strength. The amount depends upon the per cent, 
silicon, the rate of cooling, the per cent, sulphur and the per cent, 
manganese. 
t SiHcon exists in cast-iron in the form of silicides. Its chief effects 
are through its action on the carbon. Increasing the silicon decreases 
the total carbon because it replaces carbon in the molten solution. 
Increasing the silicon increases the graphite because it replaces car- 
bon in the solid solution, the displaced carbon being precipitated as 
graphite. 
Y Phosphorus exists in cast-iron as the phosphide FesP which is 
insoluble in the solid iron-carbon solution. Phosphorus decreases 
the total carbon. According to Upton, the effect of phosphorus on 
carbon is to slightly increase graphite and decrease total carbon. 
"5 Sulphur exists in cast-iron as iron sulphide and manganese sul- 
phide. Iron sulphide forms a eutectic with iron melting at 1 780 deg. 
Fahr. and insoluble in the solid iron-carbon solution. It therefore 
forms films between the iron crystals and causes brittleness. 
iJ. Manganese sulphide does not form these films and is less detri- 
mental. Manganese has a greater aiSnity than iron for sulphur and 
with enough manganese all the sulphur will be in combination with it. 
\,- Sulphur has a greater tendency to segregate than any other con- 
stituent of cast-iron. This tendency is greatest with manganese 
sulphide. Sulphur tends to decrease graphite and increase combined 
carbon. 

The presence of silicon decreases the amount of sulphur which 
cast-iron can take up. Much sulphur reduces the total carbon, and 
vice versa. 

Manganese may exist in cast-iron as manganese sulphide or as 
manganese carbide. It tends to harden iron. It can neutralize 
sulphur and will also remove dissolved oxide at high temperatures, 
as in the blast-furnace. 

Traces of copper are common in pig iron. Its effects on cast-iron 
are poorly understood. Cast-iron will take up only about 5 per cent, 
copper and this does not affect the casting properties. Copper 
accentuates the red-shortness due to sulphur. Copper prevents a 
complete evolution of sulphur in iron analysis. 

Small amounts of nickel occur in many pig irons. Its effects on 
the strength and ductility of cast-iron are relatively unimportant. 

The strength of cast-iron is dependent upon nine factors: i, per 
cent, of graphite; 2, size of graphite flakes; 3, per cent, of combined 
carbon; 4, size of primary crystals of solid solution, Fe-C-Si; 5, 
amount of dissolved oxide; 6, per cent, of phosphorus; 7, per cent, of 
sulphur; 8, per cent, of silicon; 9, per cent, of manganese. 



'*' The size of graphite flakes accounts for many cases of difference 
in strength of irons of the same composition. The factors influencing 
the size are very poorly understood. 
{■, The effect of dissolved oxide is probably important. To reduce 
oxide we may get the best brands of pig iron, avoid oxidizing con- 
ditions in the cupola, and use deoxidizing agents. 

te Phosphorus lessens strength, particularly resistance to shock. 1 
One per cent, produces a marked effect. 

1 Sulphur may indirectly strengthen iron through decreasing the 
graphite, but is more likely to weaken it through causing blowholes 
and high shrinkage. 

% Silicon and manganese act chiefly indirectly. Silicon should be' 
kept as low as possible and still have the necessary softness. Man- 
ganese should be high, but if too high produces weakness. 

Of the elastic properties only toughness and elasticity are important 
in cast-iron. The sum of these properties is given by the deflection. 
The factors influencing them are about the same as those influencing 
strength. 

Maximum rigidity with the least sacrifice of strength and. 
toughness is obtained through the use of manganese and combined 
carbon. 

Hardness is due both to combined carbon and gamma solid solution. 
The latter explains the cases of hard cast-iron which are yet low in 
combined carbon. 

Phosphorus has only a slight hardening effect.7 Manganese may 
soften iron through its action on the sulphur, but in larger amounts 
will harden it. Sulphur is an energetic hardening agent. Silicon 
softens iron due to its action in decreasing combined carbon up to a 
certain point. Beyond this point it hardens, due to its direct action. 
Combined carbon is the chief hardening agent in cast-iron. 

In chilled iron the factors influencing the depth and quality of the 
chill are, pouring temperature, and percentage of silicon, sulphur, 
phosphorus and total carbon. The higher the pouring temperature 
the deeper the chiU. Sulphur causes a brittle chill and is undesirable. 
Phosphorus injures the strength of chill and causes a sharp line be- 
tween the white and gray portions. '7 Manganese increases the hard- 
ness of the chill and its resistance to heat strains. 
'\ ^ The grain structure and porosity depend on the size and percentage 
of the graphite. 3 The fusibility of cast-iron depends primarily on 
combined carbon, and to a less extent on the phosphorus. Graphite 
affects the melting-point only in so far as it dissolves in the iron at 
temperatures below the melting-point. 

Fluidity is determined by per cent, sihcon, per cent, phosphorus, 
freedom from dissolved oxide and temperature above the freezing- 
point. 

The following Table i of classified castings is taken by Dr. Porter 
partly from published results but chiefly from rephes to inquiries. 
Thickness is taken into consideration since this largely determines 
the percentage of silicon necessary, and it has been the aim to sub- 
divide the various classes according to section wherever possible. 
In this respect the endeavor has been to follow the definitions of the 
American Society for Testing Materials, who have grouped castings 
according to thickness as follows: 

" Castings having any section less than 5 in. thick shall be known 
as Ught castings. 

" Castings in which no section is less than 2 ins. thick shall be known 
as heavy castings. 

"Medium castings are those not included in the above defi- 
nitions." 
361 



\ 



362 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table i. — Chemical Composition of Iron Castings for Various Purposes 

The last analysis under each head is preceded by the word " Sug." (abbreviated from suggested) and is the tentative standard or probable best analysis 
suggested by the committee. Under is abbreviated by "und. " 



Silicon Sulphur 


Phos. 


Mang. 


Comb, 
carb. 


Total 
carb. 


Silicon 


Sulphur 


Phos. 


Mang. 


Comb, 
carb. 


Total 
carb. 


Acid Resisting Castings: 










Bed Plates: 












1.00% .050% 


.50% 






3 • 00 % 


2.20% 


.090% 


.5S% 


.50% 






2.30 low 


.20 


.41% 




3.60 


1.32 


.090 


.40 


.60 






.80-2.00 .02-. 03 


.40-. 60 


1 . 00-2 . 00 




3 • 00-3 . 50 


1.65 




.28 


.92 


.72% 




Sug. 1.00-2.00 und. .05 


und. .40 


I. 00-1.50 




3 . 00-3 . 50 


1.8s 


.080 


.60 


■ 55 


.50 


3-25- 

3-50% 


Agricultural Machinery, 


Ordinary: 








1.80-2.20 


.04-. 06 


•45-. 55 


.40-. 50 


,40-. 50 


3 . 40-3 . 60 


2. 20-2. 80% und. .085 


% und. 70% 


und. .70% 






1.65-1.85 


.070 


.65-. 80 


.60-. 75 




3.85 


2.65 .050 


.81 


.70 


.15% 


3-50% 


Sug. I.2S-I.75 


und. .10 


.30-. so 


.60-. 80 






2.25 .070 


.70 


.80 


.30 


3.50 














2.10 .068 


.73 


.45 


■ 47 


3-42 


Boiler Castings 












2.00 .089 


.89 


.46 


.50 


3-39 


2 . so % 


und. .07% 


und. .20% 


.80-1.0% 






Sug. 2.00-2.50 .06-. 08 


.60-. 80 


.60-. 80 






2.25 
Sug. 2.00-2.50 


.060 
und. .06 


.62 
und. .20 


• 59 
.60-1.0 






Agricultural Machinery, 


Very Thin: 




















2.90% .050% 


•85% 


.70% 


.10% 


3.50% 


Brake Shoes: 












2.50 .080 


.65 


.60 


.30 


3-50. 


I • so % 




low 






low 



Air Cylinders: 












1.20-1.50% und. .09% 


.35-. 60% 


.50-. 80% 






1.90 


.074 


.50 


.65 






1 . 12 


.085 


.40 


.70 


.70% 


3.50% 


• 95 


.100 


.30 


.90 


.80 


3 40 


2.00 


.070 


.30 


.60 


.40 




Sug. 1. 00-1.75 


und. .09 


.30-. 50 


.70-. 90 




3.00-3.30 


Ammonia Cylinders: 










1 . 20-1.90% und. .095% und. .70% 


.6o-.«o% 






Sug. 1. 00-1.75 


und. .09 


.30-. 50 


.70-. 90 


3 


.00-3.30% 


Annealing Boxes, Pots and 


Pans: 




' 




1.20% 


. 060 % 


■ 10% 


.40% 






1.80 


■ 03 


.70. 


.60 




2.90% 


1-53 


.04 


.33 


1.08 


■ ss 


3-68 


Sug. 1. 40-1. 60 


und. .06 


und. .20 


.60-1.00 




low 


Automobile Castings: 










1.80% 


• 030 % 


.50% 


.70% 


.60% 


3 ■ SO % 


1. 6s 


.076 


■ 45 


.65 


■ 55 




2.3S 


.072 


.60 


.70 


.40 




Sug. I. 75-2. 25 


und. .08 


•40-. so 


.60-. 80 






Automobile Cyli 


nders: 










1.65% 


.076% 


.45% 


.65% 


• 55% 




2.31 


• 094 


• SO 


•43 


• SI 


3^35% 


2.70 


• 053 


•46 


• 23 


.44. 


3.02 


2.4s 


. 102 


• 72 


•41 


.41 


3.47 


2. 59 


• 083 


• 57 


•47 


. II 


3-35 


2.5s 


.104 


.82 


• 32 


■ 09 


3 04 


2.98 


.047 


.89 


• 27 


• 14 


3.19 


2.67 


.III 


.73 


.38 


. 10 


3.24 


2.30 


.084 


.81 


• 52 


• 59 


335 


1.60 


• 083 


■ S4 


.42 


.66 


3-75 


3.26 


• IS9 


• 93 


• 44 


• 03 


3.87 


1.72 


.091 


■ 58 


• 48 


.62 


2.52 


1.67 


.068 


■ 44 


.82 


.62 


3-91 


1.38 


.093 


.62 


.52 


.76 


3^61 


1.47 


• 075 


■ 13 


.60 






I. SO 


.103 


.86 


■ 43 






1.99 


.130 


■ 6S 


■ 39 


•45 


3^17 


1.89 


.090 


.70 


■ 39 


• 77 


3-34 


2.29 


.090 


■ 83 


.60 


• 90 


4. 16 


Sug. 1.75-2.00 


und. . 08 


. 40-^50 


.60-. 80 


.5S-.65 


3 •00-3^25 


Automobile Fly- 


wheels: 










2.3S% 


• 072 % 


.60% 


.70% 


.40% 




3-10 


.045 


■ 35 


■ SS 


.27 




Sug. 2.25-2.50 


und. .07 


.40-. 50 


■SO-. 70 






Balls for Ball Mills: 










1.00% 


.100% 


■ 30% 


■ 50% 




low 


Sug. 1. 00-1.25 


und. .08 


und. .20 


.60-1.00 




low 



low 








lo 


und. .70% 


und. 


.70% 






und. . 70 


und. 


.70 






.50-. 80 


• 45- 


.60 


.40-. 65% 


3 


1.93 




.33 


1.22 


3 


.30 


.50- 


■ 70 




1 



2 . 00-2 . SO % und. . IS % 
2.00-2 .50 und. .IS 

1. 40-1. 80 .06-. 08 .50-. 80 .45-. 60 .40-. 65% 3.50% 

1.86 .183 

Sug. 1. 40-1. 60 .08-. 10 

Car Castings, Gray Iron. See also Brake Shoe and Car Wheels: 
2.20-2.80% und. .085% und. .70% und. .70% 
1.40-1.80 .06-. 08 .50-. 80 .45-. 60 .40-. 65% 

2.25 050 .60 .75 

1.75 .070 .85 .60 

Sug. 1.50-2.25 und. .08 .40-. 60 .60-. 80 

Car Wheels, Chilled: 

.50-. 70% .05-. 07% .35-. 45% .30-. 50% .50-. 75% 3-50% 



low 



350% 
3^S0 



.58-. 68 


.05-. 08 


■ 25-^45 


■15-.27 


.63-1.0 




■ 73 


.080 


■43 


■ 44 


1.25 


4.31 


.86 


.127 


■ 35 


• 49 


.92 


3.47 


.70 


.08 


■ SO 


.40 


.60 


3.50 


■ 58 


.141 


.38 


.48 


.90 


3.63 


■ 57 


.101 


.41 


.42 






.68 


.188 


.36 


.53 






.67 


.170 


■ 38 


.81 


.74 


3.66 


.50-. 60 


.08-. 10 


.30-. 40 


.45-. 55 


.70-. 80 


350 


.60-. 70 


.08-. 10 


.30-. 40 


.50-. 60 


.60-. 80 


3. 50-3 ■ 70 



Sug^ 



Car Wheels, Unchilled. See Wheels. 



Chilled Castings: 

.80-1.00% .09-. 11% 



Sug. 



1 .20-1 .40 
1. 00 
1.3s 

■ 50 
1.20 
1.20 

■ 75 
.75-1.25 



Chills: 
Sug. 



2.07% 

.7S-2.2S 



.08 
.117 
.200 
.090 
.080 
.090 
.08-1.0 



.073% 
und. .07 



.50% 
low 
.40 
.60 

■ 45 
.30 

■ 30 
.30 

.20-. 40 



.31% 
.20-. 40 



Collars and Couplings for Shafting: 

1.60% .040% .55% 

Sug. 1.75-2.00 und. .08 .40-. so 



.50% 

■ 7S 

■ 54 
I. SO 

• SO 

1.25 

.30 

.80-1.2 



.48% 
.60-1.0 



.55% 
.60-. 80 



Cotton Machinery. See also Machinery Castings: 

2.20-2.30% und. .09% .70% .60% 

Sug. 2. 00-2. 25 und. .08 .60-. 80 .60-. 80 



Crusher Jaws: 

.80-1.00% .09-. 11% .50% 

I . 00 .080 .40 

.50 .20 .45 

Sug. .80-1.00 .08-. 10 .20-. 40 



low 

3-25% 

. 65 % 3 • 00 

3 . 00 3 . 00 

1.20 3.20 

3. SO 
3.00 3.20 



. 23 % 2 . 64 % 



.30% 3.57% 



.45% 3^45% 



.50% 






■ 75 




3.25% 


I. SO 


3.00% 


3.00 


.80-1 .2 







CAST-IRON 



363 



Table i. — Chemical Composition of Iron Castings for Various Purposes — {Continued) 



Silicon 


Sulphur 


Phos. 


M 


ang. 


Comb, 
carb. 


Total 
carb. 


Silicon 


Sulphur 


Phos. 


Mang. 


Comb, 
carb. 


Total 
carb. 


Cutting Tools, 


Chilled Cast-iron: 








Gears, Medium: 














1.35% 


.117% 


.60% 


.54% 


.65% 


3 • 00 % 


1 . 50-2 . 00 % 


und. . 08 % 


■ 35-^6o% 


.50-. 80% 








Sug. I.00-I.2S 


und. .08 


.20-. 40 


.60-. 80 






1.90 
2.30 


.060 
.060 


. 10 
.60 


.40 
.60 






3.75% 


Dies for Drop Hammers: 










1 .90 


. 100 


.69 


.58 




55% 


3.83 


1 . 40 % 


.060% 


.10% 


.40% 






Sug. 1.50-2.00 


und. , 09 


. 40- . 60 


.70-. 90 








1 .40 


.090 


.40 


.70 


1 . 00 % 


3^20% 
















Sug. I.2S-I.S0 


und. .07 


und. .20 


.60-. 80 




low 


Gears, Small: 
3 ■ 43 % 




1.42% 


■ 90% 








Diamond Polishing Wheels- 










2.00 


. 100% 


.50 


.70 






3^50% 


2.70% 


. 063 % 


• 30% 


.44% 


1 . 60 % 


2.97% 


Sug. 2.00-2.50 


und. . 08 


.50-. 70 


.60-. 80 








Dyanmo and Motor Frames, 


Bases and 


Spiders, Large: 




Grate Bars: 














I ■ 95 % 


. 042 % 


.40% 


.39% 


■59% 


3^82% 


2.75% 


low 


low 










1 .90 


.08 


■ 47 


.60 


.64 


3-79 


2 .00 


. 085 % 


• 35% 


■ 53% 








2. IS 


.070 


■ 75 


.60 


•55 


3 •So 


Sug. 2.00-2.30 


und. . 06 


und. . 20 


.60-1 . 


unc 


. .30 


low 


2. 10 


.070 


■ 55 


■ 40 




350 
















Sug. 2.00-2.50 


und. .08 


.50-. 80 


.30-. 40 


.20-. 30 


low 


Grinding Machinery, Chilled Castings 


for: 




















■ 50% 


.200% 


• 45% 


1.50% 


3 


00% 


3 00% 


Dynamo and Motor Frames, 


Bases and 


Spiders, Small: 




Sug. .50-. 75 


. 15-20 


.20-. 40 


I ■ 5-2 . 








3.19% 


.075% 


.89% 


■ 35% 


.06% 


2.95% 
















2.30 


. 070 


• 55 


.40 




3.50 


Gun Carriages: 














2.50 


.070 


.75 


.60 


•55 


395 


■ 94% 


. 050 % 


.44% 


.31% 




.63% 


3^03% 


Sug. 2 . S0-3 . 00 


und. . 08 


.50-. 80 


.30-. 46 


.20-. 30 


low 


I . 00 

Sug. 1. 00-1.25 


.050 
und. .06 


.30 

. 20-. 30 


.60 
.80-1 .0 


I 


10 


2.50 
low 


Electrical Castings: 
























3.19% 


.075% 


.89% 


.35% 


.06% 


2.95% 


Gun Iron: 














1.95 


.042 


.40 




39 


■ 59 


3.82 


1.34% 


. 003 % 


.08% 


1 . 00 % 




.93% 


3.12% 


1 .90 


.080 


• 47 




60 


.64 


3.79 


1. 19 


.OSS 


.41 


.42 


I 


13 


3^18 


2. IS 


.070 


.75 




60 


■ 55 


3 80 


I 53 


.050 


.29 


■45 




.42 


3.43 


2.50 


. 070 


75 




60 


• 55 


3^95 


.98 


.06 


•43 


•43 




.75 


1^74 


2. 10 


.070 


55 




40 




3 SO 


• 30 




.44 


3-55 


I 


.70 


3 90 


2.30 


.070 


■ 55 




40 




3. SO 


1.20 


. 100 


.30 


.80 


I 


.00 


3.00 


Sug. 2.00-3.00 


und. .08 


.50-. 80 


■30-. 40 


.20-. 30 


low 


Sug. 1. 00-1.25 


und. .06 


.20-. 30 




.80-1.0 


low 


Eccentric Straps. See Locomotive Castings and Mach 


inery Castt 


ngs: 


Hangers for Shafting: 












Engine Frames. 


See also Machinery Castings: 






1.60% 


. 040 % 


.55% 


• 55% 




.30% 


3^57% 


2.25% 


. 080 % 


.55%! 


.60% 






Sug. 1 . 50-2 . 00 


und. . 08 


.40-. 50 


.60-. 80 








1.60 


.090 


.50 


.60 




















1.32 


.100 


.40 


.60 






Hardware, Light: 












Sug. 1.25-2.00 


und. . 09 


.30-. 50 


.60-1.0 






1.84% 
2.20 




• 58% 

• 74 


1.04% 
1 . 10 








Farm Implements: 










2.50 




I. 21 


1. 16 








2 . 00 % 


. 089 % 


.89% 


.46% 


.50% 


3.39% 


2. SI 


.110% 


.62 


•41 




.24% 


3^18% 


2 . 10 


.068 


.68 


• 45 


• 47 


332 


2.70 


.030 


.60 


• SO 




.40 


3 60 


Sug. 2.00-2.50 


.06-. 08 


.50-. 80 


.60-. 80 






2.50 

2.00-2.25 


und. .050 
.050 


.60 

.85 


• 70 
.40 






3 ■85-4^00 


Fire Pots: 












Sug. 2.25-2.75 


und. .08 


.50-. 80 


.50-. 70 








2.50% 


und. .07% 


und. .20% 


.80-1.0% 




















Sug. 2 . 00-2 . 50 


und. . 06 


und. .20 


.60-1.0 




low 


Heat Resistant Iron: 
























1.20% 


. 060 % 


.10% 


.40% 








Fly-wheels. See also Automobile Fly-wheels and Mach 


nery Castings: 


1.67 


.032 


.09 


.29 




.43% 


3.87% 


2. 20% 


.090% 


.55% 


-50% 






2. IS 


.086 


1.26 


• 41 




.13 


3 30 


I. so 


.090 


.50 


.60 






2.02 


.070 


.89 


• 29 




.84 


3^6o 


Sug. 1.50-2.25 


und. .08 


. 40- . 60 


•SO-. 70 






r.S3 


.040 


.33 


1.08 




.58 


3^68 


Friction Clutches: 










2.07 


.073 


•31 


.48 




• 23 


2.64 


2.00-2.50% und. .15% 


und. .70% 


und. .70% 






1.80 
2.7s 


.030 

low 


.70 

low 


.60 








Sug. 1.75-2.00 


.08-. 10 


und. .30 


.50-. 70 




low 


2.50 


und. . 07 


und. .20 


.80-1 .0 








Furnace Castings: 










1.76 


• 075 


.63 


• 79 




• S6 


3^68 


2.50% 


und. .07% 


und. .20% 


.80-1 .0% 






2 .00 


• 030 


.70 










2.00 


.08s 


■ 35 


■53 






Sug. 1.25-2.50 


und. .06 


und. . 20 


.60-1 .00 


unc 


- ^30 


low 


1.8s 


.090 


■ 70 


.60 






Hollow Ware: 














Sug. 2.00-2.50 


und. .06 


und. .20 


.6o-i .00 




low 


2.51% 


. 110% 


.62% 


• 41 % 




.24% 


3 •18% 


Gas Engine Cylinders: 










Sug. 2.25-2.75 


und. .08 


•50-. 70 


•50-. 70 








1 . 45 % 






.65% 




















1 .98 


. 090% 


.84% 
.40 


.63 






Housings for Rolling Mills 












I. 21 


.117 


• 35 


1 . 40 % 


3 74% 


1.00-1.25% 


. 085 % 


• 65% 


.75% 






low 


1 .00-1 . 25 


.04-. 08 


. 20-. 40 


.70-. 80 


. 60- . 80 


3.00-3. 10 


Sug. 1. 00-1.25 


und. .08 


.20-. 30 


.80-1.0 






low 


Sug. I.00-I.7S 


und. ,08 


.20-. 40 


.70-. 90 




3.00-3.30 




























Hydraulic Cylinders, Heavy: 










Gears, Heavy: 












1 . 00 % 


• 050 % 


.30% 


.60% 


I 


• 10% 


2^50% 


1 . 40 % 


. 060 % 


• 10% 


.40% 






.90 


.136 


• 39 


.25 


I 


44 


3 34 


.94 


.150 


■43 


.31 


1.47% 




.80-1.50 


.07-. II 


•35-50 










1.60 


.080 


.40 


.60 




3.50% 


1. 12 


.085 


.40 


.70 




.70 


3 50 


I .50-1.75 


.080 


.40-. 60 


.50-. 70 






• 95 


. 100 


.30 


.90 




.80 


3 40 


I.00-I.2S 


.075 


.40 


.80-1.0 




very low 


115 


und. .08 


• SO 


.60 


1 


15 




I .40-1 .60 


.04-. 08 


.30-. 50 


.40-. 60 


.50-. 80 


3 ■ 20-3 . 40 


.90-1 .20 


.06-. 08 


. 30-^50 


.80-1 .0 


. 80-1 .0 


2.90-3. 10 


Sug. 1,00-1.50 


.08-. 10 


.30-. 50 


.8 


3-1.0 




low 


Sug. .80-1.20 


und. , 10 


.20-. 40 


.80-1 .0 






low 



364 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table i. — Chemical Composition 

Silicon Sulphur Phos. Mang. 

Hydraulic Cylinders, Medium: 



Sug. 



1 . 40 % 


.060% 


.10% 


.40% 


1.90 


.074 


• 50 


.6s 


1.62 


.08 


• SO 


.60 


1-75 


.070 


.40 


.ss 


20-1 .60 


und. .09 


.30-. so 


.70-. 90 


rfOLDS AND 


Stools: 






1.20% 


. 060 % 


.10% 


.40% 


1.67 


.032 


.09 


.29 


25-1.50 


und. .06 


und. .20 


.60-1.0 



Sug. 



Locomotive Castings, Heavy: 

1.40-2.00% und. .085% und. .60% und. .70% 



SITION 


OF Iron 


Castings for Various Purposes — 


[Continued) 






Comb, 
carb. 


Total 
carb. 


Silicon Sulphur 
Ornamental Work: 


Phos. 


Mang. 


Comb, 
carb. 


Total 
carb. 






4.19% .080% 


1.24% 


.67%. 


.03% 


2.88% 






2.51 .110 


.62 


.41 


.24 


3.18 






2.2s 


.60-. 90 








.50% 


low 


Sug. 2.25-2.7S und. .08 
Permanent Molds: 


.60-1.0 


.50-. 70 




3.30% 






2.15% .086% 


1.26% 


.41% 


.13% 






2.02 .070 


.89 


■ 29 


.84 


3.60 






Sug. 2. 00-2. 75 und. .07 


.20-. 40 


. 6o-i .0 






• 43% 


3.87% 


Permanent Mold Castings 
2.00-3 .00% 






3 


00-4.00% 



Sug. 1.S0-3.00 und. .06% 
Piano Plates: 



und. .40% 



I. 25-1. 50 


.06-. 08 


.40-. 60 


.45- -60 


.50-. 70% 


3-50% 


2 . 00%, 


low 


.40% 


.60% 






1.62 


.098 


.40 


.49 






Sug. 2.00-2.25 


und. .07 


.40-. 60 


.60-. 80 






Sug. 1.25-1.50 


und. .08 


.30-. 50 


.70-. 90 






Pillow Blocks: 
























1 .60 % 


.040% 


.55% 


■ 55% 


.30% 


3.50% 


Locomotive Castings, Light: 










Sug. I. 50-1. 75 und. .08 


.40-. 50 


.60-. 80 






1 . 40-2 . 00 % 


und. .085% 


und. .60% 


und. .70% 


















1.50-2.00 


. 06- . 08 


.40-. 60 


.45-. 60 


• 45-. 55% 


3^50% 


Pipe: 












Sug. 1.50-2.00 


und. .08 


.40-. 60 


.60-. 80 






2.00% 
2.00 


. 060 % 
.060 


.60% 

I. 00 


.60% 
.60 






Locomotive Cyli 


MDERS: 










Sug. 1.50-2.00 


und. . 10 


.50-. 80 


.60-. 80 






i.2S-i.75% 
1.40-2.00 


und. .10% 
und. .085 


und. . 90 % 
und. .60 


und. .70% 






Pipe Fittings: 
2.88% 












1. 25-1. 50 


.06-. 08 


.40-. 60 


.45- -60 


.50-. 70% 


3 • 50 % 




.41% 


1 . 10% 






I . oo-i . 40 


und. .11 


.40-. 90 


.40-. 90 






1.70 


.058 


• 50 


■ 73 


1. 16 


4.18 


1 .41 


.092 


.38 


.39 






2.51 


.110 


.62 


.41 


.24 


3.18 


1.56 


.061 


.45 


.78 






Sug. 1.75-2.50 


und. .08 


.50-. 80 


. 60- . 80 






Sug. 1. 00-1.50 


.08-. 10 


.30-. 50 


.80-1.0 






Pipe Fittings for Superheated Steam Lines: 


















1-72% 


.085% 


.89% 


.48% 


.17% 


2^45% 


Machinery Castings, Heavy 










I. 40-1. 60 


. 06- . 09 


.20-. 40 


•45-- 75 




3. 00-3. 25 


1 . OS % 


.110% 


.54% 


.35% 


.33% 


2.98% 


Sug. 1. 50-1. 75 


und. 08 


.20-. 40 


.70-. 90 




low 


.85 


.030 


.35 


.92 






Piston Rings: 












.80-1.50 


.030-. 050 


.35- -SO 




















.90-1.50 


.09-1.2 


.15- -40 


.20-. 80 


.10-. 30 


2 .50-2.90 


1.35% 






■ 40% 






1.85 


.100 


■ SO 


.60 




3-50 


1.60 


.08% 


1.15% 


■ 35 


.60% 




1.30 


.090 


.40 


.60 






1.50-2.00 


. 06- . 08 


. 40- . 60 


.45- 60 


■ 45-^55 


3. so 


1.85 


. 120 


.60 


• 45 




3-40-3.55 


Sug. 1.50-2.00 


und. :o8 


.30-. 50 


.40-. 60 




low 


1-75 


. 100 


.50 


.70 


.80 


3.65 


Plow Points, Chilled: 










Sug. i.oo-i.so 


und. . 10 


.30-. 50 


. 80-1 . 




low 


1 . 20-1 . 40 % 




low 






low 














1.20 


. 090 % 


■ 30% 


■ 50% 


1.20 


3-20% 


Machinery Castings, Medium: 








■ 75 


.090 


• 30 


■ 30 


3 00 


3^20 


1.83% 


.078% 


.50% 


.31% 


.43% 


2.93% 


1.20 


.080 


.30 


I 25 




3 50 


2.25 


.080 


.SS 


.60 






Sug. .75-1.25 


und. .08 


.20-. 30 


. 80-1 . 






I .60 


.060 


.66 
































Propeller Wheels: 










2. 29 


.071 


.66 


.49 


















1.60 


.090 


■ SO 


.60 






i.iS% 




.32% 


.51% 


.60% 




2. 10 


. no 


.67 


• SO 




3.40-3.55 


1.40 


low 


.20 


.40 






2.25 


.060 


.75 


.55 






Sug. 1. 00-1.75 


und. .10% 


. 20-. 40 


■ 60-1 . 




low 


2.00 


.100 


.75 


.50 


■ 75 


350 


Pulleys, Heavy: 












1.76 


.075 


.63 


.79 


.56 


3.68 


1.75% 


. 040 % 


■ 55% 


.55% 


■ 30% 


3^57% 


2.00 


.100 


.50 


• SO 


■50 


3.60 


2.40 


.060 


.60 


.60 




3^75 


2.35 


.075 


• 4S 


.6s 


.30 




Sug. 1.75-2.25 


und. . 09 


.50-. 70 


.60-. 80 






1.80 


.060 


.80 


• SO 


■ 70 




Pulleys, Light: 












2.06 


.075 


.78 


.47 




3.45 


2.20-2.80% 


und. .08% 


und. .70% 


und. . 70 % 






1.40 


low 


.20 


.40 






2.40 


und. .08 


■ 95 


■ 70 






2.00 


.030 


.70 








2.72 


.040 


• 50 


.66 






1.8s 


.08 


.60 


.50-. 60 


■ SO 


3 ■25-3^50 


2.S2 


.075 


• 77 


.68 




3-37% 


I.S0-2.I0 


.08-. 09 


.40-. 80 


.20-. 60 


.10-. 40 


2 . 60-3 . 20 


3.35 


.089 


.70 


• 47 




3.42 


1.80-2. 10 


und. . 09 


.40-. 90 


.40-. 90 






2.25 


.040 


■ 55 


• 55 


■30 


3.57 


Sug. 1.50-2.00 


und. .09 


.40-. 60 


.60-. 80 






2.15 


.080 


• 70 


.60 


■ 40 


3.SS 


Machinery Castings, Light: 










Sug. 2.25-2.75 


und. .08 


.60-. 80 


.50-. 70 






2 . 04 % 


. 044 % 


.58% 


.39% 


■ 32% 


3.84% 


Pumps, Hand: 












2.2s 


.080 


.70 


• SO 


.20 


3-55 


2.30-2.75% 


und. .08% 


. 60-1 . % 


.30-.S0% 






2.76 


.037 


1.19 




■ 13 


3.66 


Sug. 2.00-2.25 


und. .08 


.60-. 80 


.50-. 70 






2.49 


.097 


.90 


.42 




3-40 


Radiators: 












2. SI 


.084 


.62 


.61 




3.46 


2.15% 


low 


.80% 


■ 45% 


• 50% 


3 • SO % 


2. SO 


.100 


.60 
.65 


.70 




350 


2.4s 


.104% 


■44 


.40 


■ 35 


3 40 


3.00 


.060 


.50 




3.50 


Sug. 2.00-2.25 


und. .08 


.60-. 80 


.50-. 70 


.50-. 60 




2.40 


.oso 


.47 


.59 


















2.8s 


.064 


.67 


.65 






Railroad Castings: 










2. 52 


.062 


.66 


.68 






2.20-2.80% 


und. .08% 


und. .70% 


und. .70% 






3.1s 


.050 










I. 40-1. 80 


.06-. 08 


.50-. 80 


.45-. 60 


.40-.6s% 


3.50% 


2. so 


.100 


.70 


.60 




3 ■40-3- 55 


2.25 


.050 


.60 


■ 75 






2.30-2.80 


.06-. 08 


.60-. 1.3 


.20-. 40 


. IO-.60 


3 . 00-3 . 60 


1.75 


.070 


.85 


.60 






Sug. 2. 00-2. SO 


und. .08 


.50-. 70 


•so-. 70 






Sug. 1.50-2.25 


und. .08 


. 40- . 60 


.60-. 80 







CAST-IRON 



365 





Table 


I . — Chemical Composition 


OF Iron C 


ASTiNGS FOR VARIOUS PURPOSES — {Continued) 






Silicon 


Sulphur 


Phos. 


Mang. 


Comb, 
carb. 


Total 
carb. 


Silicon 


Sulphur 


Phos. 


Mang. 


Comb, 
carb. 


Total 
carb. 


Rolls, Chilled: 












Steam Cylinders, Medium- 


-{Continued) : 








. 50-1 . 00 % 


.01-. 06% 


.20-. 80% 


.15-1. 5% 


2.60-3.25% 


2.00 


.070 


.30 


.60 






.80 


. 100 


.88 


.16 


.91 


2.84% 


I. SO 


.070 


.75 


.70 




3.50 


.71 


.058 


■ 54 


.39 


1.38 


3.00 


I.S9 


.109 


.60 


.38 




3.34 


.6s 


.050 


.25 


1.50 


.63 


3. SO 


1.86 




.29 


.55 


.52 




Sug. .60-. 80 


. 06- . 08 


.20-. 40 


I . o-i . 2 




3.00-3.25 


1.90 
I.S6 


.074 
.o6i 


.50 

.45 


.65 
.78 






Rolls, Unchillkd (sand cast) 










Sug. 1.25-1.75 


und. .09 


.30-. SO 


.70-. 90 






.75% 


. 030 % 


.25% 


.66% 


1.20% 


4. 10% 


Stove Plate: 












Scales: 












2 . 90 % 




• 73% 


1.40% 






1.67% 




1.92% 


1.90% 

.80 
1 .60 






2.59 


.072% 


.62 


.37 


.35% 


3.30% 


2. 12 




.63 






3.19 


.084 


1. 16 


.38 


■33 


3.41 


1.70 








2.75 


.050 


1. 00 


.80 


.18 


3.38 


Sug. 2 . 00-2 . 30 


und. .08 


.60-1.0 


■SO-. 70 






2.79 


.077 


1.40 


.32 


.20 


3.22 


Slag Car Castings: 










2.51 


.1X0 


.62 


•41 


.24 


3.18 


1.76% 


.075% 


.63% 


.79% 


.56% 


3 . 68 % 


2.76 


.071 


.63 


.63 


.37 


3.50 


2.00 


.030 


.70 








2.76 


.084 


.65 


■54 






Sug. I.7S-2.00 


und. .07 


und. .30 


.70-. 90 






2.50 
2.60 


.060 
.050 


1. 00 

.60 


.60 
.60 






Soil, Pipe and Fittings: 










2 . S0-3 . 00 


und. . 10 


.60-. 80 


. 40- . 60 




3 . 00-4 . 00 


2.00% 


.060% 


1 . 00 % 


.60% 






Sug. 2 . 2S-2 . 7S 


und. .08 


.60-. 90 


.60-. 80 






Sug. I.7S-2.2S 


und. . 09 


.so-. 80 


.60-. 80 






Valves, Large: 












Steam Cylinders 


, Heavy: 










1 . 20-1 . so % und. . 09 % 


.35-. 60% 


.50-. 80% 






1.41% 


. 092 % 


.38% 


.39% 






1. 00 


.100 


.50 


.90 






.95 


.100 


.30 


.90 


.80% 


3.40% 


1.67 




.26 


•45 


.69% 




1. 10 


.136 


.43 


.33 


.99 


330 


Sug. I.2S-I.7S 


und. .09 


.20-. 40 


.80-1.0 






1. 00 


.080 


.20-. 30 


1. 00 


• 75 


3.00 














I. 35-1. SO 


.080 


.SO 


.75 




3.65 


Valves, Small: 












I. 30-1. 40 


. 04- . 08 


.40-. 50 


.70-. 80 


.70-. 80 


3.00-3.20 


1.70% 


.058% 


.50% 


.74% 


1.16% 


4- 18% 


.90-1.20 


.09-. 12 


.20-. 40 


.70-. 90 




und. 3.50 


2.23 


.075 


.67 


.67 






Sug. I.00-I.2S 


und. . 10 


.20-. 40 


.80-1.0 




low 


Sug. 1.75-2.25 


und. . 08 


.30-. so 


.60-. 80 




low 


Steam Cylinders 


Medium: 










Water Heaters: 












1 . 66 % 


. 06s % 


.70% 


.90% 






2.15% 


. 050 % 


.40% 


.50% 






1.60 


.063 


.72 


.85 






Sug. 2.00-2.2S 


und. . 08 


.30-. so 


.60-. 80 






1.70 


.070 


.70 


.75 






Wheels, Large: 












1.70 


.075 


.60 


.92 




3.50% 


2.10% 


. 040 % 


.40% 


.70% 






1.40-2.00 


.085 


.70 


.30-. 70 






Sug. 1.50-2.00 


und. .09 


.30-. 40 


.60-. 80 






1.50-2.00 


und. .08 


.35-. 60 


.50-. 80 


















1. 40-1. 60 


und. .09 


. 40- . 90 


. 40- . 90 






Wheels, Small: 












1. SO- I. 6s 


.080 


.60 


.60-. 70 






2. 10% 


. 050 % 


.40% 


.50% 






1. SO- I. 80 


.070 


.43 


.76 






1.60 


.083 


.60 


.39 






1.85 


.080 


.60 


.50-. 60 


.50% 


3.25-3.50 


Sug. I. 75-2. 00 


und. .08 


.40-. so 


.50-. 70 






1.75 


.100 


.65 


.55 




3. 40-3. 55 














1.32 


.136 


.43 


.33 


.99 


3.30 


White Iron Castings: 










1. 12 


.085 


.40 


.70 


.70 


3. SO 


.50% 


.150% 


.20% 


.17% 


2.90% 




2.00 


. 100 


• SO 


.70 


.40 


3.50 


.90 


.250 


.70 


.50 




2.50 



Table 2. — Tests of Malleable Castings 
Tension Tests 



Compression Tests 



Section 



Area 



Tensile 
strength, lbs. 
I per sq. in. 



Round 
Round 
Round 

Round 
Round 
Round 

Square 
Square, 
Square 

Square, 
Square 
Square 

Rect.. 
Rect.. 
Star... 

Star... 
Star... 



.793 


43100 


.817 


43000 


.801 


43400 


.219 


41130 


.202 


44700 


. 210 


43050 


.277 


36700 


.277 


38100 


.283 


37S20 


1.040 


38460 


1.030 


38000 


1.050 


37860 


.244 


37600 


.218 


37250 


.584 


34600 


■ 523 


36500 


•575 


37200 



Elongation 
in 8 ins., 
per cent. 



8.70 
S.87 
6.21 

7.70 
13.00 

S.80 

4.70 
3-72 
4.21 

4.10 
1-95 
2.38 

3.87 
3.22 
4. 20 

7. 20 
4.80 



Reduction 

area, 
per cent. 



3-75 
4.76 
3.98 

3-40 

3-S2 

2.00 
3.00 
2.71 

3.30 
2.88 
2.94 

3.80 

4.70 
3.10 

2.50 
3.50 



Section 


Area 


Length, 
ins. 


Compressive 

strength, 
lbs. per sq. in. 


Final 
area 


Round 

Round 

Round 


.83s 
.847 
.801 

• 213 
. 209 
. 204 

.282 
.263 
.254 

1.051 
1.040 
1.048 

•453 
■436 
■457 


IS 
IS 
IS 

7-S 
7-5 
7^5 

7^5 
7^5 
7-5 

IS 
15 
IS 

15 
IS 
IS 


32950 
31700 
33240 

33300 
32600 
34600 

32580 
33200 
31870 

29650 
30450 
29700 

31900 
32200 
30400 


.883 
.901 
.886 


Round 


. 222 


Round 


. 221 


Round 


. 215 


Square 


. 201 


Square 


. 272 


Square 

Square 

Square 

Square 

Star 


.278 

1.070 
1.066 
1.070 

.465 
.448 


Star 


Star 


.467 







366 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Malleable Castings 

Malleable castings, made by the Buhl Malleable Co., were tested 
and reported on by C. M. Day {Amer. Mack., Apr. s, igo6) and from 
the report the following facts are taken. Tensile and compressive 
tests were made on round, square, rectangular, and cruciform sec- 
tions. The rounds were of | and i in. diameters, the squares of | 



fact that the iron casts better in round sections or that the outer 
skin of a malleable casting is not its strongest part. Every one of the 
round bars showed a perfect fracture, with a good skin and a smooth 
velvety interior. It is generally thought that malleable iron does 
not cast well in round sections and that a section with narrow ribs 
is the strongest possible section. This is why the star-shaped section 
was made to test. In only one of the star pieces did the fracture 



37, 

30, 

.26, 
a 

"122, 

c 

" ^' 

a 15, 

.oil, 
Hi 

7, 







" 




















" 




■ 




" 














■ 








"" 


" 




1 









































— 


-" 


~ 
















■> 1 
























— 






















































'"' 
















































































































^ 
























































































































f 




































































































































































































































































































































































































































































































































































































































































































































J 


n 


,ke>f 




























s 

c 




^ 


3 




1 




=> 


c 
c 


i 




i 


i 


4 




i 


c 


1 


I 


i 






i 


t 


I 




i 


i 


s 



33,000 




■ 












- 


~ 






























^,'- 


-'■' 


















/ 


















/ 




















1 








































1 








































1, 


















1 




















1 




















1 
















S 16,500 




















a 


/ 


















J. 13,750 


/ 


















T 11 fifin 


1 






































R 9';n 


1 




















1 


















'^ ^ifin 
















































































\L 


_ 








- -i-k}Y?^^ 







PI 9 3 



Star Section, Length 8 Ins. 



Star Section, Length 15 Ins. 



37,500 

33,750 

30,000 

^ 26.250 

. 22,500 

a* 

^ 18,750 

<j 

'^15,000 

J 11.250 

7,500 

3,750 



44,000 

41,ffi0 

I 37,500 

\ 33,750 

f 30,000 

; 26,250 

'22,500 

I 18,750 

15,000 

11,250 

5,500 

3,750 















































































; 




C JL 









_ 












^ 












































^^ 












































/ 












































/ 














































4 












































/ 












































' 




























































































1 












































1 


































































1 


































































































































1 




















































































































































































lie 


B 










"J 












• S 


O C30 

i § 

auare 


i i 

Section 


O O 

, 1x1 


CO Q 

g i 

Inch, L 


o o c 

engthS 


P GO 

Ins. 


» O 


































































































































n 































































































— L 








-_ 


n 




J_ 


J 




__ 




1 











- 


- 
















-^ 


-h 


-| 


1 — 


-[- 


- 


4^ 


^ 

































^ 


— ■* 
















































































































^^ 








































^ ^ 








































^ ^' 


' 














































































,' 










































( 










































































































































































































































































































































































































































































































































































































i \ 


\ 


i 


g ■ 


%\ 


V^ 


% 


g 


s 




1 1 


i ; 


\ 


i 


i; 


1 


1 1 


8 















































































1 1 










_. 




















^ 






' 
















* 












J 


-^ 






































^ 


' 








































/ 


































99 f^n 








r 












































































20 000 
























































































/ 










































/ 




















































































/ 






































12 500 




















































































in nnn 


/ 










































/ 




















































































y 










































,/ 










































r 




















































































































































eM 

















g g 2 d 3 s 



Square Section, 1x1 Inch, Length IE Ins. 



33.000 














































































__ 


^ 


-^ 


1 — 


— 




















" "^ . 


















, 






"^ 


































30,250 














^ 




















































, 
























































/ 
























































1 






















































1 
















































H oo nnn 








/ 
























































/ 














































































































/ 


















































^ ifi f^nn 






/ 


















































p* 




























































1 
























































1 




















































■^ 11 nnn 




1 














































































































/ 
























































/ 
























































\ 
























































i 
























































' 
















































































































































_. 




^ 


,^ 


_ 


_ 







iH c^ eo 



s a a s 3 



Round Section. Diameter 1 Inch, Length 8 Ins. Bound Section, Diameter 1 Inch^ Length 15 Ins. 

Tension tests. Compression tests. 

Fig. I. — Stress-strain diagrams of malleable castings. 



and I in. sides, the rectangles JXi in., while the crosses were i in. 
wide with 4 ribs } in. thick. The results are given in Table 2. 

Mr Day makes the following comments on the tests: It was found 
that the round section gave the best results both in the i-in. and the 
2-in. sizes, as well as in both tension and compression tests. The 
round section, besides having a greater tensile strength, also had a 
greater elongation and reduction in area. This may be due to the 



show up well. In the others there were signs of shrinkage. It 
seems rather strange that these results were quite the reverse from 
what were expected; the round section being the strongest, next the 
square, then the rectangular section, and lastly the star; varying 
inversely as the perimeter exposed. 

Fig. I gives representative stress strain diagrams from the 
tests. 



CAST-IRON 367 

Following are the specifications of the Bureau of Steam Engineering The minimum tensile strength of the material must be 40,000 lbs. 

of the U. S. Navy Department for malleable castings: per sq. in., and the elongation at least 25 per cent, in 2 ins. 

Malleable-iron castings for which physical requirements are speci- Castings must be true to pattern, free from blemishes, scale, and 

fied may be made by either the open-hearth or the air-furnace process. shrinkage cracks. 

Sulphur must not exceed .06 per cent., and phosphorus must not Malleable castings must be neither "over" nor "under" annealed, 

exceed .225 per cent. They must have received their full heat in the oven at least 60 hours 

The transverse breaking load for a bar i in. square, loaded at the after re3,ching that temperature, and shall not be dumped until they 

middle and resting on supports i ft. apart, shall be not less than are at least " black hot." 
3,000 lbs., deflection being at least 2 in. 



STEEL 



For steel for springs, see also Springs. 

For steel for boilers, see Steam Boilers. 

A list of heat treatments will be found at the end of the section. 

The composition and physical properties of steel for a large variety 
of purposes are given in Table i of representative specifications by 
C. A. TuppER (Amer. Mack., Mar. 24, 1910). The table is the 
result of an inquiry extending through two years of time into the 
practice of many of the largest machinery building and structural 
work companies and has been submitted to a number of large users 
and manufacturers of steel, including engineers and chemists^ It 
represents advanced practice. 

With the table Mr. Tupper makes some observations on the trend 
of practice in these matters from which the following extracts are 
taken : 

For carbon-steel masses of considerable weight rotating at great 
speed, such as the body of the spindle of a horizontal steam turbine 
running up to, say, 3600 r.p.m., metal of high endurance, having a 
tensile strength of 85,000 to 100,000 lbs. and an elastic limit of 
55, 000 to 70,000 lbs. is now required, with elongation in 2 ins. of 25 
to 30 per cent., contraction 40 to 48 per cent. The same is also true 
of other rotating elements turning at much slower speed but subjected, 
at the same time, to heavy pressure or resistance, such as the runner 
of a hydraulic tiirbine or the impeller of a centrifugal pump. 

For pumping engines the conclusions of manufacturers and engi- 
neers vary, some being of the judgment that an ultimate tensile 
strength of 55,000 to 60,000 lbs. is ample, though an elastic limit of 
at least 30,000 to 40,000 lbs. is required. Elongation in 2 ins. (the 
present tendency apparently being to adhere to that standard) of 
20 to 28 per cent, and contraction of 35 to 45 per cent, between 25 
and 35 per cent., being considered satisfactory, are allowed for under 
these conditions. Sulphur and phosphorus should not exceed .05 
per cent, each, and some users place .02 to .03 per cent, as the limit. 

In the purchase of steel billets, manufacturers find, as a matter of 
shop economy, that it is advisable to carry, as far as possible, stocks 
of generally suitable characteristics, rather than divide specifications. 

As to the carbon content, the grade of billet best adapted to engines 
or other machinery having a reciprocating motion, where no excessive 
strains or stresses are likely to be set up, should run from 25 to 33 
points. These can be machined to better advantage than the 50- 
point carbon steel often recommended. In annealing, such steel is 
customarily heated to from 1650 to 1800 deg. Fahr. for 10 to 12 hours, 
and allowed to cool very slowly. A higher temperature of heating 
is permissible, but not above the extreme limit reached in forging 
and most manufacturers fear to go above 1850 deg. Fahr. For the 
greater tensile strengths required in the operation of such machinery, 
up to, say, 70,000 lbs., the use of 33- to 35-point carbon steel is to 
be recommended. 

It should be recognized that overheating is more injurious to high- 
carbon steel than to low; also, that if the reduction in hammering 
or pressing has been great enough so that the coarsening of the grain 
at the high temperature to which the steel has been heated prior to 
such hammering or pressing has been well effaced, the harm of over- 
heating increases not only with the distance above the final point 
of recalescence to which the temperature is raised, but also in direct 
ratio to the percentage of carbon. Steel that has become danger- 
ously crystalline may — in many cases, at least — be restored to 
specification conditions, or better, by reheating and manipulation; 
but the necessity for this ought to be avoided just as far as possible. 

For less particular service, such as ordinary shop machinery, the 



usual stock billets of 15- to 25-point carbon, having a tensile strength 
of about 5 5, 000 lbs., elastic limit practically negligible, and purchased 
with no more than ordinary physical or chemical requirements, are 
regarded as sufficient for all purposes. 

For special machinery parts subject to excessive wear, such as 
crank pins, valves, compression rods, table clutches, return cranks, 
etc., a carbon content of at least 50 to 55 points is needed. Such 
f orgings, or even castings, should also show an ultimate tensile strength 
of at least 75,000 lbs., if not annealed, and a minimum of 65,000 to 
75,000 lbs. if thoroughly annealed, with elastic limit, elongation and 
contraction correspondingly high. 

For carbon-steel shafts and axles subjected to heavy strains (the 
tendency now being, however, to use some special alloy such as vana- 
dium) the requirements of machinery and truck builders vary con- 
siderably, but a content of .35 to .45 per cent, carbon, manganese not 
above .45 to .50 per cent., silicon .05 per cent., sulphur not to exceed 
.03 per cent, and phosphorus within .04 per cent, are considered safe. 
In actual manufacturing, where such steel is used, these standards 
have not heretofore been very generally realized, but the increasing 
frequency of accidents and breakdowns, under the severe require- 
ments of modern power, miU and traction service, is compelling 
builders to draw the lines tighter and tighter. 

It should be remembered, however, that the greater the percentage 
of carbon in billets, the higher is the temperature required in forging^ — 
not less than 1650 deg. Fahr. for high-carbon steel; hence costs may 
be kept down by using billets as low in carbon as the requirements 
of the finished product will permit. 

The selection of the proper steel for crank pins has always been a 
vexed question, particularly where such parts are subjected to heavy 
stresses and the force of sudden shocks — as in the case of a reversing 
engine for rolling-mill service, when the rolls bite the ingot. 45- 
to 55-point carbon steel, with tensile strength of 75,000 lbs. will 
meet ordinary requirements of heavy duty, but for continuously 
severe operating conditions, as in the instance above cited, a special 
alloy steel such as chrome vanadium of high tensile strength and great 
toughness is desirable. An example of this will be found in the accom- 
panying table. 

For large traveling cranes a leading builder states that the grade 
of castings best suited to his requirements is of open-hearth steel, 
having an ultimate tensile strength between 66,000 and 68,000 lbs., 
elastic limit of 33,000 to 35,000 lbs., and chemical content of .24 per 
cent, carbon, .48 per cent, manganese, .20 per cent, silicon, .06 per 
cent, phosphorus, and .04 per cent, sulphur. Steel for truck wheels 
carries .03 per cent, carbon, .59 per cent, manganese, .63 per cent, 
silicon, .454 per cent, phosphorus and .152 per cent, sulphur; truck- 
wheel tires .44 per cent, carbon, .78 per cent, manganese, .28 per 
cent, silicon, .38 per cent, phosphorus and .048 per cent, sulphur. 
The figures given are taken from actual tests of steel that, all things 
considered, has proved most satisfactory. For pinions, armature 
shafts, truck axles, etc., a rolled open-hearth steel, 30 to 35 points 
carbon is used, and, for the cross shafts on the crane bridges, turned 
and ground shafting that runs high in carbon. The steel and other 
metal used in the construction of the crane-motors is such as electrical 
manufacturers employ in building high-grade motors for heavy duty. 

Some years ago attention was directed, by tests of the new armor 
plate made for naval vessels, to the great tensile strength, toughness 
and ductility of the then little-known nickel steel. Experiments 
with shafts, axles, spindles and other parts of vehicles or machinery 
rotating at considerable speeds also showed that it possessed the 
368 



STEEL 



369 



extremely valuable quality of resistance to fatigue, that it readily 
withstood shocks of all kinds as well as those of shell impact and that 
steel high in nickel was practically not subject to cracking or similar 
rupture. 

Forgings made from nickel steel have, in general, the same require- 
ments as those of ordinary carbon steel of the same class, with the 
addition of 2.5 to 3.5 per cent, nickel. This raises the ultimate 
strength anywhere from 10,000 to 80,000 lbs. per. sq. in. (depending 
upon the other characteristics of the metal) , and the elastic hmit in 
proportion, without sacrificing the ductihty. In fact, the last-named 
is usually increased. 

The proportions of nickel commercially usable in large forgings 
intended for machinery parts are limited by a curious property of 
the metal, viz., that its mixture in high-carbon steel up to and beyond 
a certain percentage increases the hardness of the steel. Between 
these two points there is a small range, the working limits of which 
are about as stated above, in which nickel steel can be machined to 
the best advantage. For low-carbon nickel steel the range is some- 
what greater, and it can be easily worked cold with nickel under 
3 to 5.5. per cent. 

Specifications for small forgings, which are subsequently to be 
reduced by grinding, may call for as much nickel as is desirable. 
Steel can be forged readily, without regard to its nickel content. 

Nickel-steel forgings, also, do not ordinarily need annealing, 
except where the latter is intended to be carried to the tempering 
stage, as there is already sufl6ceiht homogeneity in the structure. 

Allowing, however, for the truth of all that has been said above, 
it is a fact indicative of the rate of modern progress that for many 
purposes nickel steel has already had its day, and the addition or 
substituiton of other alloys, to form such combinations as chrome 
nickel, nickel vanadium and chrome vanadium steels, has become 
general. Vanadium has an even more favorable effect than nickel 
alone and increases the ductility and toughness of the steel containing 
it. This is now used for engine, locomotive and automobile parts, 
as well as in bridges, viaducts or other structures where there is 
much vibration. Chrome-nickel enters similarly into the composi- 
tion of machinery steel. Titanium appears to give results even 
better than those mentioned (although this is not generally con- 
ceded) and at the same time does not appear in the finished pro- 
duct. It apparently acts as a scavenger to remove impurities. 

Advocacy of vanadium steel is particularly strong at present 
among the expert metallurgists employed by machinery builders. 

A chrome-nickel, chrome-vanadium, chrome-nickel-vanadium or 
other special alloy steel used for the rotating parts of extremely 
compact high-speed machinery is ordinarily required to have an 
ultimate tensile strength of about 120,000 lbs., with elongation of 
16 to 30 per cent, in 2 ins. and the extremely high elastic limit of 
100,000 lbs. Where such machinery is subjected to extraordinary 
stresses, however, the tensile strength may run as high on test as 
165,000 to 175,000 lbs., with elastic limit very little under these 
figures and elongation up to 32 per cent, or beyond. 

In the forging of these alloyed steels, much more than in their 
machining, special skill is usually required, particularly when they 
are high in silicon; and the temperature must be maintained at above 
200 to 250 deg. Fahr. under the melting-point, thus necessitating, 
with a large piece, several reheatings, it being a well-recognized fact 
that forgings made at a temperature just above the final recalescence 
point are always strongest. A small variation in the proportioning 
of the alloys makes considerable difference in forging conditions; 
hence machinery builders find it necessary to check their specifica- 
tions very closely with the results of actual tests in the shop, before 
arbitrarily demanding this or that from the steel manufacturers. 
Failure to proceed very cautiously on that basis has, not infrequently, 
led to heavy losses. 

Heat treatment of steel in the shop — and there is nothing which 
will be more likely to influence specifications in future — comes under 
24 



two heads: annealing and tempering. The former, only, will be 
considered here, and without further reference to tool steel, which 
is a subject quite by itself. 

Until very recently, annealing has not been given by builders of 
high-speed or heavy machinery the attention it deserves, and, even 
to-day, the art is practised to a far less extent than the average 
reader probably supposes. 

By the annealing of steel before it leaves the mills, a uniformity 
of structure is given to billets which does much to relieve or prevent 
subsequent internal strains; and re-annealing of ordinary carbon 
steels in the shop, after forging or machining, is, as a rule highly 
desirable, for the reason that, at each of the three periods of recales- 
cence or "absorption" in heating and cooling (i.e., six periods both 
ways) an actual re-arrangement of the molecules takes place while an 
increase in temperature is temporarily arrested, and such disturbance 
as there may have been of the physical structure of the piece, tending 
to crystallization, gives way to a restoration of the desired conditions. 

In annealing, furnaces especially designed for the purpose, and 
preferably gas-fired, should be provided. The too prevalent tendency 
among machinery builders.who do their own forging, to use an ordi- 
ary forge fire in annealing, leads to some pernicious results — results 
which, nevertheless, have to be taken into account in the preparation 
of specifications. 

A slow raising of the temperature to the final point of recalescence, 
with careful observation by means of a pyrometer, and even slower 
cooling, are essential to good practice. 

Chemical analysis, which originally met with so much opposition 
when introduced in metal-working plants, has been swung to the 
other extreme, so much so that undue reliance has, of late, been placed 
upon it in many quarters. Chemistry, property applied, is, of course, 
essential in determining the characteristics of steel; but it does not 
take the place of physical tests, and microscopic, or photo-micro- 
scopic apparatus will determine things that are altogether outside 
the range of chemistry. 

In testing, the appearance of a crystalline fracture, or any trace of 
crystallization, should be sufficient to at once cause the rejection of 
a forging, if annealed steel is required; but for unannealed forgings 
crystalline fractures may be regarded as normal if the steel is high 
in carbon. A circumstance to be here observed, and one often over- 
looked by machinery builders, is the fact that forgings made in custom 
shops will not be annealed in advance unless annealing is specified; 
also that they wiU not be physically tested unless such tests are asked 
for or they are purchased under definite physical specifications. In 
testing, a bar is ordinarily severed from an end having the fuU diam- 
eter of the forging, the cut being made in an axial direction about 
half way of the radius, according to the U. S. Naval standard. 

Since the development of the so-called high-speed press, the 
forging of steel by means of continuous hydraulic pressure, rather 
than by the use of a steam hammer, has been coming more and more 
into favor among builders of machinery subjected to severe stresses, 
for the reason that it results in a more uniform, homogeneous and 
reliable product. 

No matter how powerful a steam hammer may be, the force of 
its blow is not ordinarily felt very far below the surface of a forging, 
and, even with the exercise of the greatest skill, the depth of com- 
pression forms a very irregular line around the center of the piece, 
leaving the interior far from solidified; whereas, with a press, the 
molecular structure of the forging is almost uniformly condensed, 
the compression being felt through its diameter. The press will also 
work very close to a shoulder, or even forge a squared up shoulder, 
thereby saving metal and time in machining. 

These facts are becoming so well recognized that the day of the 
large steam hammer is drawing to a close; henceforth, for heavy 
work, presses will be quite generally installed as new equipment is 
needed. They are mentioned here particularly on account of the 
influence which they are beginning to exert on specifications. 



370 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



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STEEL 



371 



One of the effects, which does not come under any of the other 
headings here treated, is a reduction (for pressed forgings) of 40 to 50 
per cent, in the area of the initial section as compared with that 
which must be specified for hammered forgings. 

Steel that has been forged by pressure and subsequently annealed 
shows greater homogeneity than hammered forgings and greater 



figures of a single analysis that is considered favorable; as it is prac- 
tically necessary for the steel manufacturer to have a certain amount 
of leeway in either direction. "We have learned," says the super- 
intendent of one prominent crucible-steel plant, "that a chemical 
specification that permits of no leeway has usually been prepared 
by a novice. If we had no range to work on we would have to go 



Table 2. — Physical and Chemical Characteristics oe Steel Forcings eor Engines of the U. S. Navy 

Note. — Class C forgings will not be tested unless there is reason to doubt that they are of a quality suitable for the purpose for which intended. Tests, if 
required, shall be made at the expense of the contractor, and may be made at the point of delivery. 



J 


Material 


Treatment 


Minimum 
tensile 

strength, 
lbs. per 
sq. in. 


Minimum 

elastic 

limit, 

lbs. per 

sq. in. 


Minimum 
elonga- 
tion, 
per cent, 
in 2 ins. 


Maximum 
percent- 
age of — 


Without 
showing 
cracks or 
flaws must 
cold bend 
about an 

inner 

diameter 

of— 


Suitable uses 


u 


P. 


S. 




H. G 

A 
B 

r 


Open-hearth 
nickel steel, 

Open-hearth, 
either nickel 
or carbon 
steel. 

Open-hearth 
carbon steel. 

Open-hearth 
or Bessemer 
steel. 


Annealed and 
oil-tempered. 

Annealed. Oil- 
oil-tempering 
optional. 

Annealed 
Annealed 


95,000 

80,000 
60,000 

52,000 


65,000 

50,000 
30,000 


21 

25 
30 

28 


• 05 

.05 
.05 


.05 

• OS 

• 05 


I in. 
through 
180°. 

I in. 
through 
180°. 

Hn. 
through 
180°. 

I in. 
through 
180°. 


Bolts and studs for all moving parts of main engines, shaft 
couplings, main bearing caps, thrust bearing side rods, 
main engine framing and moving parts of circulating 
pumps; connecting rods, caps and bolts; eccentric rods; 
main circulating pump engine working parts; piston rods; 
suspension links and link blocks; valve stems. 

Coupling bolts; crossheads and slippers; crank, thrust, 
line, stern tube, tail, and propeller shafts; main bearing 
cap bolts; outboard coupling; reverse arms and blocks; 
rotor shaft; thrust bearing side rods; turning engine worm; 
working parts, reversing gear; working parts, pumps. 

Bearer bars; Curtis turbine shaft; engine columns and tie- 
rods; H.p. relief valve stems; main steam valve stems; 
main bearing cap bolts; piston rod nuts; piston valve 
followers; pipe flanges; rotor drum and wheel; stole-plate 
wedges; swivel pins for crosshead; working levers and 
gears. 

Gland for cylinder liners; small parts of eccentrics; uptake 
and smoke-pipe forgings. 











Table 3. — Physical and Chemical Characteristics of Steel Castings for the U. S. Navy 

Note. — Class C castings will not be tested unless there are reasons to doubt that they are of a quality suitable for the purpose for which they are intended. 
Tests, if required, may be made at the building yard. The inspector will select a sufficient number of castings and have them crushed, bent, or broken, and note 
their behavior and the appearance of the fracture. 



Class 
symbol 


Chemical 
composition 


Physical requirements 




Not over — 


Minimum 
tensile 


Minimum 
yield 


Minimum 
elonga- 


Minimum 
reduction 


Bending test; 
cold bend (not 


Suitable uses 












P.- 


S. 


strength 


point 


tion 


of area 


less than) 










Lbs. per 


Lbs. per 


Per cent, in 


Per cent. 












sq. in. 


sq. m. 


2 ms. 








Special 


.04 


• 04 


90,000 


57, 000 


20 


30 


90 deg. about an 
inner diameter 
of r in. 




A 


OS 


.05 


80,000 

Maximum 


35.000 


17 


20 


90 deg. about an 
inner diameter 
of I in. 


Engine frame strongbacks; I. P. and L.P. pistons; pistons 
and followers for piston valves; reverse arms; separators; 
valve stem crosshead. 


B 


.06 


.05 


80,000 

Minimum 
60,000 


30,000 


22 


2S 


120 deg. about an 
inner diameter 
of I in. 


Boiler fittings; crosshead backing guide; cylinder and 
valve chest covers; engine bedplates; H.p. cylinder 
relief valves; I. P. and L.P. piston followers; main bear- 
ing caps and shoes; main steam valves; outboard coup- 
ling casing; pipe flanges; pipe flanges, slip joints, and 
safety and relief valves for superheated steam; piston 
rod and valve stem stuffing boxes; reducing valves; 
reverse shaft bearings; thrust horseshoes; valve stem 
guide and crosshead cap. 


C 


.06 


.05 












Unimportant castings. 















elongation. It also has greater toughness, offers more resistance to 
all manner of strains and has a higher elastic limit. 

It may be stated here also that, in all cases, permissible variations 
in the chemical analysis of steel, no matter for what purpose intended, 
should be given — rather than insisting upon strict adherence to the 



out of the business." At the same time the exercise of too great 
latitude in such matters should be carefully guarded against; and 
the proper relation both of chemical conditions and physical char- 
acteristics ought to be rigidly insisted upon where the character of 
the service demands it. 



372 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 2 gives the physical and chemical characteristics specified 
by the Bureau of Steam Engineering of the U. S. Navy Department 
for engine forgings. The specified treatment is as follows: 

All forgings shall be annealed as a final process, unless otherwise 
directed. All tempered forgings, if forged solid, and if more than 
5 ins. in diameter in any part of their lengths, not including collars, 
palms, or flanges, shall be bored through axially before tempering, 
and the bore shall be of sufiicient size to enable the manufacturer 
to get the requisite tempering effect. Forgings, such as crank 
shafts, thrust shafts, etc., may, previous to tempering, be machined 
in a manner best calculated to insure that the tempering effect 
reaches the desired portions. In this case, the inspector will decide 
upon the location of the test pieces if they cannot be taken in the 
manner hereinafter described. All forgings shall be free from slag, 
cracks, blowholes, hard spots, sand, foreign substances, and all 
other defects afiecting their value. 

Table 3 gives the physical and chemical characteristics of 
steel castings specified by the U. S. Navy Deparment. 

Steel for resisting shock should be of high carbon content. In the 
past the accepted dictum was that for this purpose a low carbon 
steel should be used, the idea being that low carbon steel is tough and 
able to stand punishment and that high carbon steel is brittle. The 
fallacy of this reasoning was first shown by experience with steam- 
hammer piston rods at the Crescent steel works about 1880 and 
the demonstration was made complete by the experience of rock- 
drill manufacturers. In rock drills low carbon steel was a complete 
failure, high carbon steel being found, early in the history of the 
industry, to be the only suitable material. 



Looking back, with the superior wisdom that comes after the 
event, the traditional view now seems absurd. It is now clear that 
what is wanted is a material that wiU absorb and give back again 
the greatest number of ft. -lbs. of eneigy without change of form; 
that is, without passing its elastic limit. That is to say, the property 
wanted is resilience and not toughness. In other words, we should 
aim at the properties of a spring and not at those of a piece of lead, 
which is equivalent to saying that we want high and not low carbon 
steel, and, by the same token, the steel should be in the tempered and 
not the annealed condition. 

The properties of steels of various carbon percentages, as regards 
the elastic resilience, have not been studied with sufficient care to 
enable the exact composition most suitable for resisting shock to be 
stated. Analogy would indicate that the composition most suitable 
for springs is the correct one for shock resistence, and the same remark 
applies to the heat treatment. At the same time, the author in his 
own experience with rock drills, made use of steel with carbon per- 
centage as high as 1.25 and with conspicious success. For a more 
complete discussion of this subject see Materials and Constructions 
for Resisting Shock. 



Steel for Cutting Tools 

The percentage of carbon suitable for carbon steel tools may be 
obtained from Table 4 {Amer. Mach., Nov.. 21, 1912). To some 
the carbon percentages will seem high but the table is the result of 
much investigation. 



Table 4. — Carbon Percentage in Carbon Steel Tools 



Anvil facing 70 to .80 

Arbor, saw 35 to .50 

Auger bit 50 to .65 

Axes of various shapes for cutting wood i .00 to i . 10 

Ball bearing races i 

Ball-peen hammer 

Band saw 

Barrel, gun 

Barrel, gun, drill for boring i 

Bar, digging 

Bar, pinch 

Bit, auger 

Bit, ax I 

Bit, stone, channeling machine i 

Bit, for stone drilling 

Blacksmith's cold chisel 

Blacksmith's hammer 

Blacksmith's hot chisel 

Blade, knife i 

Blade, pocket knife 

Blade, reamer i , 

Blade, skate 

Blade, table cutlery 

Blanking punch for files i 

Boilermaker's snap 

Boilermaker's beading tool 

Bolt dies, cold heading 

Bolt machine plunger 

Brick chisel 

Broad axe , . i 

Bucket teeth for dredges 

Button set 

Cabinet file i . 20 to i . 30 

Cant dog 90 to i . 00 



10 to I 


.20 


Soto 


.90 


70 to 


.80 


30 to 


.70 


10 to I 


. 20 


Soto 


.90 


70 to 


.80 


50 to 


.6s 


00 to I 


. 10 


00 to I 


. 20 


70 to 


.90 


70 to 


.80 


70 to 


.80 


Soto 


.90 


10 to I 


20 


90 to I 


00 


10 to I 


20 


Soto 


90 


70 to I 


. 10 


20 to I 


■30 


60 to 


.70 


70 to 


■85 


60 to 


.70 


60 to 


.70 


60 to 


.70 


00 to I 


.10 


70 to 


So 


60 to 


70 



Cant hook 

Cant-saw file i 

Cape chisel 

Car and locomotive spring 

Cartridge shell die i 

Cartridge shell punch i 

Carving knife i 

Carving fork 

Calking chisel 

Center, lathe i 

Channeling machine bit, stone i 

Chisel, blacksmith's cold .* . 

Chisel, chipping 

Chisel, brick 

Chisel, carpenter's r 

Chisel, file cutting i 

Chisel, hot '. 

Chisel, machinist's 

Chisel, railroad track 

Chisel, stone cutter's i 

Chisel, wood-working i 

Chuck jaw 

Circular saw 

Cleaver, butcher's 

Cold-heading bolt die 

Cold chisel, blacksmith's 

Cold cutting die for metal i 

Cold-punching horseshoe die i 

Cone, bicycle ' i 

Crosscut saw 

Crowbar 

Crucible machinery steel 

Cruciform drill steel 

Cutter blank, miUing i 

Cutter, flue i 



.80 to 


.90 


. 20 to I 


•30 


.80 to 


.90 


. 90 to I 


.10 


. 20 to I 


■30 


. 20 to I 


•30 


.00 to I 


.10 


•55 to 


•65 


.80 to 


.90 


.00 to I 


.10 


.00 to I 


. 20 


.70 to 


.So 


.80 to 


.90 


.60 to 


.70 


. 00 to I 


■30 


. 10 to I 


.20 


.So to 


.90 


.80 to 


.90 


.70 to 


.80 


. 10 to I 


.20 


.00 to I 


•30 


.So to 


.90 


.So to 


.90 


.80 to 


.90 


.60 to 


.70 


.70 to 


80 


. 10 to I 


20 


.00 to I 


10 


. 00 to I 


10 


. 90 to I 


00 


.70 to 


80 


■35 to 


50 


.80 to 


90 


.10 to I 


20 


. 20 to I 


30 



STEEL 



373 



Table 4. — Carbon Percentage in Carbon Steel Tools — (Continued) 



Cutter, glass i . 20 to i . 30 

Cutter, nail i . 10 to i . 20 

Cutter, pipe i . 10 to i . 20 

Cutter, horse hoof 70 to .85 

Cutter, clinch (farrier's) 80 to .90 

Cutting die, cold, for metal i . 10 to i . 20 

Cutting die, paper i . 10 to i . 20 

Cylinder, pneumatic hammer 80 to .90 

Die, cold-heading for bolts 60 to .70 

Die, cartridge shell i . 20 to i . 30 

Die, cold cutting for metal i . 10 to i . 20 

Die, drop forging 60 to .75 

Die, leather cutting 80 to .90 

Die, drop hammer 60 to .80 

Die, horseshoe cold-punching i . 00 to i . 10 

Die, nail i . 10 to i . 20 

Die, paper cutting 1. 10 to i. 20 

Die, pipe 1. 10 to 1.20 

Die, rivet 60 to .75 

Die, shoe-upper cutting 70 to .80 

Die, silversmith's, stamping i . 10 to i . 20 

Die, silver spoon, drop 80 to .90 

Die, threading i . 00 to 1 . 10 

Die, wire drawing i . 30 to i . 50 

Dog, cant 90 to i . 00 

Drift pin 80 to .90 

Digging bar 80 to .90 

Drag saw 90 to i . 00 

Dredge bucket teeth 70 to .80 

Drill, cruciform 80 to .90 

Drill for drilling tool steel i . 00 to i . 20 

Drill for shotgun barrels i . 10 to i . 20 

Drill, quarry 75 to .90 

Drill, twist 1. 10 to 1.20 

Driver, screw 60 to .70 

Drop-forging die 60 to .75 

Edge, scythe i . 00 to i . 10 

Expander roll for tubes i . 00 to i . 10 

Eyepin for tie rods 70 to .80 

Facing, anvil 70 to .80 

File, blanking punch for i . 20 to i . 30 

File, cutting chisel for i . 10 to i . 20 

Files in general i . 20 to i . 30 

Flat chisel 80 to .90 

Flatter, blacksmith's 80 to .90 

Flue cutter, boiler i . 20 to i .30 

Forging die, drop 60 to .75 

Fork, pitch 90 to i . 10 

Gang saw 90 to i. 00 

Glass cutter i . 20 to i . 30 

Glove die, leather 80 to .90 

Glut or stone wedge 60 to .70 

Grab 70 to .90 

Granite point i . 22 to i . 35 

Grass hook 60 to .70 

Gun barrel 30 to .70 

Gun-barrel reamer i . 10 to i . 20 

Hammer, blacksmith's 70 to .80 

Hammer, ball-peen 80 to .90 



Hammer, bush for stone i . 20 to i . 30 

Hammer, machinst's 90 to i . 00 

Hammer, nail machine i . 00 to i . 10 

Hammer, peen i.i5toi.2o 

Hammer, pneumatic cylinder 80 to .90 

Hardie 70 to .80 

Hatchet i . 10 to i . 20 

Hoe 80 to .90 

Hook, cant 80 to .90 

Hook, grass 60 to .70 

Horseshoe die, cold punching i .00 to i . 10 

Hot chisel 80 to .90 

Hot punch 80 to .90 

Ice plow 1 . 10 to 1 . 20 

Jaw, chuck 80 to .90 

Knife blade i . 10 to i . 20 

Knife, butcher's 75 to .90 

Knife, carving i . 00 to i . 10 

Knife, cobbler's 85 to i . 00 

Knife, farrier's 80 to .90 

Knife, machine i . 10 to i . 20 

Knife, paper i . 10 to i . 20 

Knife, pen Soto .95 

Knife, pruning 75 to .85 

Knife, putty 90 to i . 00 

Knife, drop-forging die for table 60 to .75 

Knife, shear, for paper 80 to i . 00 

Knife, wood- working i . 10 to i . 20 

Lathe tool i.oo to i . 20 

Lathe center i . 00 to i . 10 

Lawn-mower blade 90 to i . 00 

Locomotive and car spring 90 to i . 10 

Machine knife i . 10 to i . 20 

Machinery steel, crucible 35 to .50 

Machinist's hammer 90 to i . 00 

Magnet, permanent i . 20 to i . 30 

Magnet for telephone call bell 50 to .60 

Magnet for telephone 90 to i . 10 

Mandrel i. 00 to 1. 10 

Mattock 60 to .80 

Maul, railroad 70 to .80 

Maul, woodchopper's 70 to .80 

Mill pick 1 . 20 to 1 . 30 

Mill saw 1. 20 to 1.30 

Milling cutter blank i . 10 to i . 20 

Mining tools, drills, picks, etc 65 to i . 10 

Molder's hand tools i .00 to i . 10 

Mower blade, lawn 90 to i . 00 

Nail cutter i . 10 to i . 20 

Nail die i . 10 to i . 20 

Nail-machine hammer i . 00 to i . 10 

Nail puller 90 to i . 00 

Paper-cutting die 1 . 10 to i . 20 

Paper knife i . 10 to i . 20 

Paving and plug drill i . 10 to i . 20 

Peen hammer i . 1 5 to i . 20 

Pick ax 70 to .80 

Pick mill 1 . 20 to 1 . 30 



374 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 4. — Carbon Percentage in Carbon Steel Tools — {Continued) 



Pin, drift 80 to .90 

Pincers (farrier's) 80 to .90 

Pinch bar 70 to .80 

Pipe cutter i . 10 to i . 20 

Pipe die i . 10 to i . 20 

Pit saw 90 to 1 . 00 

Pitchfork 90 to i . 10 

Pitching chisel for stone i . 00 to i . 1 5 

Planer tools for metal i . 00 to i . 20 

Planer tools for stone 80 to i . 30 

Planer tools for wood i . 10 to i . 20 

Pliers 60 to .95 

Plow blade for ice i . 10 to i . 20 

Plug and paving drill i . 10 to i . 20 

Plunger for bolt machine 60 to .70 

Pneumatic-hammer cylinder 80 to .90 

Point, granite 1.22 to 1.35 

Point, clay pick 70 to .80 

Pruning shears 90 to i . 00 

Puller, nail 90 to i . 00 

Punch, boilermaker's 85 to .95 

Punch, cartridge shell i . 20 to i . 30 

Punch, for file blanks i . 20 to i . 30 

Punch, hot Soto .90 

Punch, railroad track 70 to .80 

Punch, washer 80 to i . 00 

Putty knife 90 to i . 00 

Quarry drill 75 to .90 

Railroad car and locomotive spring 90 to i . 10 

Railroad-spike maul 80 to .90 

Railroad- track chisel 70 to .80 

Ramrod 60 to .70 

Razor i . 00 to i . 1 5 

Razor blade, safety i . 00 to i . 20 

Reamer blade i . 00 to i . 20 

Reamer, hand i . 00 to i . 10 

Rivet, die 60 to .75 

Rivet set 65 to .75 

Road-scraper blade 60 to .70 

Roll,; expander . 1.00 to 1. 10 

Saws for wood in general 80 to .90 

Saw, arbor 3 S to .50 

Saw file i . 20 to i . 30 

Saw blade, band - 70 to .80 

Saw, circular ' 80 to .90 

Saw, crosscut 90 to i . 00 

Saw, drag 90 to i . 00 

Saw for steel i . 60 

Saw, gang 90 to i. 00 

Saw, mill i . 20 to i . 30 

Saw, pit 90 to 1 . 00 

Saw swage 8b to .90 

Saw teeth, inserted for wood 90 to i . 00 

Scraper blade for roads .' 60 to .70 

Heat Treatment of Steel 

By the American Vanadium Company 

A list of heat treatments will be found at the end of the section. 

By the term heat treatment is meant all those operations of heating 

and cooling to produce or develop certain definite properties which 

are desired. Broadly speaking, it could refer to all the heating con- 



20 to I 


30 


90 to I 


00 


60 to 


.70 


00 to I 


.10 


60 to 


.70 


60 to 


.70 


75 to I 


.00 


65 to 


•75 


00 to I 


.20 


80 to I 


.00 


90 to I 


00 


20 to I 


•30 


20 to I 


■ 30 


70 to 


.80 


10 to I 


.20 


60 to 


.70 


10 to I 


20 


80 to 


.90 


Soto 


.90 


70 to 


.80 


40 to 


• 50 


SO to 


.60 


Soto 


.90 


20 to I 


•30 


70 to 


■85 


10 to I 


.20 


70 to 


.90 


80 to I 


•30 


70 to 


.80 


Soto 


.90 



Scraper tube i 

Scraper, wood-working 

Screw-driver 

Scythe edge i 

Setscrew 

Set, button 

Set, mason's 

Set, rivet 

Shaper tools i 

Shear knife 

Shears, pruning 

Shell, drawing punch for cartridge i 

Shell, drawing die for cartridge i , 

Shoe die, for cutting leather 

Shotgun barrel drill i 

Shovel teeth, dredge 

Silversmith's die for stamping i 

Silver-spoon die 

Skate-blade steel 

Sledge 

Spade 

Spindle for cotton or wool 

Star drill '. 

Steel for files i 

Steel for welding 

Stonecutter's chisels i . 

Stone drilling bit 

Stone planer tools 

Stone wedge or glut 

Swage, saw 



Table-knife blade 70 to i . 10 

Tap 1 . 00 to 1 . 20 

Tooth, dredge bucket . 70 to .80 

Teeth, inserted for wood saw 90 to i . 00 

Tools, mason's 60 to 1.35 

Tools, molder's i . 00 to i . 10 

Tools, pitching i .00 to i . 15 

Track chisel, railroad 70 to .80 

Track punch 7° to .80 

Trowel, mason's 40 to .50 

Tube scraper i . 20 to 1 . 30 

Twist drill i . 10 to i . 20 

Vise, jaw 7° to .80 

Washer, punch .' 70 to .80 

Washer, punch 80 to i . 00 

Wedge, stone 7° to .80 

Well bit for stone drilling 7° to .90 

Wire-drawing die i . 30 to i . 50 

Woodchopper's maul 70 to .80 

Wood-planer blades i . 10 to i . 20 

Wood-saw inserted teeth 90 to i . 00 

Wood- working chisel i . 00 to i . 30 

Wood- working knife i . 00 to i . 20 

Wrench 7° to .80 

ditions to which steel is subjected in the course of its manufacture to 
the finished ardcle. The term is of comparatively recent origin, 
although the processes involved, annealing, hardening and tempering 
have been practised for centuries, with varying degrees of precision. 
The lead pot and the temper colors were in regular use over a century 
ago. The modern conception of heat treatment dates from about 



STEEL 



375 



1885, and is coincident with the electric pyrometer and the applica- 
tion of the microscope to the metallurgy of steel. The revelation by 
the microscope of the structural changes produced by heat and the 
ability to measure accurately by means of the electric pyrometer, 
the temperature involved, are the basis of modern heat treatment 
methods. 

If heat be applied to a bar of normal steel, it will become sensibly 
hotter with each increment of heat up to a given point. Thereupon, 
further application of heat instead of increasing the sensible tempera- 
ture, is absorbed by the steel in some molecular change or reconstruc- 
tion within the metal, lasting during a more or less protracted period. 
As this rearrangement is completed, the sensible temperature begins 
to increase again regularly. During this absorption or retardation 
of sensible temperature, the expansion of the steel bar is checked, 
its magnetic qualities disappear, and the steel develops the quality 
of becoming hard if quenched. In cooling, the reverse changes take 
place. To a certain temperature the steel cools regularly, then it 
ceases to cool for an interval, and heat is actually given off sufficient 
to cause a visible rise in temperature, after which the cooling pro- 
ceeds regularly; the steel regains its magnetic qualities and loses its 
hardening power. 

These phenomena, the retardation of sensible temperature on 
heating, and the evolution of sensible heat on cooling, are known as 
the "calescence" and " recalescence," or more generally the "critical 
point," or "absorption point" in steel. For convenience these 
points are designated Ac (calescence) and Ar (recalescence). It has 
been demonstrated that in order to obtain the retardation Ar, the 
steel must be first heated past the point Ac, and conversely the ab- 
sorption or change at Ac cannot be induced unless the steel has first 
been cooled below the point Ar. It is therefore evident that these 
two points, while not taking place at the same temperature, are 
opposite phases of the same phenomenon. Low carbon, and some 
of the alloy steels have two or three critical points, but high carbon 
steels and those of medium hardness have only one point. The 
various alloying metals — nickel, chromium, tungsten, vanadium, etc., 
have an influence on the location of these points, some raising and 
others lowering the temperature at which the retardation takes place. 

Before taking up further the application of heat treatment to 
steel it will be in order to refer briefly to the ultimate structure of 
steel as revealed by the microscope. For illustration we will consider 
a medium carbon steel, one with about .50 per cent, carbon. The 
carbon is combined chemically with a molecular proportion of iron, 
forming a definite carbide of iron, which is the structural constitu- 
ent cementite. One part of this carbide, cementite, unites or alloys 
with about seven parts of carbonless iron or ferrite, forming the 
structural constituent pearlite, which is distributed in mesh form 
through the main back-ground or net work of ferrite. In slowly 
cooled steel, pearlite has a characteristic laminated structure made up 
of thin plates alternately of ferrite and cementite. The precise 
manner in which pearlite is arranged in respect to size, plate-like 
form, regular or irregular distribution, etc., depends to a consider- 
able degree, on the nature and amount of "hot work" put on the 
steel, the rate of cooling and so forth. Under certain conditions 
pearlite loses its lamellar structure and becomes a granular mixture 
of small irregular grains of cementite and ferrite. 

Investigation by the microscope of the changes in structure pro- 
duced during the calescence and recalescence period shows that dur- 
ing the calescence the pearlite becomes broken up, its carbides going 
into solid solution in the ferrite. During the recalescence period the 
dissolved carbides are thrown out of solution and alloy with the 
ferrite to re-form pearhte. Medium and low carbon steel slowly 
cooled from a temperature just above the recalescence point presents 
a very different structure than it originally had when slowly cooled 
from a high temperature. The pearlite has been broken up into small 
areas, the plate-like structure is less distinct and the tendency is 
more to a granular structure. The original coarseness of the grain 



is removed, internal strains have been eliminated and the steel has 
become softer and more easily machined. It has become, in ordinary 
terms, Annealed. 

If the steel when heated above its calescence point (when it con- 
tains all its carbides in solid solution), be subjected to very quick 
cooling, so that no chance is given for the deposition or reprecipita- 
tion of. its dissolved carbide, a new body is formed, known by the 
generic term "martensite;" in other words, martensite may be said 
to consist of a frozen solution of carbides in ferrite. In its nature, 
this body is brittle and intensely hard. The intensity of its hardness, 
however, naturally varies both with regard to the nature and amount 
of carbides contained in the frozen solid solution, and to their rate 
of freezing. 

Martensite is not a stable "body," its equilibrium being destroyed 
very much below the calescence point; when subjected to a tempera- 
ture of about 360 deg. C, for a period of time sufficient to thor- 
oughly soak through the mass, it is decomposed, its carbides being 
deposited in situ and soft ferrite liberated as a background. 

A vanadium ferrite does not permit of the ready passage through 
it of the precipitated carbides, therefore the colonization of carbides 
in such steel is much less complete and their distribution better; 
consequently, the toughness and tenacity of the steel is increased, 
irrespective of the added toughness of the background of vanadium 
ferrite. 

From the above it is evident that the long practised operations 
of hardening and tempering of steel depend upon the formation of 
martensite by quenching from temperatures above the critical point, 
and its partial decomposition at temperatures below the critical point. 
It has been shown that martensite, the constituent that confers 
hardness upon quenched steel, is not formed below the calescence 
point. Consequently steel quenched at temperatures below the 
beginning of the calescence point are not hardened. This is illus- 
trated by the following tests made with a .50 per cent, carbon simple 
open-hearth steel. The temperature range of the calescence point 
is from about 705 deg. Cent, to 750 deg. Cent., the point of maximvun 
retardation being at about 713 deg. Cent. The range for the recales- 
cence point is from about 690 deg. Cent, to 640 deg. Cent., the maxi- 
mum retardation taking place at about 58o deg. Cent. 



Treatment 



As rolled 

Quenched in water from: 

650°C 

700° C 

72S° C 

750° c 

825° c ■ 

Heated to 825° C. and 

cooled slowly to: 

700° C. and quenched.. . 

650° C. and quenched.. . 

600° C. and quenched... 



Elastic 
limit 



49,000 

42,500 
40,000 
38,000 



Tensile 
strength 



91,000 
89,500 
96,500 
142,000 



135,500 

121,000 

89.000 



Elong- 
ation 
in 2 ins. 
per 
cent. 



25 

27 .0 

13. 5 

2 .0 



23. 5 



Reduc- 
tion 

of area 
per 
cent. 



47.5 

49 -S 

SO. 5 

42 .0 

0.0 



42 .0 



Hardness No. 



Brin- 

nell 



187 
217 
228 
302 
555 



430 
375 
187 



Sclero- 
scope 



24 

27 
30 
32 
48 
75 



63 
S6 
28 



From these examples it is evident that there can be two methods 
of procedure in the tempering, drawing back, letting down, or 
annealing of hardened steel. The article can be heated to the prede- 
termined temperature, and then allowed to cool in the air, or it can 
be quenched in oil or water. This second procedure possesses an 
advantage over the first, in that greater uniformity in results are 
obtainable in the heat treatment of a large nmnber of pieces, and 
also that it facilitates output without impairment of quality. 

Frequently, in the case of the first procedme; that is, allowing 
the piece to cool in the air after it has been heated to the drawback 
temperatures, numerous pieces are heaped together and the rate 
of cooling varies greatly for different pieces in the pile. 



376 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



It is preferable to heat the articles to be drawn back, in a fur- 
nace which is already at the desired temperature, maintaining the 
heat during the period necessary to bring the article up to this 
temperature. 

An excellent way of drawing back small quenched articles, which 
are required to be let down at a higher temperature than is consistent 
with the use of hot oil, is to immerse them in a bath of molten lead, 
or fused salts, kept at the desired heat by means of a fire. 

It should be remembered, that: i. The sti£fer steels when quenched 
form more intense martensites. 

2. Some quenching liquids are more drastic than others. 

3. The more intense the martensite, the more decomposing it 
takes, other things being equal. " 

The stiffening elements may be said to be carbon, manganese, 
chromium, and part of the vanadium. It is assumed that the phos- 
phorus and sulphur remain reasonably low in every case. 

The quenching oil is generally contained in a tank which is water- 
cooled, so that the temperature of the bath is usually about 50 deg. 
or 60 deg. Cent., or in some cases, possibly a little higher. It is not 
absolutely necessary that lard and fish oils be used alone, as it is ad- 
missible to add a considerable quantity of cotton-seed oil, etc., but 
the characteristics of such mixture of Jard and fish oils should be 
adhered to as much as possible; for example, the admixture of any 
class of medium-thin parafiin oil would not be recommended. 

A more drastic quenching liquid than the above would be cold 
water, but this is not recommended for steels containing considerable 
quantities of chromium and manganese together, as such steels are 
particularly liable to crack when quenched in water. 



tions to which it is subjected; in a way it is the most important, as 
it is generally the final one. There can be no unimportant details. 
It is essential that the work be done by skilful men, supplied with 
accurate pyrometers and well designed and constructed furnaces 
capable of maintaining a uniform heat and of being easily regulated. 

Hardness Tests 

The leading methods of testing materials for hardness are by the use 
of the Brinell test and the scleroscope. In the Brinell test a hardened- 
steel sphere is made to indent the metal under test by a measured 
pressure, the diameter of the indentation being measured under a 
microscope. The measure of the hardness is then given by the equa- 
tion: 

Hardness numeral = r-r— ] — -r—r^ 

area of indentation, sq. mm. 

The loads recommended are 500 kg. for the softer and 3000 kg. 
for the harder metals. Investigations have shown the hardness 
numeral to be, within limits, independent of the size of the sphere, 
which, commonly, is of 10 mm. diameter. Table 5 gives Brinell 
hardness numerals for this size of ball and 500 and 3000 kg. pressure. 

The scleroscope consists of a pointed hammer enclosed within a 
glass tube and made to fall a definite distance upon the material 
under test. The height of the rebound of the hammer, as measured 
on a scale behind the glass tube, is the measure of the hardness. 
The rebound from steel hardened right out is made to read 100 on 
the scale, which is then divided in simple proportion down to zero. 
There is a direct relation between the hardness numerals and the 
tensile strength of steel. 









Table 5.- 


-Brinell Hardness Numerals, Steel Ball of 10 Mm. Diameter 








Diameter 


Hardness 


Diameter 


Hardness 


Diameter 


Hardness 


Diameter 


Hardness 


Diame- 


Hardness 


of 


numeral. 


of 


numeral. 


of 


numeral, 


of 


numeral, 


ter of 


numeral, 


impression. 


pressure, kg. 


impres- 
sion, mm. 


pressure, kg. 


impres- 
sion, mm. 


pressure, kg. 


impres- 
sion, mm. 


pressure, kg. 


impres- 
sion mm. 


pressure, kg. 


mm. 


3000 


500 


3000 


500 


3000 


SCO 


3000 


500 


3000 


500 


2. — ■ 


946 


158 


3 — 


418 


70 


4- — 


228 


38 


5- — 


143 


23.8 


6.— 


95 


iS-9 


2.0s 


898 


ISO 


3 -OS 


402 


67 


4-oS 


223 


37 


5-05 


140 


23 -3 


6.05 


94 


156 


2 . 10 


857 


143 


3.10 


387 


65 


4. 10 


217 


36 


5- 10 


137 


22.8 


6. 10 


92 


iS-3 


2. IS 


817 


136 


315 


375 


63 


4-15 


212 


35 


5-15 


134 


22.3 


6.15 


90 


15 -I 


2. 20 


782 


130 


3.20 


364 


61 


4.20 


207 


34-S 


S.20 


131 


21.8 


6. 20 


89 


14.8 


2.2s 


744 


124 


3-25 


351 


59 


4-25 


202 


33-6 


5. 25 


128 


21.5 


6.2s 


87 


14-5 


2,30 


713 


119 


3-30 


340 


57 


4-3° 


196 


32.6 


5-30 


126 


21 


6.30 


86 


14-3 


2-35 


683 


114 


3 35 


332 


55 


4-35 


192 


32 


5-35 


124 


20.6 


6.3s 


84 


14 


2 . 40 


652 


109 


3 40 


321 


54 


4.40 


187 


31.2 


5-40 


121 


20. 1 


6.40 


82 


13.8 


2.4s 


627 


105 


3-45 


311 


52 


4-4S 


183 


30.4 


5-45 


118 


19.7 


6-45 


81 


I3-S 


2.50 


600 


100 


3-5° 


302 


SO 


4-50 


179 


29.7 


5-50 


116 


193 


6.50 


80 


13 -3 


2-5S 


578 


96 


3-55 


293 


49 


4-55 


174 


29.1 


5-55 


114 


19 


6.55 


79 


13 I 


2 . 60 


555 


93 


3.60 


286 


48 


4. 60 


170 


28.4 


5.60 


112 


18.6 


6.60 


77 


12.8 


2.6s 


532 


89 


3.65 


277 


46 


4-65 


166 


27.8 


5-65 


.109 


18.2 


6.6s 


76 


12.6 


2 . 70 


512 


86 


3 -70. 


269 


45 


4.70 


163 


27. 2 


5 -70 


107 


17.8 


6. 70 


74 


12.4 


2-75 


495 


83 


3-75 


262 


44 


4-75 


159 


26. s 


5-75 


105 


17-5 


6.75 


73 


12 . 2 


2.80 


477 


80 


380 


25s 


43 


4.80 


156 


25 -9 


5.80 


103 


17.2 


6.80 


71-5 


II. 9 


2.8s 


460 


77 


3.85 


248 


41 


4-8s 


153 


25-4 


5.85 


lOI 


16.9 


6.85 


70 


II. 7 


2.90 


444 


74 


3 90 


241 


40 


4.90 


149 


24.9 


5 90 


99 


16.6 


6. 90 


, 69 


II-5 


2-95 


430 


73 


3-95 


235 


39 


4 95 


146 


24.4 


5-95 


97 


16. 2 


6-95 


68 


II-3 



Due consideration must of course be given to variation in section 
of the articles being heat treated. It is evident that the rate of cool- 
ing by quenching will vary with the section, and consequently the 
amount of martensite formed will vary, and with it, of course, the 
hardness. The small section of the article therefore, should control 
the heat treatment temperatures. 

The heat treatment of steel is one of the most important opera- 



The relations of Brinell and scleroscope numbers and of the maximum 
strength of steel, according to R. R. Abbott {Trans. Am. Soc. for 
Testing Materials, 1915) are given in Table 6, which is based on 
several thousand tests and 
in which ikf = maximum strength in 1000 lbs. per sq. in., 

B — Brinell hardness number, 

S = scleroscope hardness number. 



STEEL 



377 



Table 6. — Relations or Brinell and Scleroscope Hardness 
Numbers with One Another and with Maximum 
Strength of Steel 





Equations connecting 


Kind of steel 


Max. strength 

and Brinell 

number 


Max. strength 

and scleroscope 

number 


Brinell and 

scleroscope 

numbers 




M = o.73B-28 
M = o.7iB — 32 
M = o.7iB — 29 
M = o.68S-22 
M = o.7iB — 33 
M = o.7oB — 26 


M = 4. 45-28 
M = 3.5S- 6 
M = 4. 25-21 
M = 3.75- I 
M = 3.75- 3 
M = 4.o5-iS 


£ = 5.65 + 14 


Nickel 


5 = 5.05+48 




B = s.5.S + 27 


Low chrome-nickel 

High chrome nickel 

All steels grouped together. 


5 = 5.45 + 33 
5 = 4.85 + 58 
5 = 5.55 + 28 



Composition Properties and Heat Treatment of Steel 

The following data are from the report of the iron and steel division 
of the standards committee of the Society of Automobile Engineers, 
Aug., 1915. 

A numerical index system has been adopted in the numbering of the 
metal specifications contained in this report. This system renders it 
possible to employ specification numerals on shop drawings and blue 
prints, that are partially descriptive of the quality of material covered 
by such number. The first figure indicates the class to which the 
steel belongs; thus i indicates a carbon steel, 2 nickel, 3 nickel chro- 
mium, etc. In the case of the alloy steels, the second figure generally 
indicates the approximate percentage of the predominant alloying 
element. The last two or three figures indicate the average carbon 
content in "points," or hundredths of i per cent. Thus 2340 indi- 
cates a nickel steel with approximately 3 per cent, nickel (3.25 per 
cent.-3.75 per cent.) and 0.40 per cent, carbon (.35 per cent.-.4S 
per cent.) and 51,120 indicates a chromium steel with about i per 
cent, chromium (.90 per cent.-i.io per cent.) and 1.20 per cent. 
carbon (i.io per cent.— 1.30 per cent.). 

The basic numerals for the various qualities of steels herein speci- 
fied follow: 

Carbon steels i 

Nickel steels .^ 2 

Nickel chromium steels 3 

Chromium steels 5 

Chromium vanadium steels 6 

Silico-manganese steels 9 

These steels may be of open hearth, crucible or electric manufac- 
ture, and must be homogeneous, sound and free from physical defects, 
such as pipes, seams, heavy scale or scabs and surface and internal 
defects visible to the naked eye. 

These steels wiU be purchased on the basis of chemical analysis. 
The specifications indicate the desired chemical composition. Any 
shipments not conforming to these specifications after careful check 
analysis may be rejected. 

Recognizing the wide variance in methods used for the determina- 
tion of sulphur, the final reference method shall be the gravimetric 
(aqua regia) method, by oxidation. 

Materials to be sampled shall be considered under three classes 
namely: 

1. Wire, tubing, sheet and rod metal less than i}i ins. in size shall 
be sampled across or through the entire section. 

2. Forgings or pieces of irregular shape shall be sampled by drilling 
or cutting at thickest and thinnest sections, or through or across 
entire section. 

3. Bars and billets or other shapes above ij^ ins. thick shall be 
drilled at half radius, or halfway between center and exterior sur- 
faces. 

The notes and instructions following the chemical specifications are 
not to be considered in any way a part of these Specifications. They 
are added solely for the information of the user of the steels and for 
the guidance of the purchaser in the selection of proper steels for his 
different purposes. They should not be incorporated in the speci- 



fication when ordering steel. This is especially true of the "Phys- 
ical Characteristics." Where possible, specific data are given on the 
physical properties which can be expected with the most widely used 
heat treatments. 

The materials specified in detail as S. A. E. steels include the most 
important ones available to the builder of automobiles. 

The results of physical tests, whether tension tests or otherwise, are 
largely dependent upon the mass and form of the specimen tested. 
This is particularly true of heat treated steels. For the foregoing 
reason, aU results of physical tests are comparative, and in order to 
make the comparison a proper one a uniform test specimen must be 
used. 

The committee therefore decided that recommended practice 
should be the use of the S. A. E. standard test specimen, this speci- 
men to be treated approximately in its finished form, leaving only 
sufficient stock for finish grinding after the treatment is completed, 
say .020 in. on the diameter. 

The figures for physical characteristics given for all steels following 
specification No. 1045 refer to those obtained on specimens prepared 
from sections common in automobile use, that is, bars from i in. 
round up to i}4 ins. round. 

The yield point is under control in two ways — by choice of quench- 
ing medium (oil, water or brine) and by varying the final drawing 
temperature. In the interpretation of the physical characteristic 
figures, it must be remembered that only the minimum figures as to 
toughness {i.e., reduction and elongation) can be expected with the 
highest degree of strength {i.e., yield point); and, conversely, that 
the highest degree of toughness may be expected with the lowest 
yield point. This remark applies to all heat treated steels. It would 
be manifestly impossible to obtain the highest percentage of elonga- 
tion and the highest yield point on the same specimen. 

Except in the physical property charts, the yield point is specified 
rather than the elastic limit. The yield point is measured by the 
drop of the testing machine beam and furnishes the most ready and 
widely used measure of the so-called elastic limit; results obtained 
by this method, however, are generally from 5000 to 15,000 lbs. 
higher than the true elastic limit, where this property is not in excess 
of 125,000 lbs. per sq. in. With material having a yield point in 
excess of 125,000 lbs. per sq. in. the true elastic limit should be 
obtained by means of an extensometer. 

There is little use in giving the physical characteristics of a car- 
bonized steel, inasmuch as any test must be deceptive because of the 
very high carbon exterior case which cracks and fails long before the 
soft and tough interior does. This means that the rupture is frag- 
mental and progressive and misleading. 

In addition to the usual physical characteristics the hardness tests 
have been considered, as obtained by means of the Brinell ball test 
and the Shore scleroscope. The Brinell test recommended by the 
committee is the use of the lo-mm. ball and 3000-kg. load. 
It is pointed out, however, that the Brinell test must not be used on 
soft steels less than J^ in. thick, or on areas small enough to permit 
the depression to flow toward the edges of the specimen. With hard 
steels, where the depth of the depression and the flow of metal are 
less, material as thin as }i in. can be so tested. The Brinell test 
can be fairly made on surfaces that are free from scale and smooth. 

The Shore test (scleroscope) must be used only on surfaces that 
have been carefuUy polished and free from all tool marks, file marks or 
grinding scratches. The test specimen should also be of such mass or 
be held in such manner as to give the greatest possible freedom from 
deflection when struck by the hammer. 

In interpreting the physical property curves and tabulations these 
considerations should be borne in mind: 

The figures given have been made as valuable as possible to the 
engineer by indicating what can be expected as the average product 
of a given composition when treated in the specified manner, in 
average sections prevailing in motor car work. 

At the same time the data have been so chosen as to protect makers 



378 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



of treated stock and parts from unreasonable demands. This has 
been done by taking figures low enough to be obtained with reason- 
able certainty when open market stock of medium to high grade is 
treated in commercially efficient equipment, controlled by commer- 
cially accurate instruments. 

For the sake of simplicity only average minimum figures for tensile 
strength, elastic limit, reduction of area and elongation have been 
adopted; these figures are based upon the following assumptions, 
heat treatment being kept constant: 

The lowest tensile strength and elastic limit are produced with 
steels at the bottom of a given range in carbon. 

The lowest reduction in area and elongation are produced with 
steels at the top of a given range in carbon. 

Thus, for 103 s steel, the tensile strengths and elastic limits given 
are the average minimum as of a steel containing 0.30 per cent, carbon; 
the reductions of area and elongations are the average minimum as 
of a steel containing .40 per cent, carbon. 

The figures for hardness are conventional averages for the whole 
range of compositions within any given specification. In general, 
the Brinell hardness figure is subject to fluctuations of plus or minus 
ten to fifteen points, the Shore (scleroscope) hardness of plus or 
minus five points. 

Specimens for test must comply with all the requirements given 
above. In addition, tensile test pieces are to be taken concentrically 
from bars which are treated in diameters up to and including i in. 
round or square; from larger sections the axis of the test piece should 
be made parallel to the axis of the bar at any point as nearly as 
possible 50 per cent, from the center to the exterior. 

From all the foregoing it wiU be seen that the data referred to are 
very conservative. Average results in practice wiU generally exceed 
appreciably the figures given, and this excess then becomes an 
increased factor of safety which protects both the engineer and the 
manufacturer. 

Carbon Steels 

Specification No. ioio 

.10 Carbon Steel 

Carbon 05 to . 15% (.10% desired) 

Manganese 30 to . 60% (.45% desired) 

Phosphorus, not to exceed .045% 

Sulphur, not to exceed -05% 

This is usually known in the trade as soft, basic open-hearth steel. 
It is a material commonly used for seamless tubing, pressed steel 
frames, pressed steel brake-drums, sheet steel brake-bands and pressed 
steel parts of many varieties. It is soft and ductile and will stand 
much deformation without cracking. 

This steel in a natural or annealed condition has little tenacity 
and must not be used where much strength is required. This quality 
of material is considerably stronger after cold drawing or rolling; 
that is, its yield point is raised by such working. This is important 
in view of the fact that many wire and sheet metal parts above 
mentioned are used in the cold-rolled or cold-drawn form. 

It must not be forgotten that when this steel (so cold worked) is 
-heated, as for bending, brazing, welding, or the hke, the yield 
point returns to that characteristic of the annealed material. This 
remark also applies to all materials that have an increased yield point 
produced by cold working. 

This material in a natural or annealed state does not machine 
freely. It wiU tear badly in turning, threading and broaching 
operations. Heat treatment produces but little benefit, and that 
not in strength but in toughness. It is possible to quench this grade 
of steel and put it in a condition to machine better than in the 
annealed state. 

The heat treatment which wiU produce a little stiffness is to quench 
at 1500 deg. Fahr. in oil or water. No drawing is required. 

This steel will case-harden but is not as suitable for this purpose 
as steel 1020, a note on which follows: 



Physical Characteristics 





Annealed 


Cold rolled or 
cold drawn 


Yield point, lbs. per sq. in 


28,000 


40,000 




to 


to 




36,000 


60,000^ 


Reduction of area . 


65-55 per cent. 
40-30 per cent. 


55-45 per cent. 
Unimportant 


Elongation in 2 ins 





Specification No. 1020 
.20 Carbon Steel 

Carbon 15 to . 25% (.20% desired) 

Manganese 30 to .60% (.45% desired) 

Phosphorus, not to exceed .045% 

Sulphur, not to exceed .05% 

This steel is known to the trade as .20 carbon, open-hearth steel, 
and often as machine steel. 

This quality is intended primarily for case-hardening. It forges 
well and machines well, but should not be considered as screw-machine 
stock. It may therefore be used for a very large variety of forged, 
machined and case-hardened parts of an automobile where strength 
is not paramount. 

Steel of this quality may also be drawn into tubes and roUed into 
cold-roUed forms, and, as a matter of fact, makes a better frame 
than steel loio, because of the slightly higher carbon and re- 
sulting strength. The increased carbon content has no detrimental 
effect as far as usage is concerned, and it is- only the most difficult 
of cold forming operations that cause it to crack during the form- 
ing. For automobile parts it may be safely used interchangeably 
with steel loio as far as cold pressed shapes are concerned. 

Heat treatment of this steel produces but little change as far as 
strength is concerned, but does cause a desirable refinement of grain 
after forging, and materially increases the toughness. Heat treat- 
ment, which will often help the machine qualities, is all that is 
necessary. 

Case-hardening is the most important treatment for this quality 
of steel. The character of the operation must depend upon the 
importance of the part to be treated and upon the shape and size. 
There is a certain group of parts in an automobile which are not called 
upon to carry much load or withstand any shock. The principal 
requirement is hardness. Such parts are fairly illustrated by screws 
and by rod-end pins. The simplest form of case-hardening will 
suffice, that is, heat treatment A . 

Another class of parts demands the best treatment (heat treatment 
B), such as gears, steering-wheel pivot-pins, cam-rollers, push-rods 
and many similar details of an automobile which the manufacturer 
learns by experience must be not only hard on the exterior surface 
but possess strength as well. The desired treatment is one which first 
refines and strengthens the interior and uncarbonized metal. This 
is then followed by heat treatment B which refines the exterior, car- 
bonized, or high-carbon metal. 

In the case of very important parts, the last drawing operation 
should be continued from one to three hours, to insure the full 
benefit of the operation. 

The objects of drawing are twofold: First, and not least im- 
portant, is the reUeving of all internal strains produced by quench- 
ing; second, is the decrease in hardness, which is sometimes desirable. 
The hardness begins to decrease very materially from 350 deg. Fahr. 
up, and the operation must be controlled as dictated by experience 
with any given part. 

There are certain very important pieces that demand all of these 
operations, but the last drawing operation may be omitted with a 
large number. Experience teaches what degree of hardness and 

' These high yield points can be obtained only in comparatively light or 
small sections, either in the sheet or rod form; say H in. round or }4 in. sheets 
or Rats. 



STEEL 



379 



toughness combined is necessary for any given part. It is impossible 
to lay down a general rule covering all different uses. If the funda- 
mental principle is well understood, there should be no trouble in 
developing the treatment to a proper degree. 

Following the foregoing treatment, a fractured part should show 
a fine-grained exterior, without any appearance of shiny crystals. 
The smaller the crystals the better. The interior may show a silky, 
fibrous condition or a fine crystalline condition; but it must not show 
a coarse, shiny, crystalline condition. 



4 


400 


500 


600 


Degrees-Falir. 
700 800 900 1000 


1100 


1200 


1300 




80 tnieo 












































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750 


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The accompanying data apply to ],i in. to iH in. round specimens which 
were heated from is to 30 min. at 1560 to 1580 deg. F.; quenched in oil; re- 
heated for 30 min. at temperatures indicated by the abscissae of the curves; 
and finally cooled in air. 

Fig. I.— S. a. E. Steel' No. 1020.' 



a 

80 ^160 

1 75 glSO 

"^ 70 „-140 

Sa65 1-1130 

■5° 

|'-S60 ;i20 

^|55 §110 
g« 

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sn a 

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400 500 600 



Degrees-Fahr. 
_700 800 900 1000 1100 1200 1300 



m a 
I 10 

<D 

f^ 5 





370 425 480 
Degrees-0. 



The accompanying data apply to V2 in. to 1^4 in. round specimens which 
were heated from 15 to 30 mm. at isio to 1530 deg. F.; quenched in oil; re- 
heated tor 30 mm. at temperatures indicated by the abscissse of the curves' 
and finally cooled in air. 

Fig. 3.— S. a. E. Steel No. 1035. 



Specitication No. 1025 

.25 Carbon Steel 

Carbon 20 to .30% (.25% desired) 

Manganese 50 to .80% (.65% desired) 

Phosphorus, not to exceed .045% 

Sulphur, not to exceed .05% 

This steel is used most widely for frames and for ordinary drop 
forgings where moderate ductility is desired, but high strength is not 



400 500 600 700 



80 wlOO 

S75 gl50 
*= o. 

,» o70 »140 

ga 3 

•S .2 65 t^l30 

So o 

1^ s 60 ^120 

§rt55 §110 
a <e 3 
•g »50 _§100 

W -§45 "Z 90 

^ M '^ 

oco40 S 80 

S a a 

-35370 

1-230.2 60 

S |25 I 50 



xi*^ .a 
M 5 15 -a 30 



20 



glO 
5 0, 10 



Degrees-Fahr. 

800 900 1000 nOO 1200 1300 









































































































































































































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205 250 315 370 



800 
750 
700 
650 i 
600 a 
550 f 
500 I 
450 S 
400-5 
350 » 
300° 
250 1 
200 M 
150 
100 
50 




425 480 
Desrees-0 



540 595 650 705 



The accompanying data apply to YzAn. to I'A in. round specimens which 
were heated from 15 to 30 min. at 1540 to 1560 deg. F.; quenched in oil; re- 
heated for 30 min. at temperatures indicated by the abscissse of the curves 
and finally cooled in air. 

Fig. 2.— S. a. E. Steel No. 1025/ 



80 tgieo 

1 75 £150 
2«70 ^140 
I g 65 ^130 
II 60 1 120 
||55 |110 
'2'*J50 5l00 
W545 a 90 



OCN 

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700 800 900 1000 


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700 


















































































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205 260 315 370 



425 480 
Degrees-O 



540 595 650 7.05 



The accompanying data apply to U in. to ij^ in. round specimens which 
were heated from is to 30 min. at 1490 to isiodeg. F.; quenched in oil; re- 
heated for 30 mm. at temperatures indicated by the abscissa of the curves; 
and finally cooled in air. 



per 



Fig. 4.— S. a. E. Steel No. 1045. 

Values are average minimum, except those for hardness, which are average. 
Notation: T.S. = tensile strength, E.L.= elastic limit, R.A. = reduction of area, per cent., ELONG. = elongation in 2 
cent., B.FI. = Brinell hardness, S.H. = scleroscope hardness. 

Figs, i to 4.— Physical characteristics of carbon steels when subjected to heat treatment. 

When cold rolled or cold drawn, this steel will have a yield point of essential. Heat treatment has a moderate effect on the physical 

40,000 to 7S,oooi lbs. per sq. in., and a reduction of area from 35 to 30 properties but this effect is not nearly so marked as on steel 1035. 

per cent. jj^^^j. treatment H ox D may be used for this quality of steel. 

The physical characteristics of this steel when subjected to heat Heat treatment H is the simplest form of heat treatment. The 

treatment are shown m Fig. i. drawing operation (No. 3) must be varied to suit each individual 

1 In sections not over ^ in. round or M in. sheets or flats. case. If great toughness,, and little increased strength are desired, 



380 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



the higher drawing temperatures. may be used, that is in the neighbor- 
hood of iioo deg. Fahr. to 1200 deg. Fahr. If much strength is 
desired and little toughness, the lower temperatures are available. 
Even the lowest of the temperatures given will produce a quality 
of steel, after oil quenching, that is very tough — sufficiently tough 
for many important parts. In fact, with some parts the drawing 
operation (No. 3) can be entirely omitted. 

Results better than obtainable with the above sequence of opera- 
tions can be obtained by the double treatment of heat treatment D 
which produces a refinement of grain not possible with one treatment 
and is resorted to in parts where extremely good qualities are desired. 

This quality of steel is not intended for case-hardening, but by 
careful manipulation it may be so treated. This should be done in 
emergencies only, rather than as a regular practice and, if at all, only 
with the double treatinent followed by the drawing operation; that is, 
the most painstaking form of case-hardening. 

The physical characteristics of this steel when subjected to heat 
treatment H are shown in Fig. 2. 

Specification No. 1035 
.35 Carbon Steel 

Carbon 30 to .40% (.35% desired) 

Manganese 50 to .80% (.65% desired) 

Phosphorus, not to exceed -045% 

Sulphur, not to exceed -05% 

This material is sometimes referred to in the trade as .35 carbon 
machine steel. 

It is primarily for use as a structural steel. It forges well, machines 
well and responds to heat treatment in the matter of strength as well as 
toughness; that is to say, intelligent heat treatment will produce 
marked increase in the yield point. It can be used for all forgings 
such as axles, driving-shafts, steering pivots and other structural 
parts. It is the best all-round structural steel for such use as its 
strength warrants. 

Heat treatment for toughening and strength is of importance with 
this steel. The heat treatment must be modified in accordance with 
the experience of the individual user, to suit the size of the part 
treated and the combination of strength and toughness desired. The 
steel should be heat treated in all cases where reliability is important. 

Machining may precede the heat treatment, depending somewhat 
upon convenience and the character of the treatment. If the highest 
strength is demanded, a strong quenching medium must be employed; 
for example, brine. In such case, the yield point will be correspond- 
ingly high and the steel correspondingly hard and difficult to machine. 
On the other hand, if a moderately high yield point is all that is 
desired, an oil quench wUl suffice and machining may follow without 
any difficulty whatever. 

Heat treatment H, D, or E may be used on this quality of steel. 
When heat treatment E is applied, machining may follow operation 2. 

The physical characteristics of this steel when subjected to heat 
treatment U are shown in Fig. 3. 

Specification No. 1045 ^ 

.45 Carbon Steel 

Carbon 40 to . 50% ( . 45% desired) 

Manganese 5° to .80% (.65% desired) 

Phosphorus, not to exceed .045% 

Sulphur, not to exceed .05% 

This material is ordinarily known to the trade as .45 carbon machine 
steel. This quality represents a structural steel of greater strength 
than steel 1035. Its uses are more limited and are confined in a 
general way to such parts as demand a high degree of strength and a 
considerable degree of toughness. At the same time, with proper 
heat treatment the fatigue-resisting (endurance) qualities are very 
high — higher than those of any of the foregoing steels. 



This steel is commonly used for crankshafts, driving-shafts and pro- 
peller-shafts. It has also been used for transmission gears, but it is 
not quite hard enough without case-hardening and is not tough 
enough with case-hardening to make safe transmission gears. It 
should not be used for case-hardened parts. Other specifications are 
decidedly better for this purpose. 

In a properly annealed condition it machines well — not well enough 
for screw machine work, but certainly well enough for all-round 
machine shop practice. Heat treatment E provides the annealing 
operation when needed, machining to follow operation 2; this treat- 
ment is especially adapted to crankshafts and similar parts. Heat 
treatment // is also commonly used for this quality of steel. 

The physical characteristics under this heat treatment are shown 
in Fig. 4. 

Specification No. 1095 
.95 Carbon Steel 

Carbon 90 to 1.05% (.95% desired) 

Manganese 25 to . 50% (.35% desired) 

Phosphorus, not to exceed • 04 % 

Sulphur, not to exceed . 05% 

This is a grade of steel used generally for springs. Properly heat 
treated, extremely good results are possible. 

The hardening and drawing of springs, that is, the heat treatment 
of them, is, as a rule, in the hands of the springmaker, but in case 
it is desired to treat, as for small springs, heat treatment F is 
recommended. 

It must be understood that the higher the drawing temperature 
(operation 3), the lower will be the yield point of the material. On 
the other hand, if the material be drawn at too low a temperature, 
it will be brittle. A few practical trials will locate the best temper 
for any given shape or size. 

The physical characteristics of heat treated spring steel are best 
determined by transverse test. This is because steel as hard as 
tempered spring steel is very difficult to hold firmly in the jaws of a 
tensile testing machine. There is more or less slip, and side strains 
are bound to occur, all of which tend to produce misleading results. 

The physical characteristics in the annealed condition may be 
omitted, inasmuch as this grade of steel is not ordinarily used for 
structural parts in such condition. 

Careful examination of the fracture of the treated material is 
desirable. After tempering no suitable spring steel should be coarsely 
crystalline. It should be finely crystalline, and in some cases will 
show a partly fibrous fracture. 

Physical Characteristics 
(Transverse Test) 





Heat treatment F 


Elastic limit (initial set), lbs. per sq. in 

Reduction of area 


90,000 
to 
180,000 
Not determined 


Elongation 


in transverse test 
Not determined 




in transverse test 



Screw Stock 

Specification No. 1114 

Carbon 08 to . 20% 

Manganese 30 to . 80% 

Phosphorus, not to exceed .12% 

Sulphur 06 to . 1 2% 

This steel may be made by any process. It is intended for use 
where high screw machine production is the important factor. 



STEEL 



381 



strength and toughness being secondary considerations. Its com- 
position and texture are of such nature as to permit the rapid removal 
of metal and a resulting smoothness of finish. 

Steel Castings 

Specification No. 1235 

Carbon As required for physical properties 

Phosphorus, not to exceed .05% 

Sulphur, not to exceed .05% 

In the following remarks, genuine steel castings, and not malleable 
iron or complex mixtures often found in the market masquerading 
under the name of steel, are referred to. 

All steel castings should be annealed and some shapes may be heat 
treated to great advantage. Like other castings, steel castings are 
subject to blow-holes. Consequently, they should not be used in the 
vital parts of an automobile. It is impossible to inspect against 
blow-holes, and steel castings for axles, crankshafts and steering 
spindles are used only at great risk. Freedom from blow-holes and 
proper physical condition are of more importance than the absolute 
analysis. 

On account of the great influence of varying types of foundry 
practice upon the properties of castings, it has not been found feasible 
to give a closer specification for chemical composition than that 
quoted under No. 1235. If it is desired to buy steel castings under 
precise specifications, the following, based upon the Specifications 
for Steel Castings, Class B, Serial Designation A 27-14, of the 
American Society for Testing Materials, can be used: 

The steel may be made by any process approved by the pur- 
chaser. Three grades are recognized: hard, medium, and soft. 

All castings shall be allowed to become cold; they shall then be 
reheated uniformly to the proper temperature to refine the grain, and 
allowed to cool uniformly and slowly. 

No casting, on check analysis, shall show over .05 per cent, phos- 
phorus or sulphur. The carbon content shall be suitable for the 
physical tests and service required. 

Drillings for analysis shall be so taken as to represent the average 
composition of the casting. 

The finished castings shall conform to the following minimum 
requirements as to tensile properties: 



Hard 


Medium 


Soft 


Tensile strength, lb. per sq. in 


80,000 
36,000 

20 

15 


70,000 
31.500 

25 
18 


60,000 


Yield point, lb. per sq. in 


27,000 
30 
22 


Reduction of area, per cent 


Elongation in 2 in., per cent 







The test specimen for soft castings shall bend cold through 120 
deg., and for medium castings through go deg., around a i-in. pin 
without cracking on the outside. Hard castings shall not be subject 
to bend-test requirements. 

In the case of small or unimportant castings, a test to destruction 
on three castings from a lot may be substituted for the tension and 
bend tests. This test shall show the material to be ductile, free 
from injurious defects, and suitable for the purpose intended. A lot 
shall consist of all castings from one melt, in the same annealing 
charge. In case test bars are cast separate, they shall be annealed 
with the lot they represent, the method of casting such test bars, or 
of casting test bars attached to castings, to be agreed upon by pur- 
chaser and manufacturer. 

Tension test specimens shall be machined to the standard S. A. E. 
form; bend test specimens shall be machined to i by J^ in. in section, 
with corners rounded to a radius not over J^e in. 

One tension and one bend test shall be made from each annealing 
charge. If more than one melt is represented in an annealing 
charge, one tension and one bend test shall be made from each melt. 

If any test specimen shows defective machining or develops flaws, it 



may be discarded; in which case another specimen may be selected 
by the manufacturer and the purchaser. 

A retest shall be allowed if the percentage of elongation is less than 
that specified, or if any part of the fracture is more than ^ in. from 
the center of the gage length, as indicated by scribe scratches marked 
on the specimen before testing. 

The finished castings shall conform substantially to the sizes and 
shapes of the patterns, shall be made in a workmanlike manner, and 
be free from injurious defects. 

Minor defects which do not impair the strength of the castings 
may, with the approval of the purchaser, be welded by an approved 
process. The defects shall first be cleaned out to solid metal; and 
after welding, the castings shall be annealed. 

Castings offered for inspection shall not be painted or covered with 
any substance that will hide defects, nor rusted to such an extent as to 
hide defects. 

Alloy Steels 

In connection with the purchase and use of alloy steels it should 
be borne in mind that such steels should be used in the treated 
condition only, that is, not in an annealed or natural condition. 
In the latter condition there is a slight benefit, perhaps, as compared 
with plain carbon steels, but as a rule nothing commensurate with 
the increased cost. In the heat-treated condition, however, there 
is a very marked improvement in physical characteristics. 

Nickel Steels 

Specification No. 2315 
.15 Carbon, sM Per Cent. Nickel Steel 

Carbon , 10 to .20% ( . 15% desired) 

Manganese so to . 80% ( .65% desired) 

Phosphorus, not to exceed .04% 

Sulphur, not to exceed -05% 

Nickel 3-25103.75% (3 . 50% desired) 

This quality of steel is embraced in these specifications to furnish 
a nickel steel that is suitable for carbonizing purposes. Steel of 
this character, properly carbonized and heat treated, will produce a 
part with an exceedingly tough and strong core, coupled with the 
desired high-carbon exterior. 

This steel is also available for structural purposes, but is not one 
to be selected for such purpose when ordering materials. Much 
better results will be obtained with one of the other nickel steels of 
higher carbon. It is intended for case-hardened gears, for both 
the bevel driving and transmission systems, and for such other case- 
hardened parts as demand a very tough, strong steel with a hardened 
exterior. 

The case-hardening sequence may be varied considerably, as with 
steel 1020, those parts of relatively small importance requiring a 
simpler form of treatment. As a rule, however, those parts which 
require the use of nickel steel require the best tj^ie of case- 
hardening, that is, heat treatment G. 

The second quench (operation 6) must be conducted at the lowest 
possible temperature at which the material will harden. It will be 
found that sometimes this is lower than 1300 deg. Fahr. 

In connection with certain uses it will be found possible to omit the 
final drawing (operation 7) entirely, but for parts of the highest 
importance this operation should be followed as a safeguard. Parts 
of intricate shape, with sudden changes of thickness, sharp corners 
and the like, particularly sliding gears, should always be drawn ,. in 
order to relieve the internal strains. 

Much is to be learned from the character of the fracture. The 
center should be fibrous in appearance, and the exterior, high-carbon 
metal closely crystalline, or even silky. 

When used for structural purposes, the physical characteristics will 
range about as follows: 



382 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Physical Characteristics 



Yield point, lbs. per sq. in. 



Reduction of area . . 
Elongation in 2 ins. 



Annealed 



35,000 
to 

45,000 



Heat treatment 
H OT K 



40,000 

to 
So, 000 



to J" — 

65-45 per cent . 65-40 per cent. 
35-25 per cent. | 35-15 per cent. 



Specification No. 2320 
.20 Carbon, s}4 Per Cent. Nickel Steel 

Carbon 15 to .25% (. 20% desired) 

Manganese 50 to .80% ( . 65% desired) 

Phosphorus, not to exceed .04% 

Sulphur, not to exceed .045% 

Nickel 7 3.25 to 3.75% (3.50% desired) 

This quality may be used interchangeably with steel 2315. Al- 
though intended primarily for case-hardening, it can be properly used 
for structural parts, with suitable heat treatment, and will give elas- 
tic limits somewhat higher than material provided by the preceding 
specification. For case-hardening heat treatment G should be fol- 
lowed, and for structural purposes the treatment should be in ac- 
cordance with heat treatment H or K; the quenching temperatures, 
as with other steels, being modified to meet individual cases. 



Physical 


Characteristics 








Annealed 


Heat treatment 
H OT K 


Yield point, lbs. per sq. in 




,40,000 

to 
50,000 




50,000 
to 

125,000 


Reduction of area 




65-40 per cent. 


65- 


-40 per cent. 


Elongation in 2 ins 




30-20 per cent. 


25- 


-10 per cent. 



Specification No. 2330 
.30 Carbon, 3}^ Per Cent. Nickel Steel 

Carbon 25 to .35% ( .30% desired) 

Manganese 50 to .80% ( .65% desired) 

Phosphorus, not to exceed 04% 

Sulphur, not to exceed . 045% 

Nickel 3-25 to 3.75% (3.50% desired) 

This quality of steel is primarily for heat-treated structural parts 
where strength and toughness are sought; such parts as axles, front- 
wheel sf)indles, crankshafts, driving-shafts and transmission shafts. 
Wide variations of yield point or elastic limit are possible by the 
use of different quenching mediums — oil, water or brine — and varia- 
tion in drawing temperatures, from 500 deg. Fahr. up to 1200 deg. 
Fahr. (heat treatment H). A higher refinement of this treatment 
is heat treatment K. 

Physical Characteristics 

The physical characteristics of this steel may be considered as 
practically those obtained with steel 2320, slight modifications in 
the treatment much more than ofi'setting the slight difference in 
the carbon content. 



Yield point or elastic limit, lbs. per 
sq. in. 



Reduction of area . . 
Elongation in 2 ins. 



Annealed 



40,000 
to 

50,000 
60-40 per cent. 
30-20 per cent. 



Heat treatment 
H 01 K 



60,000 

to 

130,000 

60-30 per cent. 

25-10 per cent. 



Specification No. 2335 
.35 Carbon, 3H Per Cent. Nickel Steel 

Carbon 30 to .40% ( .35% desired) 

Manganese 50 to . 80% ( . 65% desired) 

Phosphorus, not to exceed '. . . . 04% 

Sulphur, not to exceed . 045% 

Nickel 3 . 25 to 3 . 75% (3 . 50% desired) 

This quality of steel is subject to precisely the same remarks as 
steel 2330. It will respond a little more sharply to heat treat- 
ment and can be forced to higher elastic limits. The difference 
will be small except in extreme cases. 

Physical Characteristics 



Yield point or elastic limit, lbs. per 
sq. in. 



Reduction of area. . 
Elongation in 2 ins. 



Annealed 



45,000 
to 

55.000 
55-35 per cent. 
25-15 per cent. 



Heat treatment 
H or K 



65,000 

to 

160,000 

55-25 per cent. 

25-10 per cent. 



Specification No. 2340 
.40 Carbon 3 }4 Per Cent. Nickel Steel 

Carbon 3510 .4 %( .40% desired) 

Manganese 50 to .80% ( .65% desired) 

Phosphorus, not to exceed .04% 

Sulphur, not to exceed .045% 

Nickel 3.25 to 3.75% (3.50% desired) 

The above nickel steel is a quality not in wide use but available for 
certain purposes. The carbon content being higher than generally 
used, greater hardness is obtainable by quenching; and as increased 
brittleness accompanies the greater hardness, the treatments given 
must be modified to meet such condition. For example, the final 
quench may be at a considerably lower temperature, and the final 
drawing temperature, or partial annealing, must be carefully chosen, 
in order to produce the desired toughness and other physical 
characteristics. 

Physical Characteristics 



Yield point or elastic limit, lbs. per 
sq. in. 



Reduction of area . . 
Elongation in 2 ins. 



Annealed 



55.000 
to 

65,000 
50-30 per cent. 
25-15 percent. 



Heat treatment 
H 01 K 



70,000 

to 

200,000 

55-15 per cent. 

20- 5 per cent. 



Nickel Chromium Steels 

In general it can be said in the case of the nickel chromium 
steels that the heat treatments and the properties induced thereby 
are much the same as in the case of plain nickel steels, except 
that the effects of the heat treatments are somewhat augmented 
by the presence of the chromium, and further that these effects 
increase with increasing amounts of nickel and chromium. 

Specification No. 3120 
15 to .25% (.20% desired) 



■80% (.65% desired) 

•04% 

•045% 



Carbon 

Manganese 50 to 

Phosphorus, not to exceed 

Sulphur, not to exceed 

Nickel 1. 00 to 1.50% (1.25% desired) 

Chromium 45 to .75%^( .60% desired) 

1 Another grade of this type of steel is available with chromium content 
of .15 per cent, to .45 per cent. Its physical properties are somewhat lower 
than those of the grade with chromium content indicated in Specifications 
Nos. 3120, 3125, 3130, 3135 and 3140- 



STEEL 



383 



This quality of steel is intended primarily for case-hardening (heat 
treatment G). It may also be used for structural parts with suitable 
heat treatment (heat treatment H oi D). It should not be used in 
the natural or untreated condition. 

Physical Characteristics 



Steels 3135, 3140 





Annealed 


Heat treat- 
ment H ot D 


Yield point or elastic limit, lbs. per 


30,000 


40,000 


sq. in. 


to 


to 




40,000 


100,000 


Reduction of area 


55-40 per cent. 
35-25 per cent. 


65-40 per cent. 
25-15 per cent. 


Elongation in 2 ins 



Specifications Nos. 3125, 3130, 3135, 3140 

These qualities of steel are intended primarily for structural pur- 
poses in a heat-treated condition (heat treatment H, D or E). Steel 
3125 may be used for case-hardening, as also may steel 3130 if 
necessary. 

Specification No. 3125 



Carbon 20 to 

Manganese 50 to 

Phosphorus, not to exceed 

Sulphur, not to exceed 

Nickel ". . . 1 . 00 to I 

Chromium 45 to 



.30% ( .25% desired) 

• 80% ( .65% desired) 
■ 04% 

• 045% 

.50% (1.25% desired) 

• 7S%K -60% desired) 



Specification No. 3130 



Carbon 25 to 

Manganese 50 to 

Phosphorus, not to exceed 

Sulphur, not to exceed 

Nickel 1 . 00 to I 

Chromium 45 to 



■ 35% ( -30% desu-ed) 

■ 80% ( .65% desired) 

• 04% 

• 045% 

.50% (1.25% desired) 

• 75%K -60% desired) 



Specification No. 3135 

Carbon 30 to .40% ( .35% desired) 

Manganese 50 to .80% ( .65% desired) 

Phosphorus, not to exceed . 04% 

Sulphur, not to exceed . 045% 

Nickel 1 . 00 to 1 . 50% (i . 25% desired) 

Chromium 45 to .7S%K .60% desired) 

Specification No. 3140 

Carbon 35 to .45% ( .40% desired) 

Manganese 50 to . 80% ( . 65% desired) 

Phosphorus, not to exceed .04% 

Sulphur, not to exceed . 045% 

Nickel 1. 00 to 1,50% (i .25% desired) 

Chromium 45 to .7S%H .60% desired) 

Physical Characteristics 
Steels, 3125, 3130 







Heat treat- 




Annealed 


ment, H, D, or 
E 


Yield point or elastic limit, lbs. per 


40,000 


50,000 


sq. m. 


to 


to 




55,000 


125,000 


Reduction of area 


50-35 per cent. 


55-25 per cent. 


Elongation in 2 ins 


30-20 per cent. 


25-10 per cent. 



' Another grade of this type of steel is available with chromium content of 
.IS per cent, to .43 per cent. Its physical properties are somewhat lower than 
those of the grade with chromium content indicated in Specifications Nos. 
3120, 3125, 3130, 3135 and 3140. 







Heat treat- 




Annealed 


ment, H, D 
or E 



Yield point or elastic limit, lbs. per 
sq. in. 



Reduction of area . . 
Elongation in 2 ins. 



45,000 
to 

60,000 
45-30 per cent. 
25-15 percent. 



SS.ooo 
to 

150,000 
50-25 per cent. 
20- 5 per cent. 



Specification No. 3220 

Carbon 15 to .25% ( .20% desired) 

Manganese 30 to .60% ( .45% desired) 

Phosphorus, not to exceed -04% 

Sulphur, not to exceed . 04% 

Nickel 1.50 to 2.00% (1.75% desired) 

Chromium 90 to i . 25% (i . 10% desired) 

This steel is intended for case-hardened parts of nickel chromium 
steel. Case-hardened parts demanding this grade of steel also 
demand the most careful heat treatment (heat treatment G). It 
may also be used for structural purposes with heat treatment /? or D. 



Physical Characteristics 




Annealed 


Heat treat- 
ment H OT D 


Yield point or elastic limit, lbs. per 




35,000 


45,000 


sq. m. 




to 


to 






45,000 


120,000 


Reduction of area 


60 


-45 per cent. 


65-30 per cent. 


Elongation in 2 ins 


25- 


-20 per cent. 


20- 5 per cent. 



Specification No. 3230 

Carbon 25 to .35% ( .30% desired) 

Manganese 30 to .60% ( .45% desired) 

Phosphorus, not to exceed .04% 

Sulphur, not to exceed 04% 

Nickel 1 . 50 to 2 . 00% (r . 75% desired) 

Chromium go to i. 25% (i. 10% desired) 

This steel is intended for the most important structural parts and 
should be used only in a heat-treated condition (heat treatment H 
or D). 

Physical Characteristics 



Yield point or elastic limit, lbs. per 
sq. in. 



Reduction of area . . 
Elongation in 2 ins. 



Annealed 



40,000 
to 

50,000 
55-40 per cent. 
25-15 percent. 



Heat treat- 
ment H or D 



60,000 
to 

175,000 
60-30 per cent. 
20- 5 per cent. 



Specification No. 3240 

Carbon 35 to .45% ( .40% desired) 

Manganese '. . .30 to .60% ( .45% desired) 

Phosphorus, not to exceed 04% 

Sulphur, not to exceed • 04 % 

Nickel 1 .50 to 2.00% (i .75% desired) 

Chromium _ . . 90 to i . 25% (i . 10% desired) 

This quality of steel is suitable for structural parts where unusual 
strength is demanded. Higher elastic limit is possible under a given 
treatment than with material like steel 3230. The toughness will 
not be quite as great, but this does not bar the material from uses 
where toughness is not the controlling factor and where strength is. 

Heat treatment F or Z) is recommended. 



384 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Physical Characteristics 



Physical Characteristics 





Annealed 


Heat treat- 
ment H OT D 


Yield point or elastic limit, lbs. per 


4S,ooo 


65,000 


sq. in. 


to 


to 




60,000 


200,000 


Reduction of area 


50-40 per cent. 
25-15 percent. 


50-20 per cent. 
15-2 percent. 


Elongation in 2 ins. 





Specification No. 3250 

Carbon 45 to . 55% (. 50% desired) 

Manganese 30 to . 60% ( .45% desired) 

Phosphorus, not to exceed 04% 

Sulphur, not to exceed 04% 

Nickel 1.50 to 2.00% (1.75% desired) 

Chromium 90 to i. 25% (i. 10% desired) 

This steel is intended for gears where extreme strength and hard- 
ness are necessary. 

To heat treat for gears either heat treatment M or Q should be fol- 
lowed, the latter giving the better results. 



Physical Characteristics 






Annealed 


Heat treat- 
ment AI or Q 


Yield point or elastic limit, lbs. per 


50,000 


150,000 


sq. in. 


to 


to 




60,000 


250,000 


Reduction of area 


50-40 per cent. 


25-15 per cent. 


Elongation in 2 ins 


25-15 per cent. 


15-2 percent. 



Specification No. X33 15 



Carbon 10 to 

Manganese 45 to 

Phosphorus, not to exceed 

Sulphur, not to exceed 

Nickel 2 . 75 to 3 

Chromium 60 to 



.20% ( .15% desired) 

• 75% ( -60% desired) 
■ 04% 

.04% 

. 25% (3.00% desired) 

• 95% ( -80% desired) 



This steel is intended primarily for case-hardening. It is higher in 
nickel and chromium than the preceding nickel chromium steels. 
Heat treatment G should be followed. 

It is sometimes used for structural parts, when heat treatment M is 
applicable. 

Physical Characteristics 



Yield point or elastic limit, lbs. per 
sq. in. 



Reduction of area . . 
Elongation in 2 ins. 



Annealed 



35.000 

to 

45,000 

60-45 per cent. 



Heat treatment 
M 



yj^j—^^ pel cc:iiu. ^0 0^ pci i^ciiu 

25-20 per cent. 20- 5 per cent 



40,000 

to 

100,000 

65-30 per cent. 



Specification .No. X3335 



Carbon 30 to 

Manganese 45 to 

Phosphorus, not to exceed 

Sulphur, not to exceed 

Nickel r 2 . 75 to 3 

Chromium 60 to 



• 40% ( .35% desired) 

• 75% ( ^60% desired) 

• 04% 
.04% 

■ 25% (3-00% desired) 
.95% ( .80% desired) 



This steel is intended for structural parts of the most important 
character, such as crankshafts, axles, spindles, drive-shafts and trans- 
mission shafts. Heat treatment P or i? is recommended. 

This steel is not intended for case-hardening. 



Yield point or elastic limit, lbs. per 
sq. in. 



Reduction of area . . 
Elongation in 2 ins. 



Annealed 



45,000 
to 

55,000 
55-40 per cent. 
25-15 per cent. 



Heat treatment 
Por i? 



60,000 

to 

175,000 

60-30 per cent. 

20- 5 per cent. 



Specification No. X33S0 



Carbon 45 to 

Manganese 45 to 

Phosphorus, not to exceed 

Sulphur, not to exceed 

Nickel 2.75 to 3 

Chromium 60 to 



•55% ( ^50% desired) 

• 75%o ( .60% desired) 
■ 04% 

.04% 

• 25% (3.00% desired) 

• 95% ( •80% desired) 



This steel is an alternative quality for gears. The remarks made 
on steel 3250 apply to this case. The physical characteristics are 
similar to those of steel 3250. Heat treatment R should be used, 
although P is applicable. 

Specification No. 3320 

Carbon 15 to . 25% ( . 20% desired) 

Manganese 30 to .60% ( .45% desired) 

Phosphorus, not to exceed _ -04% 

Sulphur, not to exceed .04% 

Nickel 3 • 25 to 3 . 75 % (3 . 50% desired) 

Chromium i . 25 to i . 75% (i . 50% desired) 

The remarks made in connection with steel 3220 apply to this 
steel also. There is no appreciable difference in the physical char- 
acteristics. Carbonizing should follow the practice indicated under 
heat treatment L. 



Specification No. 3330 



Carbon 25 to 

Manganese 30 to 

Phosphorus, not to exceed 

Sulphur, not to exceed 

Nickel 3.25 to 3 

Chromium i . 25 to i 



• 35% ( •30% desired) 
.60% ( .45% desired) 

• 04% 

• 04% 

• 75% (3 •50%) desired) 

• 75% (^•50% desired) 



This steel, like No. 3230, is intended for very important structural 
parts. The high nickel and chromium contents make it exceedingly 
tough and strong when treated according to heat treatments P or R. 
Specification No. 3340 



Carbon 35 to 

Manganese 30 to 

Phosphorus, not to exceed 

Sulphur, not to exceed 

Nickel 3 . 25 to 3 

Chromium i . 2 5 to i 



• 45% ( ^40% desired) 
.60% ( .45% desired) 

• 04% 

• 04% 

• 75% {3-50% desired) 

• 75% (^•5o%> desired) 



This steel is suitable for gears to be hardened without carbonizing. 
The remarks made in connection with steels 3240 and 3250 apply. 
Heat treatment P or R should be used. 

Chromium Steels 
Specification No. 5120 

Carbon 15 to. 25% (. 20% desired) 

Manganese ^ 

Phosphorus, not to exceed .04% 

Sulphur, not to exceed ^045% 

Chromium 65 to. 85% (. 75 %desired) 

This steel is similar in properties to 2320 and 3120 in that it 
is a case-hardening grade of much better quality than carbon steel. 
Heat treatment B should be used. 

• See foot note ne.xt page, first column. 



STEEL 



385 



Specification No. 5140 

Carbon 35 to .45% (.40% desired) 

Manganese ' 

Phosphorus, not to exceed .04% 

Sulphur, not to exceed .045% 

Chromium 65 to . 85% ( . 75% desired) 

This grade of steel is very similar in properties to steel 3140. 
When treated according to H ox D it becomes useful for high-duty 
shafting, etc. The drawing temperature should be moderaetly high 
in order to maintain a safe degree of toughness. 

Specification No. 5195 

Carbon 90 to 1.05% ( -95% desired) 

Manganese 20 to .45% 

Phosphorus, not to exceed .03% 

Sulphur, not to exceed .03% 

Chromium 90 to 1.10% (1.00% desired) 

Specification No. 51120 

Carbon i . 10 to i . 30% (1.20% desired) 

Manganese 20 to .45% ( .35% desired) 

Phosphorus, not to exceed . 03 % 

Sulphur, not to exceed .03% 

Chromium 90 to i . 10% (i . 00% desired) 

Specification No. 5295 

Carbon 90 to 1.05% ( .95% desired) 

Manganese 20 to .45% ( .35% desired) 

Phosphorus, not to exceed . 03 % 

Sulphur, not to exceed.' .03% 

Chromium i . 10 to i . 30% (i . 20% desired) 

Specification No. 52120 

Carbon i.io to 1.30% (i. 20% desired) 

Manganese 20 to .45% ( .35% desired) 

Phosphorus, not to exceed • 03 % 

Sulphur, not to exceed .03% 

Chromium i.io to 1.30% (i. 20% desired) 

The above four grades of steel are used almost exclusively for ball- 
bearing cups and cones, where their extreme hardness is indispensable. 
The treatment of these steels is in the hands of specialists, but in a 
general way treatment P and R illustrate the procedures followed. 

Chromium Vanadium Steels 

Specification No. 6120 
.20 Carbon, Chromium Vanadium Steel 

Carbon 15 to . 25% (. 20% desired) 

Manganese 50 to . 80% ( . 65% desired) 

Phosphorus, not to exceed .04% 

Sulphur, not to exceed .04% 

Chromium 80 to 1. 10% (.95% desired) 

Vanadium, not less than . 15% (. 18% desired) 

This quality is primarily for case-hardening. It is used for the 
most important case-hardened parts; that is, case-hardened shafts, 
gears and the like. 

This steel may also be used in a heat-treated condition for struc- 
tural purposes, but for such work some of the steels following are to 
be preferred, particularly where higher strength is desired. 

The case-hardening treatment recommended is that covered by 
heat treatment S. 

' Two types of steel are available in this class, viz., one with manganese .25 
to .50 per cent. (.35 per cent, desired), and silicon not over .20 per cent.; the 
other with manganese .60 to .80 per cent. (.70 per cent, desired), and silicon 
.15 to .50 per cent. 

25 



Physical Characteristics 








Annealed 


Heat treatment 
T 


Yield point or elastic limit, lbs. per 


40,000 




S5,ooo 


sq. in. 


to 




to 




50,000 




100,000 


Reduction of area 


65-50 per cent. 
30-20 per cent. 


65- 
25- 


-45 per cent. 
-10 per cent. 


Elongation in 2 ins 



Specification No. 6125 
.25 Carbon, Chromium Vanadium Steel 

Carbon 20 to .30% (. 2 5 % desired) 

Manganese 50 to .80% 

Phosphorus, not to exceed 04% 

Sulphur, not to exceed .04% 

Chromium 80 to i . 10% 

Vanadium, not less than .15% 



(.65% desired) 



(.95% desired) 
(.18% desired) 



The difference between this and the preceding steel is very slight 
and they may be used interchangeably for structural purposes. This 
steel may be case-hardened but is not first choice for this purpose. 

The physical characteristics can be considered as practically the 
same as those given for steel 6120. 

Physical Characteristics 





Annealed 


Heat treat- 
ment T 


Yield point or elastic limit, lbs. per 




40,000 




S5,ooo 


sq. in. 




to 
50,000 




to 

100,000 


Reduction of area 


6S 


-50 per cent. 


65- 


-45 per cent. 


Elongation in 2 ins 


32 


-20 per cent. 


25- 


-10 per cent. 



Specification No. 6130 
.30 Carbon, Chromium Vanadium Steel 

Carbon 25 to .35% (. 30% desiref ) 

Manganese 50 to .80% (.65% desired; 

Phosphorus, not to exceed -04% 

Sulphur, not to exceed -04% 

Chromium 80 to 1.10% (.95% desired) 

Vanadium, not less than .15% (. 18% desired) 

This quality of steel is intermediate in the carbon range and can 
be used interchangeably with steel 6125 for structural purposes. 
It should not be used for case-hardening. When subjected to heat 
treatment T it possesses a high degree of combined strength and 
toughness. 

Physical Characteristics 





Annealed 


Heat treat- 
ment T 


Yield point or elastic limit, lbs. per 


45,000 


60,000 


sq. in. 


to 


to 




SS>ooo 


150,000 


Reduction of area 


60-50 per cent. 
25-20 per cent. 


55-25 percent. 
15- 5 per cent. 


Elongation in 2 ins 



Specification No. 6135 
.35 Carbon, Chromium Vanadium Steel 
This specification provides a fijst-rate quality of steel for structural 
parts that are to be heat treated. The fatigue-resisting (endurance) 
qualities of this material are excellent. 



386 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Carbon 30 to . 40% 

Manganese 50 to . 80% 

Phosphorus, not to exceed -04% 

Sulphur, not to exceed .04% 

Chromium 80 to i . 10% 

Vanadium, not less than .15% 

Physical Characteristics 



(.35% desired) 
(.65% desired) 



(.95% desired) 
(.18% desired) 





Annealed 


Heat treat- 
ment T 


Yield point or elastic limit, lbs. per 


4S,ooo 


60,000 


sq. in. 


to 


to 




SS>ooo 


150,000 


Reduction of area 


60-50 per cent. 


55-25 percent. 


Elongation in 2 ins 


25-20 per cent. 


15-5 percent. 



Specification No. 6140 
.40 Carbon, Chromium Vanadium Steel 

Carbon 35 to .45% (.40% desired) 

Manganese 50 to . 80% 

Phosphorus, not to exceed .04% 

Sulphur, not to exceed .04% 

Chromium 80 to i . 10% 

Vanadium, not less than . 15% 



(.65% desired) 



(.95% desired) 
(.18% desired) 



This is a very good quality of steel to be selected where a high de- 
gree of strength is desired, coupled with a good measure of toughness. 
Its fatigue-resisting qualities are very high, and it is a first-class 
material for high-duty shafts. Heat treatment T is recommended: 

Physical Characteristics 



Yield point or elastic limit, lbs. per 
sq. in. 



Reduction of area . . 
Elongation in 2 ins. 



Annealed 



50,000 
to 

60,000 
55-45 per cent. 
25-15 per cent. 



Heat treatment 
T 



65,000 

to 

175,000 

50-15 per cent. 

15-2 per cent. 



Carbon 

Manganese 

Phosphorus, not to exceed 
Sulphur, not to exceed. . . 

Chromium 

Vanadium, not less then . . 



Specification No. 6145 
.45 Carbon, Chromium Vanadium Steel 

40 to .50% (.45% desired) 

50 to . 80% 

.04% 

.04% 

.80 to 1.10% 

.15% 



(.65% desired) 



(.95% desired) 
(.18% desired) 



This quality of steel contains sufficient carbon in combination with 
chromium and vanadium to harden to a considerable degree when 
quenched at a proper temperature, and may be used for gears and 
-springs. 

For structural parts where exceedingly high strength is desirable 
heat treatment T should be followed. 

For gears this steel should be annealed after forging, and 
before machining. 

Physical Characteristics 



Annealed 



Yield point or elastic limit, lbs. per 
sq. in. 



Reduction of area. . 
Elongation in 2 ins. 



55,000 
to 

65,000 
55-40 per cent. 
25-15 percent. 



Heat treat- 
ment U 



150,000 
to 

200,000 
25-10 per cent. 
10- 2 percent. 



Specification No. 6150 

.50 Carbon, Chromium Vanadium Steel 

Carbon 45 to . 55% (.50% desired) 



Manganese 50 to 

Phosphorus, not to exceed 

Sulphur, not to exceed 

Chromium 80 to i 

Vanadium, not less than 



80% (.65% desired) 

04% 

04% 

10% (.95% desired) 

15% (.18% desired) 



Substantially the same remarks as made in regard to steel 6145 
apply to this steel. In this grade, however, we also find a material 
that is suitable for springs. With a proper sequence of heating, 
quenching and drawing, very high elastic limits are obtained. 

For spring material heat treatment U is recommended, except that 
the last drawing (operation 6) will be carried farther — probably from 
700-1100 deg. Fahr. This final drawing temperature will have to 
vary with the section of material being handled, whether light spiral 
springs or heavy flat springs. 

Physical Characteristics 



Yield point or elastic limit, lbs. per 
sq. in. 



Reduction of area . . 
Elongation in 2 ins . 



Annealed 



60,000 
to 

70,000 
50-35 per cent. 
20-15 psr cent. 



Heat treatment 
U 



150,000 
to 

225,000 
35-15 per cent. 
10-2 per cent. 



Silico-manganese Steels 

Specification No. 9250 
Carbon 45 to .55% 



( .50% desired) 
( .70% desired) 



Manganese 60 to .80% 

Phosphorus, not to exceed .045%^ 

Sulphur, not to exceed -045% 

Silicon 1.80 to 2.10% (1.95% desired) 



Specification No. 9260 



Carbon 

Manganese 

Phosphorus, not to exceed. 

Sulphur, not to exceed 

Silicon 



■55 to 
.50 to 



I . 50 to I 



.65% ( .60% desired) 

.70% ( .60% desired) 

.045%' 

• 045% 

.80% (1.65% desired) 



These steels have been standardized by usage principally as spring 
steels. No. 9260 is also used to some extent for gears. Neither 
steel is suitable for use without heat treatment. 

Both of these specifications are provided in order to meet the re- 
quirements of two groups of users: Those who believe in relatively 
low carbon and high silicon, and those who desire higher carbon and 
lower silicon. When properly treated, their physical properties will 
not differ appreciably, though steel 9250 will probably be slightly 
the tougher of the two. Heat treatment V is suitable for both gears 
and springs. 

Steel 9260 will become harder when quenched in the same 
medium as steel 9250. The latter, however, is more often quenched 
in water, while steel 9260 is generally quenched in oil — a circum- 
stance which largely counteracts the influence of the composition. 
Furthermore, variation in the temperature of drawing will sufi&ce 
to balance the properties closely. 

The exact temperature for quenching and drawing, and the proper 
medium should be determined for each case. In general, gears are 
drawn between 450 and 550 deg. Fahr. and springs between 800 and 
1000 deg. Fahr. 

1 Steel made by the acid process may contain maximum .OS per cent, 
phosphorus. 



STEEL 



387 



Physical Characteristics 






Annealed 


Heat treat- 
ment V 


Yield point or elastic limit, lbs. per 


SS, °oo 


60,000 


sq. in. 


to 


to 




65,000 


180,000 


Reduction of area. 


45-30 per cent. 


40-10 per cent. 


Elongation in 2 ins 


25-20 per cent. 


20- 5 per cent. 



Physical Properties of Spring Steel as Related to the Degree of 

Temper 

The effects of various heal trealmenls on the physical properties of 
carbon steel formed the subject of experiments at the Baldwin Loco- 
motive Works. The steel experimented upon was basic open-hearth 
spring steel of the following composition : 

Per cent. 

Carbon i.oi 

Manganese .38 

Phosphorous .032 

Sulphur . 03 2 

Silicon .13 

Ten test pieces were cut from the same bar, each 14 ins. long and 
I in. diameter. They were subjected to transverse or bending tests 
on supports 12 ins. apart and loaded at the center, the loads and 
deflections being measured. The fiber stress and modulus of elas- 
ticity were calculated by means of the formula for stresses in beams, 
the results being given in the stress strain diagrams and table of 
Fig. 5. The fiber stresses obtained by means of the beam formula 
are, of course, correct within the elastic limit only, the curves ob- 
tained beyond that point being useful for comparison only. The 
precision of measurement of small deflections involved in the deter- 
mination of the modulus of elasticity, together with the compression 
of the test specimen at the points of support, introduce uncertainties 
in that determination, the tendency being to give too small a result, 
especially in the softer specimens, the probability thus being that the 
modulus is even more constant at a value between 29,000,000 and 
30,000,000 than the tests indicate. The deflections given in the chart 
are the readings from the pieces as loaded. 

The hardening temperature (temperature of calescence) was 
determined by means of a magnet and Bristol pyrometer to be 
1360 deg. Fahr. A certain margin above this point, both for an- 
nealing and hardening or quenching, is necessary, these tempera- 
tures, based on previous experience, being : 

For annealing 1400° F. 

For quenching in oil 1450° F. 

For quenching in water 1425° F. 

The test specimens were heated in a gas heated lead bath specially 
constructed to secure control and uniformity of temperature, which 
was read by the Bristol pyrometer. For annealing, the bath was 
held at the temperature of 1400 deg. Fahr. for two hours and then 
allowed to cool off naturally with the furnace, the time required for 
cooling being fourteen hours. For hardening, the pieces were 
quenched; {a) in oil conforming to the Baldwin Locomotive Works 
specification for spring tempering oil and maintained at a tempera- 
ture of 80 deg. Fahr.; (b) in pure running water at 60 deg. Fahr. 
When quenching, the pieces were kept agitated until cooled to the 
temperature of the bath. For drawing the temper up to 600 deg. 
Fahr., the pieces were placed in a gas heated oil bath, the temperatures 
being read by a mercury thermometer. Above 600 deg. Fahr., the 
lead bath and Bristol pyrometer were used. After the temper was 
drawn to the desired temperature, the pieces were removed from the 
bath and allowed to cool naturally in the air. 

The results of the tests are shown in Fig. 5 and the accompanying 



table. In general, the modulus of elasticity is shown to be unaffected 
by the heat treatment while the limit of elasticity is markedly 
affected, varying between 78,500 and 240,800 lbs. per sq. in. The 
difference in the effect of quenching in oil and water is clearly brought 
out as is the lowering of the elastic limit with increase of drawing 
temperature. In particular it is apparent that steel of this carbon 
content, quenched in water and not drawn, or drawn but little, has 
no elongation or permanent set, the elastic limit and ultimate strength 
being identical. As in most cases the tests were not carried to 
destruction, the ultimate strength in these cases is not shown. 



380000 
360000 
340000 
320000 
300000 
280000 
260000 
1240000 




.5 .6 .7 .8 .9 1.0 1.1 
Defection in laclLes at Middle 



1.3 1.4 1.5 1.6 



Fig. 5. 



-Effects of various heat treatments on the mechanical 
properties of carbon spring steel. 



The term elastic limit as here used is the point at which the ratio 
of deflection to stress ceases to be apprecially constant, the deflection 
beginning to increase at a faster rate than the stress. 

S. A. E. Standard Heat Treatments 

Heat Treatment A 
After forging or machining — 

1. Carbonize at a temperature between 1600° F. and 1750° F. 

(i65o°-i7oo° F. desired). 

2. Cool slowly or quench. 

3. Reheat to i45o°-i5oo° F. and quench. 

Heat Treatment B 
After forging or machining — 

1. Carbonize at a temperature between 1600° F. and 1750° F. 

(i65o°-i7oo° F. desired). 

2. Cool slowly in the carbonizing mixture. 

3. Reheat to iS5o°-i62S° F. 

4. Quench. 

5. Reheat to i4oo°-i45o° F. 

6. Quench. 

7. Draw in hot oil at a temperature which may vary from 300° to 

450° F., depending upon the degree of hardness desired. 



388 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 

Heat Treatment P 



Heat Treatment D 
After forging or machining — 

1. Heat to 1 500°- 1 600° F. 

2. Quench. 

3. Reheat to i45o°-isoo° F. 

4. Quench. 

5. Reheat to 6oo°-i2oo° F. and cool slowly. 

Heal Treatment E 
After forging or machining — 

1. Heat to 1500°-! 550° F. 

2. Cool slowly. 

3. Reheat to 1450° to 1500° F. 

4. Quench. 

5. Reheat to 6oo°-i2oo° F. and cool slowly. 

Heat Treatment F 
After shaping or coiling — 

1. Heat to i425°-i475° F. 

2. Quench in oil. 

3. Reheat to 40o°-9oo° F., in accordance with degree of temper 

desired, and cool slowly. 

Heat Treatment G 
After forging or machining — 

1. Carbonize at a temperature between 1600° F. and 1750° F. 

(i65o°-i7oo° F. desired). 

2. Cool slowly in the carbonizing material. 

3. Reheat to i5oo°-i55o° F. 

4. Quench. 

5. Reheat to i3oo°-i4oo° F. 

6. Quench. 

7. Reheat to 25o°-5oo° F. (in accordance with the necessities of the 

case) and cool slowly. 

Heat Treatment H 
After forging or machining — 

1. Heat to 1 500°-! 600° F. 

2. Quench. 

3. Reheat to 6oo°-i2oo° F. and cool slowly. 

Heat Treatment K 
After forging or machining — 

1. Heat to 1500°-! 550° F. 

2. Quench. 

3. Reheat to i3oo°-i4oo° F. 

4. Quench. 

5. Reheat to 6oo°-i2oo° F. and cool slowly. 

Heat Treatment L 
After forging or machining — 

1. Carbonize at a temperature between 1600° F. and 1750° F. 

(1650°-! 700° F. desired). 

2. Cool slowly in the carbonizing mixture. 

3. Reheat to 1400°-! 500° F. 

4. Quench. 

5. Reheat to i3oo°-i4oo° F. 

6. Quench. 

7. Reheat to 2So°-5oo° F. and cool slowly. 

Heat Treatment M 
After forging or machining — 

1. Heat to i45o°-isoo° F. 

2. Quench. 

3. Reheat to 5oo°-i25o° F. and cool slowly. 



After forging or machining — 

1. Heat to 1 450°- 1 500° F. 

2. Quench. 

3. Reheat to i37S°-i45o° F. 

4. Quench. 

5. Reheat to soo°-i25o° F. and cool slowly. 

Heat Treatment Q 
After forging — 

1. Heat to I47s°-i525° F. (Hold at this temperature one-half 

hour to "insure thorough heating.) 

2. Cool slowly. 

3. Machine. 

4. Reheat to I37s°-i425° F. 

5. Quench. 

6. Reheat to 25o°-55o° F. and cool slowly. 



Heat Treatment R 
After forging — 

1. Heat to i5oo°-i55o° F. 

2. Quench in oil. 

3. Reheat to i2oo°-i3oo° F. (Hold at this temperature three 

hours.) 

4. Cool slowly. 

5. Machine. 

6. Reheat to i3So°-i45o° F. 

7. Quench in oil. 

8. Reheat to 2So°-soo° F. and cool slowly. 



Heat Treatment S 

After forging or machining — 

1. Carbonize at a temperature between 1600° F. and 1750° F. 

(i65o°-i7oo° F. desired). 

2. Cool slowly in the carbonizing mixture. 

3. Reheat to i6so°-i7So° F. 

4. Quench. 

5. Reheat to i47S°-isso° F. 

6. Quench. 

7. Reheat to 25o°-55o° F. and cool slowly. 



Heat Treatment T 
After forging or machining — 

1. Heat to 1 650°-! 7 50° F. 

2. Quench. 

3. Reheat to 500 "-1300° F. and cool slowly. 

Heat Treatment U 
After forging — 

1. Heat to 1 525°-! 600° F. (Hold for about one-half hour.) 

2. Cool slowly. 

3. Machine. 

4. Reheat to 1650°-! 700° F. 

5. Quench. 

6. Reheat to 3So°-S5o° F. and cool slowly. 

Heat Treatment V 
After forging or machining — 

1. Heat to 1650°-! 750° F. 

2. Quench. 

3. Reheat to 400°-i20o° F. and cool slowly. 



STEEL 



389 



I, 



Table 7. — Colors of Heated Steel in Diffused Day- 
light 



Table 8. — Temperature Equivalents of Temper Colors 



Colors 


Degrees 
Centigrade 


Degrees 
Fahrenheit 


Dark blood red, black red. . . . 
Dark red, blood red or low red 

Dark cherry red 

Medium cherry red 


S3 2 
566 

635 
677 
746 

843 

899 

940 

996 

1080 

1204 


990 
1050 

117s 
i2i;o 


Cherry, full red 


IS7'? 


Light cherry, bright cherry, 
scaling heat^ . 


1550 


Salmon, orange, free scaling 
heat 


Light salmon, light orange 

Yellow 


172s 
182s 


Light yellow 


107'? 


White 


2200 







1 Scaling heat — scales just form but do not fall away when cooling 



Colors 



Very faint yellow. . 
Very pale yellow. . . 

Light yellpw 

Pale straw yellow. . 

Straw yellow 

Deep straw yellow. 

Dark yellow 

Yellow brown 

Brown yellow 

Spotted red brown. 

Brown purple 

Light purple 

Full purple 

Dark purple 

Full blue 

Dark blue 

Very dark blue 



Degrees 


Degrees 


Centigrade 


Fahrenheit 


216 


420 


221 


430 


227 


440 


232 


450 


238 


460 


243 


470 


249 


480 


254 


490 


260 


500 


266 


Sio 


271 


S20 


277 


530 


282 


54° 


288 


SSo 


293 


560 


299 


570 


316 


600 



\ 



I 



ALLOYS 



For alloys for bearings see Index. 

Copper-Tin-Zinc Alloys 

The tensile strengths of copper-tin-zinc alloys are given in Fig. i 
from The Materials of Construction by Prof. J. B. Johnson. The 
location of any point within the triangle indicates the composition. 
Thus, point a stands for 40 per cent, copper, 20 per cent, zinc, and 
40 per cent. tin. Again, the contour lines give the tensile strengths 
for the useful alloys. The composition and strength of copper-tin 
or copper-zinc alloys may in like manner be read from the sides of 
the triangle. As put by Professor Johnson, "So much depends on 
the purity of the ingredients and on the manipulation of the process 
of melting and casting, that this chart, or any similar record, must be 



Sheet brass shall be furnished annealed or hard rolled. Annealed 
brass is to be designated as light annealed, or soft. Hard rolled brass 
shall be furnished in the following tempers, and the amount of reduc- 
tion in thickness from the annealed sheet shall be as follows, expressed 
in Brown & Sharpe gages: 

^ B. & S. 

Temper. 

numbers 

Quarter hard i 

Half hard 2 

Hard 4 

Extra hard 6 

Spring 8 




yS .^^^'&a^ Y W»\^ 6OO00 

'\^ xWvwvv 

J, 50000 

50 

Mo/ 

^0 /\ /\ A "k A C7 

\<0 \ / 30 

5000 \_Z— — Sy — i 

/,j50^^*'7 ^^ / \ / \ //A ?0 

■i V v \/ 10 V ly \/ . yo 

TV A ' T\ A A -^ X 10000 — X „, , 

='""'/\__/\ /\/\ — i^ 

^oO A/ V \/ \/ V \/ \/ V^ — V ^og 

Tin Zinc 

Fig. I. — Composition and strength of copper-tin-zinc alloys. 

taken as showing what may be obtained rather than what will be Over s Over 8 Over n 

obtained from the use of these particular mixtures." ^p *°.S ins. wide ins. wide ins. wide 

~, j: 11 • J ! \ r 1 1 , . '"s. Wide to 8 Ins. to 11 ms. to 14 ins. 

The followmg data are from the report of the sheet metals division Thickness. Limits, inclusive, inclusive, inclusive, inclusive, 

of the Standards Committee of the Society of Automobile Engineers, (B. &S. gage) ins. ins. ins. ins. ins. 

January, 1 91 2. They are approximate and should be used as a guide No. 0000 to No. o inc.(.46oo-.3248) ±.0044 ±.0048 ±.0031 =fc.ooss 

only. If the figures are of particular interest to an engineer, a special Bejow o to No. 4 inc.(.3248-.2043) -.0039 -.0043 ^.0046 ^.0050 

, ,j , 1 .,, r . ... Below 4 to No. 8 inc.(.2043-.i284) ±.0034 ±.0038 ±.0041 ±.0045 

inquiry should be sent to the mill manufacturing, giving size, temper, gelow 8 to No. 14 inc.(. 1284-.0640) ± . 0029 ^ . 0033 * . 0036 ^ . 0040 

etc., with a request for tensile strength and elongation figures covering Below 14 to No. 18 inc. (.0640-. 0403) ±.0025 ±.0029 ±.0033 ±.0037 

the particular requirements. Below 18 to No. 24 inc. (.0403-. 0201) ±.0020 ±.0024 ±.0028 ±.0032 

Below 24 to No. 28 inc.(.020i-.oi26) ±.0016 ±.0020 ±.0024 ±.0028 

Standard Sheet Brass Below 28 to No. 32 inc.(.oi26-.oo79) ±.0013 ±.0017 ±.0020 ±.0024 

Specification No %x Below32 to No. 3s inc.(.oo79-.oos6) ±.0010 ±.0014 ±.0017 ±.0022 

^T^l_ r 11 • ...,., Below 35 to No. 38 inc.(.0056-.0039) ±.0008 ±.0012 ±.0015 ±.0019 

irie lollowing composition is desired: 

^°PP^'' 64.00 to 67,00 per cent. Standard sheet brass is for use in the manufacture of lamps, 

^^'i'^ 33.00 to 36.00 per cent. horns, flexible tubes, and ornamental work in general.— Tensile 

Lead not to exceed . So per cent. strength, hard, about 60,000 lbs. per sq. in.; elongation, about 5 per 

Iron not to exceed . 10 per cent. cent, in 2 ins. Tensile strength, soft, about 48,000 lbs. per sq. in.; 

390 



ALLOYS 



391 



elongation, about 50 per cent, in 2 ins. Drawing brass and spinning 
brass are special qualities of brass for the operations indicated by the 
name. 

Low Brass 

Used on account of color, resistance to corrosion and atmospheric 
changes, and on account of superior ductility. 

Speclfication No. 34 
The following composition is desired: 

Copper 78 . 00 to 81 . 00 per cent. 

Zinc 19 . 00 to 2 2 . 00 per cent. 

Lead not to exceed .20 per cent. 

Iron not to exceed .10 per cent. 

Specifications for temper, gage variation, etc., shall be the same 
as for sheet brass. Tensile strength, hard, about 75,000 lbs. per sq. 
in.; elongation, about 5 per cent, in 2 ins. Tensile strength, soft, 
about 42,000 lbs. per sq. in.; elongation, about 50 per cent, in 2 ins. 

Brazing Brass 

Specification No. 35 
The following composition is desired : 

Copper 74 . 00 to 76 . 00 per cent. 

Zinc 24.00 to 26.00 per cent. 

Lead not to exceed .25 per cent. 

Iron not to exceed .10 per cent. 

Specifications for temper, gauge variation, etc., shall be the same as 
for sheet brass. This material is used for parts where brazing or 
silver soldering is required. This material has about the same 
physical properties as low brass. 

Free Cutting Brass 

Specification No. 36 
The following composition is desired : 

Copper 61 . 00 to 64 . 00 per cent. 

Zinc 33 . 00 to 38 . 00 per cent. 

Lead i . 25 to 2 . 00 per cent. 

Iron not to exceed .10 per cent. 

This grade of material contains lead, which makes it free cutting 
and suitable for work on which machining is to be done. It does 
not bend or form readily, because of its "shortness." Specifications 
for temper, gage variation, etc., shall be the same as for sheet brass. 
It has a tensile strength when hard of about 75,000 lbs. per sq. in., 
with an elongation of about 3 per cent, in 2 ins. When soft, its 
tensile strength is about 50,000 lbs. per sq. in., with an elongation 
of about 35 per cent, in 2 ins. 

Red Metal or Commercial Bronze 

Specification No. 37 
The following composition is desired: 

Copper 88 . 00 to 91 . 00 per cent. 

Zinc 9 . 00 to 1 2 . 00 per cent. 

Lead not to exceed .20 per cent. 

Iron not to exceed .10 per cent. 

Specifications for temper, gage, variation, etc., shall be the same 
as for sheet brass. This material has a rich gold color and is used 
for screen wires, radiators and in other places subject to corrosion. 
It is also used for ornamental parts where its color is desired. Its 
tensile strength, hard, is about 55,000 lbs per sq. in., with an elon- 
gation of about 5 per cent, in 2 ins. Soft, it has a tensile strength 
of about 37,000 lbs. per sq. in., and an elongation of about 40 
per cent, in 2 ins. 



Gilding Metal 

Specification No. 38 
The following composition is desired: 

Copper 94 . 00 to 96 . 00 per cent. 

Zinc 4 . 00 to 6 . 00 per cent. 

Lead not to exceed .15 per cent. 

Iron not to exceed .06 per cent. 

Specifications for temper, gage variation, etc., shall be the same 
as for sheet brass. This material is used for radiators. It has a 
tensile strength of about 45,000 to 55,000 lbs. per sq. in., with an 
elongation of about 5 per cent, in 2 ins. when hard. Annealed soft 
its tensile strength is about 35,000 lbs. per sq. in., with an elongation 
of about 35 per cent, in 2 ins. 

Phosphor Bronze 

Phosphor bronze is composed of copper, tin and phosphorus in pro- 
portions varied to suit the requirements of the trade. Specifications 
for temper, gage variation, etc., shall be the same as for sheet brass. 

Copper Sheets and Strips 

Copper sheets and strips shall be at least 99.50 per cent, pure, and 
shall be either soft or furnished with such roller temper as may be 
specified. 

For Copper in Rolls. — ^Less than .060 in. thick, variation .002 in. 
under and .001 in. over gage; .060 in. and thicker, variation .003 in. 
under and .003 in. over gage. 

For Copper in Sheets. — Up to and including 48 ins. wide the varia- 
tion in thickness may be 5 per cent, under or over gage. Over 48 
ins. in width, up to and including 60 ins. wide, the variation in thick- 
ness may be 7 per cent, under or over. Test specimens cut from 
soft copper sheet shall have a minimum tensile strength of 30,000 
lbs. per sq. in., with an elongation of at least 25 per cent, in two (2) 
inches for gages ncffc less than .030 in. thick. 

German Silver 

German silver in rolls and sheets is to be specified according to 
color and service required in the following standard grades: 5 per 
cent., 15 per cent., 18 per cent., 20 per cent., 25 per cent., 30 per cent, 
nickel, the balance being copper and zinc. It will be supplied soft 
or with such roller temper as may be required. 

Brass Rods 

For Cold Heading. — The material shall be suitable for cold work- 
ing, such as the heading of rivets and the rolling of threads for screws. 

Specification No. 39 
The following composition is desired: 

Copper 61 . 50 to 64 . 50 per cent. 

Zinc 35-5° to 38.50 per cent. 

Lead Not to exceed . 50 per cent. 

Iron Not to exceed. 10 per cent. 

The temper shall be produced by annealing sufficiently to give the 
metal the softness required for heading. The material should be 
ordered for heading, and the order accompanied by a sample or draw- 
ing to show the mechanical operations required. This material has 
a tensile strength of about 35,000 to 40,000 lbs. per sq. in., with an 
elongation of about 50 per cent, in 2 ins. 

Free-cutting Brass Rod. 

Material suitable for automatic screw-machine work. 

Specification No. 40 
The following composition is desired: 

Copper 6 1 . 50 to 64 . 50 per cent. 

Zinc 31.50 to 35.50 per cent. 

Lead 2.25 to 3.50 per cent. 

Iron Not to exceed. 10 per cent. 



392 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



All free cutting brass rods shall be furnished hard drawn, unless 
otherwise specified for when ordered. 

Rods shall not vary in diameter more than the amount specified 
in the following table: 

Up to and including f in., .0015 over or under required 

diameter. 
From I in. to and including i in., .002 over or under required 

diameter. 
From I in. to and including 3 ins., .0025 over or under required 

diameter. 
This material is suitable for automatic screw machine work. Its 
tensile strength is about 65,000 lbs. per sq. in., with about 15 per 
cent, elongation in 2 ins. 

Tobin Bronze. 

Turned and straightened rods for various purposes where strength 
and resistance to corrosion are required; also for hot forging. Rods 
up to and including i in. in diameter shall have a tensUe strength 
of not less than 62,000 lbs. per sq. in. Rods larger than i in. and 
up to and including 7 ins. in diameter shall have a tensile strength 
of 60,000 lbs. per sq. in. 

All rods not larger than i in. in diameter shall have an elongation 
of at least 25 per cent, in 2 ins. All rods larger than i in. in diameter 
shall have an elongation of at least 28 per cent, in 2 ins. The elastic 
limit, or the point at which rapid elongation begins, shall be at least 
30,000 lbs. per sq. in. for all sizes. 

Tubing 

Tubing can be furnished in copper and the commercial alloys of 
copper and zinc, such as high brass, bronze, phosphor bronze, and 
Tobin bronze. The composition shall be as specified to meet the re- 
quirements of use. The temper of the tubing shall be as specified 
in the order, and may be hard, half hard or annealed. If annealed, 
the tubing may be soft, or light annealed. 

The following variation on inside and outside diameter and the 
thickness of the walls shall be allowed on all commercial tubing : 

Outside and Inside Dimensions 

Up to 5 in. inclusive 002 in. over or under 

Over 5 in. to and including f in 0025 in. over or under 

Over I in. to and including i in 003 in. over or under 

Over I in. to and including i\ ins 0035 i"^- over or under 

Over I J ins. to and including i \ ins 004 in. over or under 

Over 1 1 ins. to and including if ins 0045 in. over or under 

Over if ins. to and including 2 ins 005 in. over or under 

Over 2 ins 3 of i per cent, over or under 

No combination of variations on the same tube shall make the 
thickness of the wall vary from the nominal by more than the follow- 
ing amounts: 

Thickness of Wall 

Up to and including ^ in 001 in. over or under 

Over xi in. to and including ^2 in 002 in. over or under 

Over X2 in. to and including re in 003 in. over or under 

Over xs in. to and including | in 005 in. over or under 

Over \ in. to and including } in 008 in. over or under 

Over ; in. to and including j^ in 0125 in. over or under 

Over x^ in. to and including | in 015 in. over or under 

On all stock where the above commercial variations are not per- 
missible limits shall be specified in the order. 

Brass Casting Metals 

Red Brass 
Specification No. 27 

Copper 8s . 00 per cent. 

Tin S • 00 per cent. 

Lead 5 ■ 00 per cent. 

Zinc 5 • 00 per cent. 



A tolerance of i per cent, plus or minus will be allowed in the above. 
Impurities of over .25 per cent, will not be permitted. 

Note. — A high grade of composition metal, and an excellent 
bearing where speed and pressure are not excessive. Largely used 
for light castings, and possesses good machining qualities. 

Yellow Brass 
Specification No. 28 

Copper 62 . 00 to 65 . 00 per cent. 

Lead 2 . 00 to 4 . 00 per cent. 

Zinc 36 . 00 to 3 1 . 00 per cent. 

Total impurities in excess of .50 per cent. wUl not be permitted. 

Note. — This alloy represents a high grade of yellow brass; is 
tough and possesses good machining quaUties. Its use is suggested 
in preference to ordinary commercial yellow brass castings, which 
are, generally speaking, a miscellaneous assortment of mixtures, some 
of them containing considerable amounts of iron (from i to 3 per 
cent.). This is very undesirable, as it renders the castings liable to 
blow-holes, hard spots and, in some cases, small particles of metallic 
iron. 

Cast Manganese Bronze 

Specification No. 29 
Manganese bronze is understood to mean a metal constituted 
principally of copper and zinc in the approximate proportion of 
60 to 40, iron being present in small and manganese in variable 
quantities. Main dependence will be placed upon physical specifica- 
tions. 

Tensile strength 60,000 lbs. per sq. in. 

Yield point 30,000 lbs. per sq. in. 

Elongation in 2 ins 20 per cent. 

Note. — Manganese bronze is of value for castings where strength 
and toughness are required. Specifications are not severe, being 
easily met by all makers of quality castings. Test coupons should be 
attached to castings made in the sand, the use of chills, special sand 
or artificial methods of cooling being prohibited. This precaution 
prevents the use of inferior metals. 

Alxuninum Alloys 

No. I 
Specification No. 30 

Aluminum, not less than go. 00 per cent. 

Copper 8 . 50 to 7 . 00 per cent. 

Total impurities shall not exceed 1.7 per cent, of which not over 
.2 shall be zinc. No other impurities than carbon, silicon, iron 
manganese and zinc shall be allowed. 

Note. — This is one of the lightest of the aluminum alloys, possess 
ing a high degree of strength, and can be used where a touga, light 
alloy of these characteristics is required in automobile construction. 

No. 2 
Specification No. 31 

Aluminum, not less then 80 . 00 per cent. 

Zinc, not over 1 5 . 00 per cent. 

Copper, between 2 . 00 and 3 . 00 per cent. 

Manganese, not to exceed 40 per cent. 

Total impurities shall not exceed 1.65 per cent., of which not more 
than .50 per cent, should be silicon, not more than i.oo per cent, iron, 
and not more than .15 per cent. lead. 

Note. — This mixture possesses strength, closeness of grain, and 
can be cast solid and free from blowholes. It is a light metal, its 
specific gravity being in the neighborhood of 3.00. 



ALLOYS 



393 



No. 3 
Specification N"o. 32 

Aluminum 65 . 00 per cent. 

Zinc 35 ■ 00 per cent. 

Total impurities in excess of 1.65 per cent, will not be permitted. 

Note. — This is a mixture that can be used where cheap castings 
not to be subjected to any great strains are desired. It is a desirable 
mixture for flat plates, foot-boards, running boards, etc. It is quite 
brittle and will not equal in toughness or strength specifications 30 
and 31. 

On aluminum alloys the standard specimen of reference shall be 
the same as indicated for standard steel tensile test-specimen. 
Test piece shall be tested with the skin on. We recommend a test 
bar J in. in diameter at the breaking section and filleted to a f in. 
diameter threaded end. FUlet should extend for at least f in. 
Test bar to be attached to casting, use of chills or artificial means 
of cooling being prohibited. 



100 


\ 
















90 




\ 
















N 


\ 












Per Cent. < 
sible Strer 








N 


















\^ 


S^ 








PL* 

70 












\ 


^ 


^ 



















100 200 300 400 

Degrees above Lowest Possible 
Pouring Temperature 
Fig. 2. — Effect of pouring temperature on the strength of aluminum 

castings. 

Aluminum Alloys 

The design of parts to he made of aluminum castings is subject to 
restrictions which are thus explained by H. W. Gillett {Society of 
Automobile Engineers 1911): 

Owing to certain physical properties of aluminum, such as its 
high contraction on cooling and its weakness when just solidified — 
that is, its hot shortness — aluminum castings require more careful 
design than almost any other casting metal. 

In passing from the molten to the solid state, aluminum contracts 
a good deal; when a heavy and a thin section come next to each other, 
the thin place will freeze first. If the thin section is so situated as 
to lie between a heavy section and a gate or riser the supply of metal 
is thereby cut off from the molten mass in what is to be the heavy 
part of the casting. The contraction of freezing has to take place, 
and instead of taking place uniformly over this heavy part and main- 
taining the exact shape of the mold, it will often draw away from a 
corner and produce a shrink. We can induce the heavy portion to 
freeze more quickly by placing the chill in the mold at that point, 
but it is difficult to accomplish the end completely by this method; 
it greatly increases the time required to put up the mold and produces 
unsightly chill marks on the casting. 

The ideal casting, therefore, is one of as nearly uniform section 
throughout as is practical, since that means that the whole casting 
solidifies at the same time, so that contraction is uniform. 

On account of the hot shortness of aluminum the shrinkage strains 
set up when a heavy section joins a thin one often cause the metal 
to give away entirely at that point, and a crack appears. If it is 
inevitable that light and heavy sections come together, the cooling 
strain should be distributed by joining the sections by a smooth 
curve, that is, a liberal fillet. 



There is no one factor in foundry practice that more gravely 
affects the strength of the casting than the pouring temperature. 
The reason for this again, is, the speed of crystallization. The cooler 
the metal can be poured into the mold the more quickly it solidifies 
and the less time the crystals have to grow or arrange themselves, 
and the result is a mass of closely interlocking crystals forming a 
strong fine-grained material. 

The effect of pouring temperature was well shown by a set of test 
bars, all of which were cast from the same pot of metal with exactly 
similar molds, the only variable being the pouring temperature. 
The average results obtained in this series of tests are given in Fig. 
2, which shows that the lower the pouring temperature the stronger 
the casting. 

This has a distinct bearing on design, since the lowest temperature 
at which a casting can be poured is that to which the thinnest sec- 
tion will just escape a misrun. If the casting is so designed that this 
crucial section compels hot pouring, all of the thicker parts will freeze 
too slowly and will be weaker than they should be. By slightly 
increasing the section of the thinnest parts, a casting can often be 
poured 100 deg. colder and the strength of the whole casting be in- 
creased at least 10 per cent. If the bulk of a casting is from \ to 
5 in. thick, one little part \ in. thick will give a resultant casting, 
on account of the high pouring temperature required, whose average 
strength is about 16,000 lbs. per sq. in. instead of 18,000 lbs. or 
over. The call for lightness has led many designers to overlook 
this vital point. 

The great influence of the pouring temperature is the reason why 
separately cast test bars show only the quality of the ingot metal 
and nothing at all as to the strength of the corresponding casting, 
even though the test bar a,nd casting may be poured from the same 
pot of metal. Aluminum test bars should be made on the castings. 
Were this stipulation not made the foundryman who wishes can 
pour the casting as hot as he pleases, allow his metal to cool way 
down and then pour separate test bars which will then show an utterly 
fictitious strength in comparison with the casting. 

The general lack of attention to pouring temperatures, not only 
in commercial practice, but in most of the investigations on aluminum, 
vitiates many of the published data on aluminum alloys and accounts 
for a great many irregulatities and seeming contradictions in the 
results. In comparing the different aluminum alloys, really compar- 
able results can only be obtained by pouring at the same number of 
degrees above the melting-point of the particular alloy in question 
in all cases, thus allowing the same time for crystallization and 
producing an analogous condition. 

Core work always means trouble. It takes time to set cores in 
the mold correctly, and if a lot of small cores are used the danger of 
shifts is greatly increased. If, on the other hand, large cores are 
used, they must be made hard enough to allow handling them and 
setting them in the mold, which requires not only a solid core, but 
one reinforced by iron rods and wires. This makes them hard to 
crush, and on large cores inside of thin walls of metal, introduces 
danger of cracking. When we have a core completely surrounded 
by walls of metal it is a question whether the tensile strength of the 
metal as it solidifies is greater than the compressive strength of the 
core. Let the core be ever so slightly too hard and the casting is 
inevitably ruined. 

If cores must be used, the core prints should be large and deep, 
so as to anchor the cores firmly without the use of chaplets to hold 
the cores in place, since it is impossible for the molten metal to fuse 
a chaplet into the body of the casting, without pouring at a tem- 
perature far above that necessary to give the greatest strength. 
When a job requires cores, the first question that should be asked by 
the patternmaker is if that pattern cannot be made so as to allow 
the use of green-sand core, or at least a green-sand half. Green 
sand will crush and give away when the casting contracts on cooling, 
where a hard, dry sand core will not crush and will crack the casting. 



394 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Casting Alloys Specified by the Bureau of Steam Engineering, U. S. Navy 



Name 



Composition by percentage 



Copper 



Tin 



Zinc 



Iron, 
maxi- 
mum 



Lead, 
maxi- 
mum 



Miscellan- 
eous 



Purposes for which suitable 



Tensile 
strength, 
mini- 
mum 



Yield 
point, min- 
imum 



Elongation 

in 2 ins. 

(minimum) 

per cent. 



Commercial brass 

Muntz metal 

Brazing metal. . . . 

Gun bronze 



Journal bronze. 



Manganese bronze 

Cast naval brass 

Phosphor-bronze 

Screwpipe fittings.brass 



64-68 



59-62 
84-86 

87-89 



57-60 
59-63 

80-90 
77-80 



12. 5- 

14. S 



6-8 
4 



39-41 
Rem. 



Rem. 



Rem. 
13-19 



2. 



.06 
.06 



.06 



.06 



.06 
. I 



3.0 



Aluminum, 
0.5; man- 
ganese 0.3. 



.6 



Phosphorus, 
•3- 



Heads, shapes, and water 
chests. 



Name and number plates; cases for 
instruments; oil cups; distributing 
boxes. 

Brazing metal, and all flanges and fit- 
tings that are to be brazed. 

All composition valves 4 ins. in dia- 
meter and above; expansion joints, 
flanged pipe fittings, gear wheels, bolts 
and nuts, miscellaneous brass castings, 
all parts where strength is required of 
brass castings or where subjected to 
salt water, and for all purposes where 
no other alloy is specified. 

Composition valves: Safety and relief, 
feed check and stop, surface blow, 
drain, air, and water cocks, main stop, 
throttle, reducing, sea, safety sluice, 
and manifolds at pumps. 

Condenser 

Distiller 

Feed- water 
heater. 

Oil cooler 

Pumps: Air-pump casing, valve seats, 
buckets, main circulating, water cylin- 
ders, valve boxes, water pistons stuff- 
ing boxes, followers, glands, in general 
the water end of pumps complete ex- 
cept as specified. 

Stuffiing boxes: Glands, bushings for 
iron or steel boxes. 

Blowers: Bearing boxes. 

Journal boxes: Distance pieces. 

Miscellaneous: Grease extractors; steam 
strainers, separators, casing for stern 
tube and propeller shafts, propeller hub 
caps. 

Bearings: Main, stern tube, strut and 
spring. 

Spring bearings: Glands and bafaes 

Reciprocating engine: Intermediate and 
low pressure relief valves and casings, 
crosshead brasses, crank pin brasses, 
eccentric straps and distance pieces. 

Journal boxes, guide gibs, bushings, 
sleeves, slippers, etc. 

Reciprocating engine: Valve stem cross- 
head bottom brass; link block gibs, 
suspension link brasses. 

Propeller hubs, blades, engine framing, 
and composition castings requiring 
great strength. 

Valve handwheels, hand-rail fittings, 
ornamental and miscellaneous castings, 
and valves in water chests of conden- 
sers. 

Castings where strength and incorrod- 
ibility are required. 

For composition screwed fittings 



60,000 



30,000 



Fusible Alloys 



Alloys 


Bismuth 


Lead 


Tin 


Cadmium 


Melting- 
point 


Newton's... . 

Rose's 

Darcet's. . . . 


50.0 
50.0 
50.0 
50.0 
50.0 


31-25 
28.10 
25.00 
24.00 
27.00 


18.7s 
24.64 
25.00 
14.00 
13.00 


■ 


Deg. Fahr. 
204 
212 
200 


Wood's 

Lupowitz's . . 


12.00 
10.00 


160 
140 



WEIGHT OF MATERIALS 



Table i. — Specific Gravity and Weight of Metals 



Table 2. — Specific Gravity and Weight of Wood 



Material 



Aluminum — cast 

Aluminum — wrought 

Aluminum — bronze 

Antimony 

Arsenic 

Bismuth 

I from 
Brass — cast -j to 

[ average 

Brass — Muntz-metal 

Brass — naval (rolled) 

Brass — sheet 

Brass — wire 

I from 
Bronze (gun-metal) -j to 

[ average 

Copper — cast 

Copper — hammered 

Copper — sheet 

Copper— wire 

Gold (pure) 

Gold standard 22 carat fine 

(Gold II — Copper i) 

r from 
Iron — cast -j to 

[ average 

f from 
Iron — wrought. -j to 

[ average 

Lead — cast 

Lead — sheet 

Manganese 

Nickel — cast 

Nickel — rolled 

Platinum 

Silver 

I from 
Steel < to 

[ average 

Tin 

White Metal (Babbitt's) 

Zinc — cast 

Zinc — sheet 



Specific 
gravity 



2.569 
2-. 681 
7.787 
6.712 
5.748 
9.827 
7.868 
8.430 
8. 109 

8.221 

8. SIC 
8.462 
8.5S8 
8.478 
8.863 
8.735 
8.622 
8.927 
8.815 
8.89s 
19.316 
17 502 

6.904 

7.386 

7 . 209 
7.547 
7.803 
7.707 
11.368 
11.432 

8.012 

8.285 
8.687 
21.516 
10.517 
7.820 
7.916 
7.868 
7.418 
7.322 
6.872 
7.209 



Weight in lbs. of 
one 



Cu. ft. 



160 
167 
48s 
418 
3S8 
612 
490 
S25 
50s 
512 
530 
527 
533 
528 
552 
544 
537 
556 
549 
554 
1203 
1090 

430 
499 
464 
470 
486 
480 
708 
712 
499 
516 
541 
1340 
65.5 
487 
493 
490 
462 
456 
428 
449 



Cu. in, 



.093 
.097 
.281 
.242 
.207 

• 354 
.284 

• 304 
.292 
.296 

• 307 
.305 
.308 
.306 

• 319 
.315 
.311 
.322 
.318 
.321 
.696 
.631 

.249 
.266 
.260 
.272 
.281 
.278 
.410 
.412 
.289 
.299 

• 313 
.775 
.379 
.282 
.285 
.284 
.267 
.264 
.248 
. 260 



Cu. ins. 

in one 

lb. 



10. 80 

10.35 

3 56 



Alder 

Apple 

Ash 

Bamboo 

Beech 

Birch 

Boe 

Cedar 

Cherry 

Chestnut 

Cork 

Cypress 

Dogwood 

Ebony 

Elm 

Fir 

Gum 

Hackmatack. 

Hemlock 

Hickory 

Holly 

Hornbeam . . . 

Juniper 

Larch 

Lignum vitae, 

Linden 

Locust 

Mahogany. . . 

Maple 

Mulberry. . . . 
Oak, Live. . . . 
Oak, White. . 
Oak, Red. . . , 
Pine, White. . 
Pine, Yellow. 

Poplar 

Spruce 

Sycamore. . . . 

Teak 

Walnut 

Willow 



Specific gravity 



.56 to 

■ 73 to 
.60 to 
.31 to 
.62 to 
.56 to 
.91 to 
.49 to 
.61 to 
,46 to 
.24 
.41 to 

.76 

I . 13 to 

■ 55 to 
.48 to 
.84 to 

• 59 
.36 to 
.69 to 

■ 76 

■ 76 

• 56 

• 56 

• 65 to 
.604 
.728 

.56 to 

■ 57 to 
.56 to 
.96 to 
.69 to 
.73 to 

• 35 to 
.46 to 
.38 to 
.40 to 

■ 59 to 
.66 to 
.50 to 
.49 to 



.80 
.79 
.84. 
.40 
.85 
.74 
1-33 
.75 
.72 
.66 

.66 

1.33 

• 78 

• 70 
I. 00 

• 41 

• 94 



:.o6 

• 79 

• 90 
:.26 

.86 

• 75 
■55 

■ 76 

■ 58 
.50 
.62 

• 98 

• 67 
•59 



Average 



.68 

• 76 

• 72 

• 35 

• 73 

• 65 

I . 12 
.62 

.66 

• 56 
.24 

• 53 
•76 

[.23 
.61 

• 59 

• 92 
.59 
.38 
•77 

• 76 

• 76 
•56 

• 56 
[. 00 



.81 

.68 
•73 
I. II 
• 77 
•74 
•45 
.61 
.48 
•45 
.60 
.82 
•58 
•54 



Weight per cu . 
ft. lbs., average 



42 
47 
45 
22 
46 
41 
70 
39 
41 
35 
IS 
33 
47 
76 
38 
37 
57 
37 
24 
48 
47 
47 
35 
3S 
62 
37 
46 
SI 
42 
46 
69 
48 
46 
28 
38 
30 
28 
37 
SI 
36 
34 



3-79 
4.04 
3.85 



Table 3. — Weights of Iron, Brass, and Copper Wire 
Birmingham or Stubbs gage 



No. of 


Dia. in 
in. 


Weight 


in lbs. per 1000 


linear ft. 


No. of 
gage 


Dia. in 
in. 


Weight 


n lbs. per 1000 


inear ft. 


gage 


Iron 


Brass 


Copper 


Iron 


Brass 


Copper 


0000 


■454 


546.21 


589.29 


623. 2 


17 


.058 


8.92 


9.62 


10. 17 


000 


• 42s 


478.6s 


516.41 


546.1 


18 


■ 049 


6.36 


6.86 


7.259 


CO 


.380 


382.66 


412.84 


436.6 


19 


.042 


4.67 


5. 04 


5333 





• 340 


306.34 


330.50 


349.5 


20 


.03s 


3 25 


3. 52 


3.704 


I 


.300 


238.50 


257.31 


272. 1 


21 


.032 


2.71 


2.93 


3.096 


2 


.284 


213.74 


230.60 


243 9 


22 


.028 


2.08 


2.24 


2.370 


3 


■ 259 


177^77 


191.79 


202.8 


23 


.025 


1.66 


1.79 


1.890 


4 


■ 238 


150^11 


161. 95 


171. 3 


24 


.022 


1.28 


1.39 


1.463 


S 


.220 


128.26 


138.37 


146.3 


25 


,020 


1.06 


1. 14 


1.209 


6 


.203 


109. 20 


117.82 


124.6 


26 


.018 


.863 


.926 


.979 


7 


. 180 


85.86 


92.63 


97.96 


27 


.016 


.680 


.732 


• 774 


8 


.165 


72.14 


77.83 


82.31 


28 


. 014 


.529 


.560 


• 592 


9 


.148 


58.05 


62.62 


66,23 


29 


.013 


• 438 


.483 


• Sli 


10 


■134 


47^58 


51^34 


54.29 


30 


,012 


.382 


.412 


• 435 


II 


.120 


38.16 


41.17 


43^54 


31 


.010 


.266 


.286 


.302 


12 


. 109 


31.49 


33.97 


35 92 


32 


,009 


.212 


.232 


.244 


13 


.095 


23,92 


25.80 


27.29 


33 


.008 


,167 


.183 


.193 


14 


.083 


18.26 


1970 


20.83 


34 


.007 


.133 


. 140 


.148 


IS 


.072 


13.73 


14.82 


15.67 


35 


.005 


.066 


.071 


.075 


16 


.065 


II. 19 


12. 08 


12.77 


36 


.004 


.052 


.046 


.048 



395 



396 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 4. — Weights of Seamless Brass Tubing per Linear Foot, Lbs. 
f to 25 Outside Diameter. Nos. i to 25 Stubbs Iron Gage 



No. of gage 



Thickness 
in ins. 



I 
2 
3 
4 
S 

6 

7 
8 

9 

10 

II 
12 
13 
14 
15 

16 

17 
18 

19 



21 
22 

23 
24 

25 



.300 
.284 

■ 259 

.238 
. 220 

.203 
.180 
.165 
.148 
• 134 

. 120 
. 109 
•09s 
.083 
.072 

.065 
.058 
.049 
.042 
■035 

.032 
.028 

.025 

.022 
.020 



■ 04s 
,044 

• 043 

.040 

.036 

• 034 
.031 
,029 
,026 
,024 



.096 



.092 

.087 
.078 

.070 
.062 

■OS7 
.051 
.047 
.042 
■039 



. I 
.177 
. 170 
. 160 
.144 

.138 
.128 
•113 

. lOI 

.086 
.081 

.072 

.066 
.058 
.052 



26 
256 

■ 237 

. 220 
. 201 

.186 

.169 

.150 
.130 

• 113 

. 104 

• 093 

.082 
,074 
068 



40 

39 

.38 

36 
.334 
.306 
.280 
•251 

.232 
. 212 
.183 
.161 
.136 

.128 
.112 
. 102 
,090 
.081 



•52 
•49 
.46 

.43 
•413 

■ 377 
• 340 
•303 

.279 

•2S5 
. 220 

■193 
.163 

•151 

■ 133 
.119 
.107 
.096 



64 
61 
S8 

53 
491 

■ 445 
,400 

'355 

,326 
•29s 
•255 
. 221 
.188 

■173 
,152 
,136 
, 121 
no 



•77 
•71 
•65 

.61 

•570 

•514 

.460 

.409 

•372 
•338 
. 291 
.252 
.214 

. 196 
■174 
•15s 
•137 
. 126 



1 .00 
.90 
.87 
.82 

•77 

.70 

.650 

.580 

.520 

.461 

.420 
.380 

•325 
.282 
• 238 

. 219 
. 192 

•173 
•154 
. 140 



1 . 29 
1. 19 
I. II 
1.04 
.96 

• 87 
.808 
.717 
.640 

• 564 

•515 

• 463 
•397 
•343 
. 2 

.266 

■233 
. 209 
. I 
. 169 



1^85 
1.76 
1.68 

1.58 
1.44 

^•35 

I. 25 

1. 16 

T--05 

• 96s 
.85s 
.760 
.667 

.609 
.548 
.467 
.404 
•339 

.312 
•275 
•245 
.218 
.197 



2 .42 

2-35 
2. 22 
2. 10 
1.99 

1.88 
I. 71 

^•59 
1.46 

1^35 

1 . 22 

1 . 12 

1 .00 

.88 

•77 

.70 
.64 
•54 
.46 
•39 

•357 
•315 
.281 
.249 
. 226 



3^28 
3.16 
2.98 
2.79 
2.60 

2.46 
2.23 
2.07 
1.89 
1.74 

1^57 
1-43 
1 . 27 
1. 12 
•99 



.80 
.67 
•58 
•49 

• 450 
•396 
•354 
.312 
.285 



4. 10 
4-03 
3^72 
3 49 
3.26 

3 •04 

2.77 

2.54 
2.32 
2. 12 

I. 91 
I. 76 

1-55 
1.36 
1. 19 

1 .07 

•97 
.82 

■71 
•59 

• 542 
•477 
.426 
■376 
•342 



5-03 
4.80 
4.48 
4.18 
3^89 

3.60 
3-28 
3.02 
2.75 

2^51 

2. 27 
2 .07 
1.82 
I. 61 
1.40 

1.26 

I. 14 

.96 

•83 
.69 

•635 
•556 

•497 
.438 
•399 



5^88 
5^57 
5^23 
4.82 

4^53 

4. 20 
3^78 
3 SO 
3-17 
2 .90 

2 . 16 

2^39 
2.09 
1.84 
I. 61 

^•45 
130 
1. 10 

■95 
.80 

• 727 

• 638 

■571 
.502 

•457 



6^75 
6^45 
5^96 
S-Si 
S-I4 

4.80 
433 
3^98 
3.60 
3^28 

2.96 
2. 70 

2^37 
2.08 
1. 81 

1 .64 
1.48 
1.24 
1.07 
•893 

.820 

•71 
.641 
.566 
.516 



7.62 
7.26 
6. 72 
6.25 
5^8o 

5-39 
4^83 
4.46 

4^03 
3^67 

331 
3.01 
2.65 
2.32 
2 .02 

1.82 
1.64 

I 39 
1. 19 
1 .00 

913 
799 
714 
629 

573 



For weights of seamless copper tubing, add 5 per cent, to the weights above. 



Table 5.- 


—Weights 


OF Steel Hexagon 


AND Octagon Bars 


Dia. or dis- 
tance across 


Weight per ft., lbs. 


Dia. or dis- 
tance across 
flats 


Weight per ft., lbs. 


flats 


Hexagon 


Octagon 


Hexagon 


Octagon 


A 


.012 


.011 


lA 


7.195 


6.905 


i 


.046 


.044 


If 


7.776 


7.446 


A 


.103 


• 099 


lii 


8.392 


8.027 


i 


.185 


• 177 


li 


9.025 


8.635 


A 


.288 


.277 


iW 


9.682 


9.264 


i 


.414 


.398 


li 


10.36 


9.918 


A 


.564 


.542 


li* 


11.06 


10.58 


i 


.737 


.708 


2 


11.79 


11.28 


A 


.932 


.896 


2i 


1331 


12.71 


f 


I.ISI 


1 . 107 


2i 


14.92 


14.24 


U 


1.393 


1. 331 


2i 


16.62 


15-88 


J 


I.6S8 


1.584 


2i 


18.42 


17.65 


U 


1.944 


1.860 


2i 


20.31 


19.45 


i 


2.256 


2. 156 


2} 


22. 29 


21.28 


'A 


2. 591 


2.482 


2i 


24.36 


23.28 


I 


2.947 


2.817 


3 


26.53 


25.36 


lA 


3.327 


3. 182 


3i 


28.78 


27.50 


li 


3 730 


3.568 


3i 


31. 10 


29. 28 


lA 


4^I56 


3.977 


3l 


33.57 


32. 10 


li 


4.60s 


4.407 


3i 


36. 10 


34-56 


lA 


S.0-'7 


4.858 


3l 


38.73 


37.05 


i| 


S.S7I 


S.331 


3 I 


41-45 


39-68 


lA 


6.091 


5-827 


3i 


44.26 


42-35 


li 


6.631 


6.344 


4 


47.16 


45- 12 



Table 6. — Weight of 


Spheres 


OF Various Metals 


Diameter 
in ins. 






Weight in 


pounds 






Steel 


Wrought 
iron 


Cast-iron 


Copper 


Brass 


Lead 


I 


. 146 


.142 


• 134 


.166 


.155 


.2X3 


ij 


• 495 


.481 


.564 


.563 


.526 


.723 


2 


1. 13 


I.I 


1.07 


1-33 


I. 25 


1. 71 


2j 


2.26 


2.2 


2. 1 


2.6 


2.4 


3-3 


3 


3-9 


3.8 


3.6 


4-5 


4.2 


5-8 


3J 


6.2 


6.1 


5-8 


7.146 


6.7 


9-2 


4 


9.27 


9 


8.6 


10.6 


9-9 


13.6 


4i 


13-3 


13 


12.3 


15.2 


14.2 


19-5 


5 


18. 5 


18 


17 


21 


19-5 


27 


5i 


24.4 


23.7 


22.6 


27.7 


25-9 


35.5 


6 


31.9 


31 


29. 1 


36 


33-6 


46 


6J 


40 


39 


36 


45.7 


42.7 


58.7 


7 


SO. 5 


49 


46.2 


57 


53.3 


73 


7i 


62 


60.2 


57 


70.3 


65.7 


90.3 


8 


75.2 


73 


69 


85 


79-4 


109 


Si 


90 


87.5 


83 


102.3 


95-6 


131. 4 


9 


107 


104 


98.1 


121 


113 


155 


9J 


126 


122.4 


116 


143 


133-6 


183.7 


10 


146 


142 


134.5 


166 


155 


213 


loj 


169 


165 


156.5 


193 


180 


284 


II 


I9S 


190 


180 


222 


207. 5 


286 


Hi 


222 


216.5 


205 


253 


236.4 


32s 


12 


-254 


247 


233-S 


288 


270 


370 



WEIGHTS OF MATERIALS 



397 



Table 7. — Weights of Sheet Iron, Steel, Copper and Brass 





Birmingh 


im Gage 










American 


or Brown & Sharpe gage 








Thickness in 
ins. 


Weight per sq. ft. | 


No of 
gage 


Thickness 
in ins. 


Weight per sq. ft. 


No of gage 


Steel 


Iron 1 Copper 


Brass 


Steel 


Iron 


Copper 


Brass 


0000 


■ 454 


18.5232 


18.16 


20.5662 


19-4312 


0000 


.460000 


18.7680 


18.4000 


20.8380 


19-6880 


000 


.425 


17-3400 


17.00 


19.2525 


18. 1900 


000 


.409642 


16.7134 


16.3857 


18.5568 


17-5327 


00 


.380 


15-5040 


15-20 


17.2140 


16.2640 


00 


.364796 


14-8837 


14-5918 


16.5253 


IS- 6133 





.340 


13-8720 


13-60 


15.4020 


14.5520 





.324861 


13-2543 


12.9944 


14.7162 


13-9041 


I > 


.300 


12. 2400 


12. 00 


13.5900 


12.8400 


I 


.289297 


11-8033 


11.S7I9 


13-1052 


12.3819 


2 


.284 


11.5872 


11.36 


12.8652 


12.1552 


2 


.257627 


10.5112 


10.3051 


11-6705 


11.0264 


3 


.259 


10.5672 


10.36 


11.7327 


1 1 . «85 2 


3 


.229423 


9.360s 


9.1769 


10.3929 


9.8193 


4 


.238 


9.7104 


9.52 


10.7814 


10. 1864 


4 


.204307 


8.3357 


8.1723 


9.2551 


8.7443 


s 


.220 


8.9760 


8.80 


9.966 


9.4160 


5 


. 181940 


7.4232 


7.2776 


8.2419 


7.7870 


6 


.203 


8.2824 


8.12 


9.1959 


8.6884 


6 


. 162023 


6.61OS 


6.4809 


7.3396 


6.9346 


7 


. 180 


7-3440 


7.20 


8.1540 


7.7040 


7 


.144285 


5.8868 


S.7714 


6.5361 


6.1754 


8 


.165 


6.7320 


6.60 


7.474s 


7.0620 


8 


. 128490 


5-2424 


5.1396 


S.8206 


5 - 4994 


9 


.148 


6.0384 


5-92 


6.7044 


6.3344 


9 


.114423 


4.6685 


4.5769 


S.1834 


4-8973 


10 


.134 


5.4672 


S-36 


6.0702 


5-7352 


10 


. 101897 


4-1574 


4-0759 


4.6159 


4-3612 


II 


. 120 


4.8960 


4.80 


5.4360 


5.1360 


II 


.090742 


3-7023 


3.6297 


4. 1106 


3.8838 


12 


.109 


4-4472 


4-36 


4-9377 


4-6652 


12 


.08080S 


3.2970 


3.2323 


3 . 6606 


3.4586 


13 


.095 


3-8760 


3-80 


4-3035 


4.0660 


13 


. 071962 


2.9360 


2.878s 


3-2599 


3.0800 


14 


.083 


3-3864 


3-32 


3-7599 


3-5524 


14 


. 064084 


2.6146 


2.5634 


2.9030 


2.7428 


IS 


.072 


2-9376 


2.88 


3, 2616 


3.0816 


IS 


.057068 


2.3284 


2.2827 


2.5852 


2.442s 


16 


.06s 


2.6520 


2.60 


2.9445 


2.7820 


16 


.050821 


2.0735 


2.0328 


2.3022 


2.17S1 


17 


.058 


2.3664 


2.32 


2.6274 


2.4824 


17 


.045257 


1.8465 


I.8I03 


• 2.0501 


1.9370 


18 


.049 


1.9992 


1.96 


2.2197 


2.0972 


18 


-040303 


1 . 6444 


1.6121 


1.8257 


1.7250 


19 


.042 


I. 7 136 


1.68 


1.9026 


1.7976 


19 


-035890 


1 • 4643 


I.43S6 


1.6258 


1.5361 


20 


.035 


1.4280 


1.40 


1.5855 


1.4980 


20 


.031961 


1.3040 


1.2784 


1.4478 


1.3679 


21 


.032 


1-3056 


1.28 


1 - 4496 


1.3696 


21 


. 028462 


1. 1612 


1.1385 


1.2893 


1.2182 


22 


.028 


I - 1424 


1.12 


1.2684 


I. 1984 


22 


.025346 


I -0341 


1.0138 


I. 1482 


1.0848 


23 


.025 


1-0200 


I. 00 


1.1325 


1.0700 


23 


.022572 


.92094 


.90288 


1.0225 


.96608 


24 


.022 


.8976 


.88 


.9966 


.9416 


24 


.020101 


.82012 


. 80404 


.91058 


.86032 


25 


. 020 


.8160 


.80 


.9060 


.8560 


25 


.017900 


.73032 


.71600 


.81087 


.76612 


26 


.018 


■ 7344 


.72 


■ 8154 


.7704 


26 


-015941 


.65039 


.63764 


.72213 


.68227 


27 


.016 


.6528 


.64 


.7248 


.6848 


27 


.014195 


■57916 


.56780 


.64303 


.60755 


28 


.014 


.5712 


.56 


.6342 


.5992 


28 


. 012641 


.51575 


.50564 


.57264 


-54103 


29 


.013 


• 5304 


.52 


.5889 


.5564 


29 


.011257 


.45929 


.45028 


.50994 


.48180 


30 


.012 


.4896 


.48 


.5436 


.5136 


30 


.010025 


.40902 


.40100 


.45413 


.42907 


31 


.010 


.4080 


.40 


■ 4530 


.4280 


31 


. 008928 


.36426 


.35712 


• 40444 


.38212 


32 


.009 


.3672 


.36 


.4077 


.3852 


32 


.007950 


■32436 


.31800 


.36014 


.34026 


33 


.008 


.3264 


.32 


.3624 


.3424 


33 


.007080 


■28886 


.28320 


.32072 


.30302 


34 


.007 


.2856 


.28 


.3171 


.2996 


34 


. 00630s 


■25724 


.25220 


.28562 


.26985 


35 


.005 


.2040 


.20 


.2265 


.3140 


35 


.005615 


. 22909 


. 22460 


.25436 


.24032 


36 


.004 


.1632 


.16 


. 1812 


. 1712 


36 


.005000 


. 20400 


. 20000 


.22650 


.21400 














37 


.004453 


. 18168 


. I7812 


. 20172 


.19059 


Specific gravities. . . 




7.8s 


7.70 


8.72 


8.24 


38 


.00396s 


.16177 


.15860 


.17961 


. 16970 






Weight of a cubic fo 


ot 


489.6 


480.0 


543-6 


513-6 


39 


.003531 


. 14406 


.14124 


.15995 






- 15 1 13 


Weight of a cubic in 


ch 


.2833 


.2778 


.3146 


.2972 


40 


.003144 


. 12828 


.12576 


.14242 


-13456 





Table 8. — Weights op Flat Sizes of Steel in Pounds per Linear Foot 



It 



If 



2* 



2r 



213 

42s 
638 
744 



.266 
.399 
.533 
.66s 
.798 

.931 
1.07 
1 . 20 



.320 

.480 
.640 
.800 
.960 

1. 12 
1.28 
1.44 
1 . 60 
1.76 



.372 
.558 
.743 
.929 
1. 12 

1.30 
1.49 
1.67 
1.86 
2.04 

2.23 
2.41 



.426 

•639 
.852 
1.06 
1.28 

1.49 
1.70 
1. 91 
2. 12 
2-34 

2-55 
2. 76 
2.98 
3-19 



-479 
.718 
.958 
1.20 
1.43 

1.67 
1. 91 
2.1s 
2-39 
2.63 

2.86 
3." 
3-34 
3-59 
3-82 



.530 
.790 
1.06 
1.33 
1. 59 

1.86 
2.13 
2.39 
2.66 
2.92 

3.19 

3-45 
3-72 
3.98 
4.2s 

4.78 



.585 
.878 
1. 17 
1.46 
1.7s 

2.05 

2.34 
2.63 
2.92 

3-22 

3-50 
3-80 
4.09 
4-38 
4.68 



5-27 
S-8S 
7.02 



.640 
.960 
1.28 
1.60 
1. 91 

2.23 

2.SS 

2.87 
3-19 
3.51 

3.83 
4.14 
4.46 
4.78 

S.io 

S.74 
6.38 

7-67 



.745 
1. 12 

1-49 
1.86 
2.23 

2.60 
2.98 
3-35 
3-72 
4.09 

4.46 
4-83 
5.21 

5.58 
5 -96 

6.71 
7-45 
8-94 



.850 
1.28 
I. 70 
2.13 
2.55 

2.98 
3.40 
3.83 
4. 26 
4.68 

S.IO 
5-53 
5.96 
6.38 
6.80 

7-65 

8- so 

10. 20 



-955 
1-43 
1. 91 

2.39 
2.87 

3-35 
3.83 
4.30 
4-79 
5.26 

S.74 
6.22 
6.70 
7.17 
7.66 

8.61 

9-57 
11.49 



1.07 
1.60 
2.13 
2.66 
3.20 

3-72 
4.26 
4.78 
5.32 
5.84 

6.40 
6.91 
7.46 
7-97 

8.52 

9 59 

10.65 
12.78 



1. 18 
1.76 

2.34 
2.92' 

3.51 

4.09 
4.68 
5-26 
5-86 
6-43 

7- 02 
7.60 
8.19 
8.77 
9-36 

10- 54 
II . 71 
14.04 



1.28 
1.92 
2.56 
3-19 
3S3 

4.46 
S.IO 
5.74 
6-39 
7.01 

7-65 
8.29 

8.94 

9-56 

10. 20 

11.48 
12. 76 
15.30 



1-49 
2.24 
2.98 
3.72 
4.46 

5.21 
5.96 
6.69 

7-44 
8.18 

8.92 

9.67 

10.42 

II . 20 

II .92 

13-41 
14.90 
17.88 



I. 70 
2.55 
3.40 
4.25 
5.10 

5-95 
6.80 

7.65 
8-52 
9.35 

10. 20 

11 . 10 
11.92 
12.80 
13.60 

15-30 
17 .00 
20.40 



2.13 
3.20 
4.26 
5.32 
6.40 

7.44 

8.52 

9-56 

10. 64 

11.70 

12.80 
13-80 
14-92 
15-90 
17.04 

19.17 
21.30 
25-56 



2.56 
3.83 
5-11 
6.38 
7.66 

8.92 
10. 20 
11.50 
12.78 
14.00 

15-30 
16. 60 
17.88 
19. 10 
20.40 

22.9s 
25.61 
30- 



398 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table g. — Weights and Sectional Areas op Square and Round Steel Bars. By the Carnegie Steel Co. 



Thickness or 

diameter in 

ins. 



Weight of 

□ bar I ft. 

long 



Weight of 

O bar I ft. 

long 



.013 

■OS3 
.119 

. 212 
•333 
.478 
.651 

.850 
1 .076 
1.328 
1.608 

1-913 

2.24s 
C.603 
2.989 

3.400 
3.838 
4.303 
4 -795 

5. 312 
5. 857 
6.428 
7.026 

7.650 
8.301 
8.978 
9.682 

10.41 

11 . 17 

11-95 

12. 76 

13.60 
14.46 

15. 35 

16. 27 

17. 22 
18.19 
19.18 
20. 20 

21.25 
22.33 

23.43 
24.56 

25.71 
26.90 
28.10 
29.34 

30.60 
31.89 
33 20 



.010 
.042 
.094 

. 167 
. 261 
■375 
-Sii 

.667 

■84s 

1 .043 

1 . 262 

1.502 

1.763 
2.044 

2.347 

2.670 
3.014 
3.379 
3.766 

4.173 
4.600 

5.049 
S-S18 

6.008 
6.520 
7-051 
7.604 

8.178 

8.773 
9.388 



10.68 
11.36 
12.06 
12.78 

13.52 
14.28 
15.07 
15.86 

16. 69 

17.53 
18.40 
19.29 

20. 20 

21 . 12 
22 .07 
23.04 

24.03 
25.04 
26.08 



Area of 
□ bar in 
sq. ins. 



Area of 
O Bar in 
sq. ins. 



Thickness or 

diameter in 

ins. 



Weight of 

□ bar I ft. 

long 



Weight of 

O bar I ft. 

long 



.0039 
.0156 
■0352 

.0625 

.0977 
. 1406 
.1914 

.2500 
.3164 
.3906 

•4727 

■5625 
.6602 
.7656 
.8789 

I . 0000 
I. 1289 
I . 2656 
I .4102 

1.5625 
I . 7227 
I . 8906 
2.0664 

2. 2500 
2.4414 
2 . 6406 
2.8477 

3.0625 
3.2852 
3.5156 
3.7539" 

4 . 0000 
4.2539 
4.5156 
4.7852 

5.0625 

5.3477 
5-6406 
5-9414 

6.2500 
6.5664 
6.8906 

7 . 2227 

7-5625 
7.9102 
8.2656 
8.6289 

9 . 0000 

9-3789 
9.7656 



.0031 
.0123 
.0276 

.0491 
.0767 
.1104 
-1503 

.1963 
.2485 
-3068 
.3712 

.4418 
.5185 
.6013 
.6903 

■7854 

.8866 

.9940 

1.107s 

1 . 2272 
1.3530 
I . 4849 
I .6230 

I. 7671 

I. 9175 
2.0739 
2.2365 

2 4053 
2.5802 
2 . 7612 
2 . 9483 

3.1416 
3.3410 
3.5466 
3.7583 

3.9761 
4. 2000 
4.4301 
4 . 6664 

4.9087 
5.1572 
S.4119 
5.6727 

S.9396 
6.2126 
6.4918 

6.7771 

7.0686 
7.3662 
7.6699 



34. 


55 


35. 


92 


37. 


31 


38. 


73 


40. 


18 


41. 


65 


43. 


14 


44- 


68 


46. 


24 


47- 


82 


49 


42 


SI 


05 


52 


71 


54 


40 


56 


II 


57 


85 


59 


62 


61 


41 


63 


23 


65 


08 


66 


95 


68 


85 


70 


78 


72 


73 


74 


70 


76 


71 


78 


74 


80 


81 


82 


89 


85 


00 


87 


14 


89 


30 


91 


49 


93 


72 


95 


96 


98 


23 


100 


5 


102 


8 


105 


2 


107 


6 


no 




112 


4 


114 


9 


117 


4 


119 


9 


122.4 


12 


5 


12 


7.6 


130-2 


13 


2.8 


135-5 



27.13 

28.20 

29.30 
30.42 
31.56 

32.71 
33-90 
35-09 
36-31 

37-56 
38-81 

40. 10 

41.40 

42.73 
44.07 

45-44 
46.83 

48. 24 
49.66 

51. 11 
52-58 

54-07 

55.59 
57.12 
58.67 

60.25 
61.84 
63.46 
65.10 

66.76 
68.44 
70.14 
71.86 

73.60 
75.37 
77.15 
78.95 

80 77 
82.62 
84.49 
86.38 

88.29 
90. 22 
92.17 
94.14 

96.14 
98.14 
100. 2 
102. 2 

104.3 
106.4 



Area of 
□ bar in 
sq. ins. 



10. 160 
10.563 
10.973 
II. 391 
II. 816 

1 2 . 2 50 
12.691 
13. 141 
13.598 

14.063 

14.53s 
15.016 

15.504 

16.000 

16.504 
17.016 
17.535 

18.063 
18.598 
19. 141 
19.691 

20.250 
20.816 
21.391 
21.973 

22.563 
23. 160 
23.766 
24.379 

25 .000 
25.629 
26. 266 
26.910 

27.563 
28. 223 
28.891 
29.566 

30-250 

30.941 
31.641 
32.348 

33.063 

33.785 
34.516 

35.254 

36.000 
36.754 
37.516 
38.285 

39.063 
39.848 



WEIGHTS OF MATERIALS 



399 



Table g. — Weights and Sectional Areas of Square and Round Steel Bars. By the Carnegie Steel Co. — {Contimied) 



Thickness or 


Weight of 


Weight of 


Area of 


Area of 


diameter in 


D bar I ft. 


Obar I ft. 


D bar in 


Obar in 


ins. 


long 


long 


sq. ins. 


sq. ins. 


1 


138.2 


108.5 


40.641 


31-919 


A 


140.9 


no. 7 


41.441 


32-548 


h 


143.6 


112. 8 


42.250 


33-183 


A 


146. s 


114. 9 


43.066 


33-824 


f 


149.2 


117 . 2 


43-891 


34-472 


H 


152.1 


119. 4 


44-723 


35-125 


3 

4, 


IS4-9 


121 . 7 


45-563 


35-785 


H 


IS7-8 


123.9 


46.410 


36.450 


7 


160.8 


126.2 


47.266 


37.122 


If 


163.6 


128. 5 


48. 129 


37.800 


7 


166.6 


130.9 


49 . 000 


38.485 


^ 


169.6 


133-2 


49.879 


39.17s 


i 


172.6 


135-6 


50.766 


39.871 


A 


175-6 


137-9 


51.660 


40.574 


i 


178.7 


140.4 


52-563 


41.282 


1^ 


181. 8 


142.8 


Si ■ 473 


41.997 


1 


184.9 


I4S-3 


54-391 


42.718 


A 


188. 1 


147-7 


55.316 


43-445 


i 


191-3 


150. 2 


56.250 


44- 179 


A 


194.4 


152.7 


57.191 


44.918 


f 


197.7 


IS5-2 


58.141 


45-664 


H 


200. 9 


157.8 


59.098 


46.415 


1 


204.2 


160.3 


60 . 063 


47-173 


H 


207.6 


163 


61.03s 


47-937 


1 


210.8 


165.6 


62 .016 


48.707 


if 


214.2 


168.2 


63 - 004 


49 • 483 


8 


217,6 


171. 


64 . 000 


50.265 


1^ 


221 .0 


173.6 


65 . 004 


51-054 


i 
s 


224.5 


176.3 


66.016 


51.849 


1^ 


228.0 


179.0 


67-035 


52.649 


i 


231.4 


181. 8 


68.063 


53.456 


5 
16 


234-9 


184.5 


69 . 098 


54.269 


f 


238 -5 


187-3 


70.141 


55.088 


7 
16 


242.0 


190. 1 


71.191 


55.914 


1 
2 


245.6 


193.0 


72.250 


56.745 


^ 


249-3 


195.7 


73-316 


57.583 


1 


252-9 


198.7 


74-391 


58.426 


^ 


256.6 


201.6 


75-473 


59.276 


1 


260.3 


204.4 


76-563 


60. 132 


if 


264. 1 


207.4 


77.660 


60.994 


7 
8 


267.9 


210.3 


78.766 


61.862 


if 


271.6 


213-3 


79.879 


62.737 


9 


275-4 


216.3 


81.000 


63.617 


-i_ 

16 


2793 


219.3 


82. 129 


64.505 


i 


283.2 


222.4 


83.266 


65.397 


.3_ 
16 


287.0 


225-4 


84.410 


66. 296 



Thickness or 


Weight of 


Weight of 


Area of 


Area of 


diameter in 


D bar I ft. 


bar 1 ft. 


n bar in 


Obar in 


ins. 


long 


long 


sq. ins. 


sq. ins. 


1 

i 


290. 9 


228. s 


85-563 


67, 201 


A 


294.9 


231-5 


86.723 


68.112 


3 

8 


298.9 


234-7 


87.891 


69.029 


1^ 


302.8 


237-9 


89.066 


69.953 


f 


306.8 


241.0 


1 

90-250^ 


70.882 


9 
16 


310.9 


244.2 


91,441 


71.818 


1 


315.0 


247-4 


92,641 


72. 760 


11 

16 


319.1 


250.6 


93-848 


73 . 708 


3 
4 


323.2 


253-9 


95-063 


74,662 


if 


327.4 


257-1 


i 96.285 


75.622 


7 

8 


331.6 


260.4 


97-516 


76.589 


15 
16 


335.8 


263.7 


98-754 


77.561 


10 


340.0 


267.0 


100.00 


78.540 


A 


344.3 


270.4 


101.25 


79.525 


1 
8 


348.5 


273.8 


102.52 


80.516 


A 


352.9 


277,1 


103 - 79 


81.513 


1 
1 


357.2 


280,6 


105.06 


82.516 


A 


361,6 


284,0 


106.3s 


83.525 


3 

s 


366,0 


287.4 


107.64 


84.541 


A 


370.4 


290,9 


108 . 94 


85.562 


4 


374.9 


294,4 


1 

110.25 


86.590 


9 
16 


379.4 


297.9 


"1-57 


87.624 


5 
6 


383.8 


301.4 


112.89 


88 . 664 


11 
16 


388.3 


305.0 


114. 22 

j 


89.710 


f 


392.9 


308,6 


115-56 


90.763 


13 
16 


397.5 


312,2 


116.91 


91.821 


7 
8 


402.1 


315.8 


118 27 


92.886 


15 
16 


406.8 


319.5 


119.63 


93.956 


11 


411.4 


323.1 


121.00 


95.033 


16 


416.1 


326,8 


122.38 


96. 116 


i 


420.9 


330.5 


123.77 


97.205 


A 


425.5 


334.3 


125. j6 


98,301 


\ 


430.3 


337.9 


126.56 


99.402 


A 


435.1 


341.7 


127.97 


100.51 


• 1 


439.9 


345.5 


129.39 


101 ,62 


7 

16 


444,8 


349.4 


130,82 


102.74 


h 


449.6 


353.1 


132.25 


103.87 


9 
16 


454. 5 


357.0 


133.69 


105.00 


5 
8 


459.5 


360,9 


135.14 


106. 14 


H 


464,4 


364.8 


136, 60 


107. 28 


I 


469.4 


368,6 


138,06 


108.43 


13 
16 


474-4 


372-6 


139 . 54 


109.59 


7 
8 


479-5 


376.6 


141 ,02 


no. 75 


if 


484.5 


380.6 


142,50 


III .92 



400 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table lo. — Weights of Brass, Copper and Aluminum Bars 



Dia. or 
dis- 




Brass 




Copper 


Aluminum 


tance 


Weight per ft., 


lbs. 


Weight per ft. lbs.. 


Weight per ft. lbs.. 


across 
flats 


Round 


Square 


Hexagon 


Round 


Square 


Round 


Square 


A 


.oil 


. 014 


-013 


.012 


.015 


.003 


.004 


i 


• 04S 


• OSS 


.048 


.047 


.060 


. 014 


.018 


A 


. 100 


.125 


.108 


. 106 


-135 


.032 


.041 


i 


.I7S 


.225 


-194 


.189 


.241 


• 057 


.072 


A 


.275 


• 350 


.301 


.296 


.377 


.089 


• 114 


f 


• 395 


.510 


• 436 


.426 


■ 542 


.128 


.163 


A 


■S40 


.690 


.592 


.579 


.737 


.174 


. 222 


1 


.710 


■ 90s 


.773 


■ 757 


.964 


. 227 


. 290 


A 


.900 


I -15 


.978 


.958 


I. 22 


.288 


-367 


i 


I. 10 


1 . 40 


1.24 


I. 18 


1.51 


.356 


-453 


a 


1-35 


1.72 


I -45 


1-43 


1.82 


.430 


■ 548 


i 


1.66 


2.05 


1-73 


1.70 


2.17 


■ 516 


■ 652 


a 


1.8s 


2.40 


2.03 


2.00 


2-54 


.601 


.766 


i 


2. IS 


2. 75 


2.36 


2.32 


2-95 


■ 697 


.888 


H 


2.48 


315 


2.71 


2.66 


3-39 


.800 


1.02 


I 


2.85 


3.6s 


3-10 


3-03 


3-86 


.911 


1. 16 


lA 


3-20 


4.08 


3-49 


3-42 


4-35 


1^03 


t-31 


li 


357 


4-55 


3.91 


3.81 


4.88 


I IS 


1-47 


lA 


3-97 


5 08 


4.38 


4.27 


5-44 


1.28 


1.64 


li 


4.41 


5-65 


4.82 


4-72 


6.01 


1.42 


I. 81 


lA 


4.86 


6. 22 


5-33 


S-2I 


6.63 


1.57 


2.00 


If 


S.3S 


6.81 


5 -76 


5.72 


7.24 


1.72 


2.19 


I A 


S.86 


7.45 


6.38 


6.26 


7.97 


1.88 


2.40 


i^ 


6.37 


8.13 


6.92 


6.81 


8.67 


2.05 


2.61 


lA 


6.92 


8.83 


7-54 


7.39 


9.41 


2. 22 


2.83 


if 


7.48 


9.55 


8.15 


7.99 


10.18 


2.41 


3.06 


i}4 


8.05 


10. 27 


8.80 


8.45 


10.73 


2.59 


3-30 


li 


8.6s 


II . 00 


9-47 


9.27 


11.80 


2.79 


3.SS 


il* 


9.29 


11.82 


10. IS 


9-76 


12.43 


2.99 


3.81 


i| 


9-95 


12.68 


10.86 


10.64 


13 -5S 


3-20 


4.08 


iH 


10. S8 


13.50 


11.68 


11. II 


14-15 


3-41 


4-35 


2 


II. 2S 


14-35 


12.36 


12. II 


IS -42 


3-64 


4-64 


2S 


12.78 


16. 27 


13.92 


13.67 


17-42 


4. II 


5-24 


2j 


14-32 


18. 24 


lS-72 


15-33 


19. SI 


4.61 


5-87 


2i- 


IS .96 


20.32 


17-52 


17-08 


21.74 


S-14 


6-54 


2j 


17.68 


22. S3 


19-44 


18.92 


24.09 


5-69 


7-25 


2s 


19 -SO 


24-83 


21 . 24 


20.86 


26. S6 


6. 27 


7-99 


2 J 


21 .40 


27,2s 


23.40 


22.89 


29.0s 


6.89 


8-53 


25 


23-39 


29.78 


25-82 


2S.02 


31.86 


7.52 


9-S8 


3 


25-47 


32.43 


27-84 


27.24 


34 69 


8.20 


10.44 


3i 


30.45 


38.77 


32.76 


31-97 


40.71 


9.62 


12. 2S 


35 


35-31 


44 96 


37-80 


37-08 


47-22 


II . 16 


14. 21 


3i 


40.07 


51.01 


43-56 


42. II 


S3-6I 


12.81 


16.31 


4 


46. 12 


58.73 


49-44 


48.43 


61.67 


14^56 


18.56 



Table 12.— Weights of Flat Rolled Strips, Hoop or Band Steel 

Pounds per Lineal Foot 

Thicknesses by Birmingham Wire Gage 

One cu. ft. of steel weighs 489.6 lbs. 

For widths from J in. to J in. and thicknesses from No. 19 to No. 11 

B. W. G. 



Width 
in ins. 


o,.S 


It 


1° 


a 
° 'S 


•0 c 

2 ° 


4.g 

. '^ 
00 

•z °. 


. ^ 

c> 

2 ° 


. 

^ ". 


M 6 

. 
" 


i 


■ 036 


.042 


.049 


-05s 


.061 


.071 


■ 081 


.093 


.102 


H 


■ 038 


.044 


.052 


.059 


.065 


.07s 


.086 


.098 


.108 


A 


■ 040 


.047 


.055 


.062 


.069 


.079 


.091 


.104 


■ IIS 


.i_9 


.042 


■ 049 


-0S9 


.066 


.073 


.084 


.096 


.110 


. 121 


A 


.045 


■ 052 


.062 


.069 


.077 


.088 


■ lOI 


.116 


.128 


li 


.047 


• OSS 


.065 


.073 


.080 


.093 


■ 106 


.122 


.134 


¥1 


.049 


.057 


.068 


.076 


.084 


.097 


■ III 


.127 


.140 


15 


.051 


.060 


.071 


.079 


.088 


. lOI 


.116 


.133 


•147 


1 


.054 


.062 


■ 074 


■ 083 


.092 


.106 


.121 


•139 


.153 


H 


.056 


.065 


.077 


.086 


.096 


.110 


.126 


■14s 


.159 


M 


.058 


.068 


.080 


.090 


.099 


.lis 


.131 


■ ISI 


.166 


K 


.060 


.070 


.083 


.093 


.103 


.119 


.136 


.IS6 


.172 


A 


.062 


-073 


.086 


.097 


.107 


.123 


.141 


.162 


.179 


a 


.t>6s 


-075 


.089 


. 100 


. Ill 


.128 


. 146 


.168 


.18S 


a 


.067 


.078 


■ 092 


. 104 


.lis 


.132 


.ISI 


.174 


.191 


ii. 


.069 


.081 


.096 


.107 


.119 


.137 


.136 


.180 


.198 


i 


.071 


.083 


.099 


. Ill 


. 122 


■ 141 


. 162 


.185 


. 204 




.074 


.086 


. 102 


.114 


. 126 


. 146 


.167 


.191 


.210 


M 


.076 


.089 


.105 


.117 


.130 


.ISO 


.172 


.197 


.217 


U 


.078 


. 091 


. 108 


. 121 


.134 


.154 


-177 


.203 


.223 


A 


.080 


• 094 


■ III 


.124 


.138 


.159 


.182 


■ 208 


.230 


a 


.083 


.096 


■ 114 


.128 


.142 


.163 


.187 


.214 


.236 


ii 


.085 


.099 


■ 117 


.131 


.145 


.168 


. 192 


.220 


. 242 


a 


.087 


. 102 


. 120 


.135 


.149 


.172 


.197 


.226 


.249 


f 


.089 


. 104 


.123 


.138 


.153 


.176 


.202 


.232 


.255 


n 


.091 


.107 


. 126 


.142 


.157 


.181 


.207 


.237 


.261 


a 


.094 


. 109 


. 129 


.145 


.161 


.185 


.212 


■ 243 


.268 


a 


.096 


.112 


.132 


.148 


. 164 


. 190 


.217 


■ 249 


.274 


H 


.098 


■ 115 


.136 


.152 


.168 


.194 


.222 


■ 2SS 


.281 


li 


.100 


.117 


•139 


.ISS 


.172 


. 198 


.227 


.261 


.287 


a 


.103 


.120 


.142 


• 159 


.176 


.203 


.232 


.266 


.293 


u 


.105 


■ 122 


-145 


.162 


. 180 


. 207 


.237 


.272 


.300 


! 


.107 


.I2S 


.148 


.166 


. 184 


. 212 


.242 


.278 


.306 



To compute the weight of sheet iron on the basis of 480 lbs. per cu. ft. 
divide the thickness expressed in thousandths by 25. The result is the 
weight in lbs. per sq. ft. 



Table ii. — Weight of Steel Plates per Sq. Ft. Both Theoretical 
AND with Commercial Overweight Allowance 



Thickness 


Theoretical 


Allowance for 


Adjusted 


weight, 


overweight, plates 


weight, 




lbs. 


SO to 75 ins. wide 


lbs. 






Per cent. 




3 

T6 


7.6s 


10 


8.42 


1 

1 


10. 207 


10 


11.23 


^ 


12.75 


8 


13.78 


i 


15.31 


7 


16.38 


^ 


17.86 


6 


18.93 


i 


20.41 


5 


21.44 


-9_ 
16 


22.96 


4i 


24.00 


5 
8 


2551 


4 


26.5 


H 


28.07 


3l 


29. 20 


3 

4 


30.62 


3i 


30.80 


a 


33-17 


3i 


34 ■so 



I 

^^f The centigrade thermometer scale is a case of the blind worship 
of decimals. It possesses no advantage that can be discovered by 
any except its devotees, while the confusion due to its existence 
overbalances a hundredfold all the advantages that its advocates 
imagine they see in it. It has introduced two sets of temperature 
observations where there might have been one; it has made necessary 
countless conversions between observations where there might have 
been none, and to offset this it has introduced no compensating ad- 
vantage whatever. Every application of the following conversion 
formulas and tables is an illustration of the harm done by this fussy 
and amateurish attempt to improve a thing that did not need 



HEAT 



improvement as well as of the uniform result when metric and 
other hobby riders endeavor to change established standards of 
measurement. 

Conversions between the Fahrenheit and Centigrade scales may be 
made by the following formulas: 

in which F = reading by Fahrenheit scale, 
C = reading by Centrigade scale. 









Table i. — 


Equivalent Temperatures — Centigrade to 


Fahrenheit. 








c.° 





10 


20 


30 


40 


50 


60 


70 


80 


90 






F 


F. 


F. 


F. 


F. 


F. 


F. 


F. 


F. 


F. 




— 200 


-328 


-346 
-166 


-364 
-184 


-382 
— 202 


— 400 


— 418 


—4^6 


-454 
-274 








— lOO 


*■" 

-148 


— 220 


-238 


'TO" 
-256 


— 292 


-310 




— 


+32 


+ 14 


-4 


— 22 


-40 


-S8 


-76 


-94 


— 112 


-130 







32 


so 


68 


86 


104 


122 


140 


158 


176 


194 




100 


212 


230 


248 


266 


284 


302 


320 


338 


356. 


374 




200 


392 


410 


428 


446 


464 


482 


SCO 


518 


536 


554 




300 


572 


590 


608 


626 


644 


662 


680 


698 


716 


734 




400 


752 


770 


788 


806 


824 


842 


860 


878 


896 


914 




500 


932 


95° 


968 


986 


1004 


1022 


1040 


1058 


1076 


1094 




600 


1112 


1130 


1148 


1166 


1184 


1202 


1220 


1238 


1256 


1274 




700 


1292 


1310 


1328 


1346 


1364 


1382 


1400 


1418 


1436 


1454 




800 


1472 


1490 


1508 


1526 


1544 


1562 


1580 


1598 


1616 


1634 




900 


1652 


1670 


1688 


1706 


1724 


1742 


1760 


1778 


1796 


1814 




1000 


1832 


1850 


1868 


1886 


1904 


1922 


1940 


1958 


1976 


1994 




1 100 


2012 


2030 


2048 


2066 


2084 


2102 


2120 


2138 


2156 


2174 




1200 


2192 


2210 


2228 


2246 


2264 


2282 


2300 


2318 


2336 


2354 


C.° 


F.° 


1300 


2372 


2390 


2408 


2426 


2444 


2462 


2480 


2498 


2516 


2534 


I 


1.8 


1400 


2552 


2570 


2588 


2606 


2624 


2642 


2660 


2678 


2696 


2714 


2 


3-6 


1500 


2732 


2750 


2768 


2786 


2804 


2822 


2840 


2858 


2876 


2894 


3 


5-4 


1600 


2912 


2930 


2948 


2966 


2984 


3002 


3020 


3038 


3056 


3074 


4 


7.2 


1700 


3092 


3110 


3128 


3146 


3164 


3182 


3200 


3218 


3236 


3254 


5 


9.0 


1800 


3272 


3290 


3308 


3326 


3344 


3362 


3380 


3398 


3416 


3434 


6 


10.8 


1900 


3452 


3470 


3488 


3506 


3524 


3542 


3560 


3578 


3596 


3614 


7 


12 .6 


2000 


3632 


365° 


3668 


3686 


3704 


3722 


3740 


3758 


3776 


3794 


8 


14.4 


2100 


3812 


3830 


3848 


3866 


3884 


3902 


3920 


3938 


3956 


3974 


9 


16 . 2 
18.0 


2200 


3992 


4010 


4028 


4046 


4064 


4082 


4100 


4118 


4136 


4154 


10 


2300 


4172 


4190 


4208 


4226 


4244 


4262 


4280 


4298 


4316 


4334 




2400 


4352 


4370 


4388 


4406 


4424 


4442 


4460 


4478 


4496 


4514 




2500 


4532 


4550 


4568 


4586 


4604 


4622 


4640 


4658 


4676 


4694 




2600 


4712 


4730 


4748 


4766 


4784 


4802 


4820 


4838 


4856 


4874 




2700 


4892 


4910 


4928 


4946 


4964 


4982 


5000 


5018 


5036 


5054 




2800 


5072 


5090 


5108 


5126 


5144 


5162 


S180 


5198 


5216 


5234 




2900 


5252 


5270 


5288 


5306 


5324 


5342 


5360 


5378 


5396 


5414 




3000 


S432 


5450 


5468 


5486 


5504 


5522 


S540 


5558 


5576 


5594 




3100 


5612 


5630 


5648 


5666 


5684 


5702 


5720 


5738 


5756 


5774 




3200 


5792 


5810 


5828 


5846 


5864 


5882 


5900 


5918 


5936 


5954 




3200 


5972 . 


5990 


6008 


6026 


6044 


6062 


6080 


6098 


6116 


6134 




3400 


6152 


6170 


6188 


6206 


6224 


6242 


6260 


6278 


6296 


6314 




3500 


6332 


6350 


6368 


6386 


6404 


6422 


6440 


6458 


6476 


6494 




3600 


6512 


6530 


6548 


6566 


6584 


6602 


6620 


6638 


6656 


6674 




3700 


6692 


6710 


6728 


6746 


6764 


6782 


6800 


6818 


6836 


6854 




3800 


6872 


6890 


6908 


6926 


6944 


6962 


6980 


6998 


7016 


7034 




3900 


7052 


7070 


7088 


7106 


7124 


7142 


7160 


7178 


7196 


7214 




C.° 





10 


20 


30 


40 


50 


60 


70 


80 


90 





Example: 1347° C. 
26 



= 2444° F. +12.6° F. =2456.6° F. 



401 



402 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Tables i and 2 by Dr. Leonard Waldo {Trans. A. I. M. E., 191 1) 
are the most complete that have been prepared. Their range is 
from absolute zero to the temperature of the electric arc. The 
equivalents for lo-deg. intervals are read directly and for i-deg. 



intervals by the supplementary tables of proportional parts. The 
tables are used precisely like tables of logarithms as illustrated by 
the examples below them. 



Table 2. — Equivalent Temperatures — Fahrenheit to Centigrade 

Heavy Face Figures Indicate Recurring Decimals. 



F.° 





10 


20 


30 


40 


50 


60 


70 


80 


90 




1 




C. 


C. 


C. 


C. 


C. 


C. 


C. 


C. 


C. 


C. 




—400 
-300 


— 240.0 


-245-5 
— 190.C 


-251-I 
-195-5 


-256.6 
— 201. 1 


— 262 .2 


— 267.7 

— 212.2 












-184.4 


— 206.6 


-217.7 


-223.3 


-228.8 


-234-4 




—200 


-128.8 


-134-4 


— 140.0 


-145-5 


-151. 1 


-156.6 


— 162.2 


-167.7 


-173-3 


-178.8 




— 100 


-73-3 


-78.8 


-84-4 


—90.0 


-95-5 


— lOI.I 


106.6 


— 112. 2 


-"7-7 


-123.3 




—0 


-17.7 


-23-3 


-28.8 


-34-4 


—40.0 


-45-5 


-51-I 


-56.6 


— 62.2 


-67.7 







-17.7 


— 12.2 


-6.6 


— I.I 


+4-4 


+ 10.0 


+ 15-5 


+ 21.1 


+26.6 


+32.2 




100 


37-7 


43-3 


48.8 


54-4 


60.0 


65-5 


71. 1 


76.6 


82.2 


87.7 




200 


93-3 


98.8 


104.4 


IIO.O 


"5-5 


121 . 1 


126.6 


132.2 


137-7 


143-3 




300 


148.8 


154-4 


160.0 


165-5 


171. 1 


176.6 


182.2 


187.7 


193-3 


198.8 




400 


204.4 


210.0 


2IS-5 


221. 1 


226.6 


232.2 


237-7 


243-3 


248.8 


254-4 




500 


260.0 


265.5 


271. 1 


276.6 


282.2 


287.7 


293-3 


298.8 


304-4 


310.0 




600 


3IS-5 


321. 1 


326.6 


332.2 


337-7 


343-3 


348.8 


354-4 


360.0 


365-5 




700 


37I-I 


376-6 


382.2 


387-7 


393-3 


398.8 


404.4 


410.0 


415-5 


421. 1 




800 


426.6 


432.2 


437-7 


443-3 


448.8 


454-4 


460.0 


465-5 


471. 1 


476.6 




900 


482.2 


487-7 


493-3 


498.8 


504 -4 


510.0 


515-5 


521.1 


526.6 


532.2 




1000 


537-7 


543-3 


548.8 


554-4 


560.0 


565-5 


57I-I 


576.6 


582.2 


587-7 


F.° 


C." 


IIOO 


593-3 


598.8 


604.4 


610.0 


615-5 


621. 1 


626.6 


632.2 


637-7 


643-3 


I 


0.5 


1200 


648.8 


654-4 


660.0 


665-5 


671. 1 


676.6 


682.2 


687.7 


693-3 


698.8 


2 


I.I 


1300 


704.4 


710.0 


71S-5 


721. 1 


726.6 


732-2 


737-7 


743-3 


748.8 


754-4 


3 


1.6 


1400 


760.0 


765-5 


771. 1 


776.6 


782.2 


787-7 


793-3 


798.8 


804.4 


810.0 


4 


2.2 


1500 


815-5 


821. 1 


826.6 


832.2 


837-7 


843-3 


848.8 


854-4 


860.0 


865.5 


5 


2-7 


1600 


871. 1 


876.6 


882.2 


887-7 


893-3 


898.8 


904.4 


910.0 


915-5 


921. 1 


6 


3-3 


1700 


926 .6 


9S2.2 


937-7 


943-3 


948.8 


954-4 


960.0 


965-5 


971. 1 


976.6 


7 


3-8 


1800 


982.2 


987.7 


993-3 


998.8 


1004.4 


lOIO.O 


1015.5 


1021.1 


1026.6 


1032.2 


8 


4-4 


1900 


1037.7 


1043-3 


1048.8 


1054-4 


1060.0 


1065.5 


1071.1 


1076.6 


1082.2 


1087.7 


9 


50 


2000 


1093.3 


1098.8 


I 104. 4 


IIIO.O 


1115-5 


II2I.I 


1126.6 


1132.2 


II37-7 


1143.3 




2100 


1148.8 


1154-4 


1160.0 


1165.5 


1171.1 


II76.6 


1182.2 


1187.7 


"93-3 


1198.8 




2200 


1204.4 


1210.0 


1215-5 


1221.1 


1226.6 


1232.2 


1237.7 


1243-3 


1248.8 


I2S4-4 




2300 


1260.0 


1265.5 


1271.1 


1276.6 


1282.2 


1287.7 


1293-3 


1298.8 


1304-4 


1310.0 




2400 


1315-5 


1321.1 


1326.6 


1332.2 


1337-7 


1343-3 


1348.8 


1354-4 


1360.0 


1365-5 




2500 


1371.1 


1376.6 


1382.2 


1387-7 


1393-3 


1398-8 


1404.4 


1410.0 


1415-5 


1421.1 




2600 


1426.6 


1432.2 


1437-7 


1443-3 


1448.8 


1454-4 


1460.0 


1465-5 


1471.1 


1476.6 




2700 


1482.2 


1487.7 


1493-3 


1498.8 


1504-4 


I5IO.O 


1515-5 


1521.1 


1526.6 


1532-2 




2800 


1537-7 


1543-3 


1548.8 


1554-4 


1560.0 


1565-5 


1571-I 


1576-6 


1582.2 


1587-7 




2900 


1593-3 


1598.8 


1604.4 


1610.0 


1615-5 


162I.I 


1626.6 


1632.2 


1637.7 


1643-3 




3000 


1648.8 


1654.4 


1660.0 


1665.5 


1671.1 


1676.6 


1682.2 


1687.7 


1693-3 


1698.8 




3100 


1704.4 


1710.0 


1715-5 


1721. 1 


1726.6 


1732.2 


1737-7 


1743-3 


1748.8 


1754-4 




3200 


1760.0 


1765-5 


1771.1 


1776.6 


1782.2 


1787.7 


1793-3 


1798.8 


1804.4 


1810.0 




3300 


181S-5 


1821.1 


1826.6 


1832.2 


1837-7 


1843-3 


1848.8 


1854-4 


1860.0 


1865.5 




3400 


1871.1 


1876.6 


1882.2 


1887-7 


1893-3 


1898.8 


1904.4 


1910.0 


1915-5 


1921 . 1 




3500 


1926.6 


1932.2 


1937-7 


1943-3 


1948.8 


1954-4 


1960.0 


1965-5 


1971.1 


1976.6 




3600 


1982.2 


1987.7 


1993-3 


1998.8 


2004.4 


2010.0 


2015-5 


2021 . I 


2026.6 


2032.2 




F.° 





10 


20 


30 


40 


50 


60 


70 


80 


90 




1 



Examples: —246.0° F.= — 151. 11° C — 3. 
^e.f.0 n _!_., t^t^f.° n -i_ .,.,.,0 n = 



666' 



3-33° C.= -154. 44° C. 



40.0 r.= — 151.11 v^. — 3-33 *-•=— 15 
C. +1 .666° C.+ . 277° C. = 1328.609° C. 



3762° F. = 207i.ii° C.+i.ii° €. = 2072.22° C. 2423.5° F. = i326.- 



HEAT 



403 



Table 2. — Equivalent Temperatures — Fahrenheit to Centigrade — (^Continued) 
Heavy Faced Figures Indicate Recurring Decimals 



F.° 





10 


20 


30 


40 


50 


60 


70 


80 


90 






C. 


C. 


C. 


C. 


C. 


C. 


C. 


C. 


C. 


C. 




3700 


2037.7 


2043-3 


2048.8 


2054.4 


2060. 


2065.5 


2071. I 


2076.6 


2082.2 


2087.7 




3800 


2093.3 


2098.8 


2104.4 


2110.0 


2115.5 


2121.1 


2126.6 


2132.2 


2137.7 


2143-3 




3900 


2148.8 


2154.4 


2160.0 


2165.5 


2171. I 


2176.6 


2182.2 


2187.7 


2193-3 


2198.8 




4000 


2204.4 


2210.0 


2215.5 


2221. 1 


2226.6 


2232.2 


2237.7 


2243.3 


2248.8 


2254-4 




4100 


2260.0 


2265.5 


2271. 1 


2276.6 


2282.2 


2287.7 


2293.3 


2298.8 


2304.4 


2310.0 




4200 


2315.5 


2321. 1 


2326.6 


2332.2 


2337-7 


2343-3 


2348.8 


2354.4 


2360.0 


2365-5 




4300 


2371. 1 


2376-6 


2382.2 


2387.7 


2393-3 


2398.8 


2404.4 


2410.0 


24155 


2421. I 




4400 


2426.6 


2432.2 


2437-7 


2443 ■ 3 


2448.8 


2454.4 


2460.0 


2465.5 


2471. I 


2476.6 




4500 


2482.2 


2487.7 


2493-3 


2498 . 8 


2504.4 


2510.0 


2515.5 


2521. I 


2526.6 


2532.2 




4600 


2537.7 


2543-3 


2548.8 


2554-4 


2560.0 


2565-5 


2571. 1 


2576.6 


2582.2 


2587-7 




4700 


2593-3 


2598.8 


2604.4 


2610.0 


2615.5 


2621. I 


2626.6 


2632.2 


2637.7 


2643.3 




4800 


2648.8 


2654.4 


2660.0 


2665-5 


2671. I 


2676.6 


2682.2 


2687.7 


2693-3 


2698.8 




4900 


2704.4 


2710.0 


2715-5 


2721. 1 


2726.6 


2732.2 


2737-7 


2743.3 


2748.8 


2754-4 




5000 


2760.0 


2765-5 


2771. I 


2776.6 


2782.2 


2787.7 


2793-3 


2798.8 


2804.4 


2810.0 


F.° 


C." 


5100 


2815.5 


2821. I 


2826.6 


2832.2 


2837.7 


2843-3 


2848.8 


2854.4 


2860.0 


2865.5 


I 


0.5 


5200 


2871. 1 


2876.6 


2882.2 


2887.7 


2893 -3 


2898.8 


2904.4 


2910.0 


2915.5 


292 1. 1 


2 


i.i 


5300 


2926.6 


2932.2 


2937.7 


2943-3 


2948.8 


2954.4 


2960.0 


2965.5 


2971. I 


2976.6 


3 


1.6 


5400 


2982.2 


2987-7 


2993-3 


2998.8 


3004.4 


3010.0 


3015-5 


3021. I 


3026.6 


3032.2 


4 


2.2 


5500 


3037.7 


3043-3 


3048.8 


3054-4 


3060.0 


3065.5 


3071-1 


3076.6 


3082.2 


3087.7 


5 


2.7 


5600 


3093.3 


3098.8 


3104.4 


3110.0 


3115-5 


3121.1 


3126.6 


3132.2 


3137.7 


3143-3 


6 


3-3 


5700 


3148.8 


3154-4 


3160.0 


3165-5 


3171-1 


3176.6 


3182.2 


3187.7 


3193.3 


3198.8 


7 


3.8 


5800 


3204.4 


3210.0 


321S-5 


3221. 1 


3226.6 


3232.2 


3237-7 


3243.3 


3248.8 


3254.4 


8 


4-4 


5900 


3260.0 


3265-5 


3271-1 


3276.6 


3282.2 


3287.7 


3293-3 


3298.8 


3304 -4 


3310.0 


9 


50 


6000 


3315.5 


3321. I 


3326.6 


3332-2 


3337-7 


3343.3 


3348-8 


3354.4 


3360.0 


3365.5 




6100 


337I-I 


3376-6 


3382.2 


3387-7 


3393-3 


3398.8 


3404-4 


3410.0 


3415.5 


3421. I 




6200 


3426.6 


3432-2 


3437.7 


3443 3 


3448-8 


3454-4 


3460.0 


3465.5 


3471-1 


3476.6 




6300 


3482.2 


3487-7 


3493-3 


3498-8 


3504-4 


3510.0 


3515-5 


3521. 1 


3526.6 


3532.2 




6400 


3537-7 


3543-3 


3548.8 


3554-4 


3560.0 


3565-5 


3571-I 


3576.6 


3582.2 


3587.7 




6500 


3593-3 


3598-8 


3604.4 


3610.0 


3615-5 


3621. I 


3626.6 


3632.2 


3637.7 


3643.3 




6600 


3648.8 


3654.4 


3660.0 


3665-5 


3671. I 


3676.6 • 


3682.2 


3687.7 


3693.3 


3698.8 




6700 


3704-4 


3710.0 


3715-5 


3721. I 


3726.6 


3732.2 


.3737.7 


3743.3 


3748.8 


3754.4 




6800 


3760.0 


3765-5 


3771-I 


3776.6 


3782.2 


3787.7 


3793.3 


3798-8 


3804.4 


3810.0 




6900 


3815.5 


3821. I 


3826.6 


3832.2 


3837-7 


3843.3 


3848.8 


3854.4 


3860.0 


3865.5 




7000 


3871. 1 


3876.6 


3882.2 


3887-7 


3893-3 


3898.8 


3904-4 


3910.0 


3915-5 


3921. I 




7100 


3926.6 


3932.2 


3937-7 


3943-3 


3948.8 


3954-0 


3960.0 


3965-5 


3971-I 


3976.6 




7200 


3982.2 


3987-7 


3993-3 


3998-8 


4004 . 4 


4010.0 


4015.5 


4021. I 


4026.6 


403 2 . 2 




7300 


4037.7 


4043-3 


4048.8 


4054-4 


4060 . 


4065.5 


4071. I 


4076.6 


4082.2 


4087.7 




7400 


4093 -3 


4098 . 8 


4104.4 


4110.0 


4115-5 


4121. I 


4126.6 


4132.2 


4137-7 


4143.3 




7SOO 


4148.8 


4154-4 


4160.0 


4165-5 


4171.1 


4176.6 


4182.2 


4187.7 


4193-3 


4198.8 




7600 


4204.4 


4210.0 


4215-5 


4221 . I 


4226.6 


4232.2 


4237-7 


4243-3 


4248.8 


4254.4 




7700 


4260.0 


4265.5 


4271. I 


4276.6 


4282.2 


4287.7 


4293-3 


4298.8 


4304 -4 


4310.0 




7800 


4315-5 


4321-1 


4326.6 


4332.2 


4337-7 


4343-3 


4348.8 


4354-4 


4360.0 


4365.5 




7900 


4371-I 


4376.6 


4382.2 


4387-7 


4393-3 


4398.8 


4404-4 


4410.0 


4415-5 


4421. I 




F.° 





10 


20 


30 


40 


50 


60 


70 


80 


90 









Table 3. 


— Kilogram Calories, Equivalent to British Thermal Units 






British thermal 





I 


2 


3 


4 


5 


6 


7 


8 


9 


British thermal 







.252 


■504 


• 756 


1.008 


1.260 


I. 512 


1.764 


2.016 


2.268 





10 


2.52 


2.772 


3-024 


3.276 


3-528 


3-780 


4.032 


4.284 


4.536 


4-788 


10 


20 


5-04 


5.292 


5-544 


5-796 


6.048 


6.300 


6.552 


6.804 


7-056 


7-308 


20 


30 


7-56 


7.812 


8.064 


8.316 


8.568 


8.820 


9.072 


9-324 


9-576 


9.828 


30 


40 


10.08 


10.332 


10.584 


10.836 


11.088 


ir.340 


11.592 


11.844 


12.096 


12.348 


40 


SO 


12 . 60 


12.852 


13.104 


13.356 


13.608 


13.860 


14. 112 


14.364 


14.616 


14.868 


50 


60 


15.12 


15.372 


15.624 


15-876 


16.128 


16.380 


16.632 


16.884 


17.136 


17-388 


60 


70 


17.64 


17.892 


18. 144 


18.396 


18.648 


18.900 


19.152 


19.404 


19.656 


19 . 908 


70 


80 


20. 16 


20.412 


20. 664 


20.916 


21.168 


21 .420 


21 . 672 


21.924 


22. 176 


22.428 


80 


90 


22.68 


22.932 


23-184 


23436 


23.688 


23.940 


24.192 


24 - 444 


24.696 


24.948 


90 



I British thermal unit = 251.996 therms, or gram calories. 



404 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 4. — Melting-points of Metals and Other Siibstances 

These melting-points were collected by Dr. G. K. Burgess, of the Bureau 
of Standards, Washington, D. C. Those shown in CAPITALS are accepted 
by the Bureau as standard at this time (igii). 

These melting-points were obtained on the purest metals obtainable. 
Lower melting-points may be expected with metals of less purity. 



Table 5. — Lineal Expansion of Solids at Ordinary 
Temperatures 

British Board of Trade — from Clark's Manual of Rules, Tables and Data 



ALUMINUM . . . . 
ANTIMONY.... 

Arsenic 

Bismuth 

CADMIUM 

Calcium 

Chromium 

COBALT 

COPPER 

GOLD 

Iridium (?) 

IRON 

LEAD 

Magnesium 

Manganese 

MERCURY 

Molybdenum (?) . 

NICKEL 

PALLADIUM... 

Phosphorus 

PLATINUM .... 
POTASSIUM . . . 

Rhodium (?) 

Silicon 

SILVER 

SODIUM 

Tantalum (?).... 

TIN 

Titanium (?).... 
TUNGSTEN.. .. 

Uranium 

Vanadium (?)... 
ZINC 



Fahrenheit 
degrees 



Centigrade 
degrees 



(?) Doubtful 



Some Other Melting-points 



GLASS 

GLASS, LEAD FREE 

DELTA METAL 

BARIUM CHLORIDE . . . . 
POTASSIUM CHLORIDE 
SODIUM CHLORIDE .... 
Sulphur 



Fusible metals: 

I tin, 2 lead 

I tin, I lead 

3 tin, 2 lead 

4 tin, 4 lead, i bismuth 
3 tin, 5 lead, 8 bismuth 




658 
630 
800 
270 
321 
80s 

1505 
1490 
1083 
1063 
2300 
1520 
327 
651 
1225 

39 
2 SCO 
1450 
1550 

44 
175s 

62 

1900 

1420 

961 

97 
2900 

232 
i8sc 
3000 
2400 

1750 
419 



1000 
1200 

950 
891 
718 
800 
114 
120 

183. 
151 
13s 
128 
100 



Aluminum (cast) 

Brass (cast) 

Brass (plate) 

Bronze (copper, tin 2j, zinc i) 

Bismuth 

Cement, Portland (mixed) pure 

Concrete (cement, mortar and pebbles) 

Copper 

Glass, English flint 

Glass, thermometer 

Glass, hard 

Gold, pure 

Iron, wrought 

Iron, cast 

Lead 

Mercury (cubical expansion) 

Nickel 

Platinum 

Platinum 8s % ; iridium 15 % 

Porcelain 

Silver, pure 

Steel, cast 

Steel, tempered 

Tin 

Wood, pine 

Zinc 

Zinc 8; tin i 



For I deg. 


For I deg. 


Fahr. 


Cent. 


(Length — i) 


.00001234 


.00002221 


.00000957 


.00001722 


.00001052 


.00001894 


. 00000986 


.00001774 


.00000975 


.0000175s 


. 00000594 


.00001070 


. 00000795 


.00001430 


.00000887 


.00001596 


,00000451 


.00000812 


. 00000499 


.00000897 


.00000397 


.00000714 


. 00000786 


.0000141S 


.00000648 


.00001166 


.00000556 


.00001001 


.00001571 


.00002828 


. 00009984 


.00017971 


. 00000695 


.00001251 


.00000479 


.00000863 


, 00000453 


.0000081S 


.00000200 


.00000360 


.00001079 


.00001943 


.00000636 


.00001144 


.00000689 


.00001240 


.00001163 


.00002094 


. 00000276 


. 00000496 


.00001407 


. 00002532 


. 00001496 


.00002692 



The transmission of heat through metallic tubes, according to the 
report of the Research Committee A. S. M. E. (Trans. A. S. M. E., 
Vol. :iz) may be determined from the following table of constants: 





B.t.u. transmitted 




per sq. ft. per 


Conditions 


hour, per deg. 




Fahr. difference 




of temperature 


From flue gas to water under the conditions existing in 


2 


economizers. 




From flue gas to boiling water under boiler conditions 


10 


(feed heated to steam temperature). 




From flue gas to steam or air under superheater or air-heater 


4 


conditions. 




From hot air to colder water under air-cooler conditions. . . 


4 


From condensing steam to water under surface condensing 


1500 


conditions, no air present. 




When an ordinary amount of air is present .... 


500 to 600 







The heat transfer capacity of metallic tubes is so largely in ex- 
cess of the heat that can be brought in contact with the tube surfaces, 
that the material of the tube has little to do with the amount of 
heat transferred. 



STEAM BOILERS 



The horse-power of boilers is, in a sense, a misnomer, as that term is 
a measure properly applicable only to dynamic effect. But as 
boilers are necessary to drive steam engines, the same measure ap- 
plied to steam engines has come to be universally applied to the 
boiler, and cannot well be discarded. 

The standard adopted by the judges at the Centennial Exhibition, 
of 30 lbs. water per hour, evaporated, at 70 lbs. pressure, from 
100 deg., for each horse-power, is a fair one for both boilers and 
engines, and has been favorably received by engineers and steam 
users generally. The Centennial standard is practically equivalent 
to the evaporation of 345 lbs. of water from and at 212 deg. Fahr. 
per hour. Expressed in this form it has been endorsed by the 
American Society of Mechanical Engineers. 

Square feet of heating surface is no criterion as between different 
styles of boilers — a square foot under some circumstances being 
many times as efficient as in others; but when an average rate of 
evaporation per sq. ft. for any given boiler has been fixed upon 
by experiment, there is no more convenient way of rating the power 
of others of the same style. The following table gives an approxi- 
mate list of sq. ft. of heating surface per h.p. in different styles 
of boilers, and various other data for comparison: 

Heating Surface Per H.p. of Steam Boilers 
From Steam by Permission of the Babcock and Wilcox Co. 



Type of boiler 


Sq. ft. 

of 

heating 

surface 

for I 

h.p. 


Coal per 

sq. ft. 

H.S. per 

hour 


Relative 
economy 


Relative 
rapidity 
of steam- 
ing 


Authority 


Water-tube 


10 to 12 

14 to 18 
8 to 12 
6 to 10 

12 to 16 

15 to 20 


. 1 


I .00 
.91 
■ 79 
.69 
.85 
.80 


1 .00 
.50 

.2S 

.20 

• ss 
.60 


Isherwood 


Tubular 

Flue 




25 

4 
5 

275 
25 


Isherwood 
Prof. Trow- 


Locomotive 

Vertical tubular 





The following specifications for material and rules for the propor- 
tions of riveted joints and other details of and safety valves for steam 
boilers are extracted from the 1915 report of the A. S. M. E. Boiler 
Code Committee. The complete report includes also specifications 
regarding the testing of materials and boilers, manner of marking, 
inspection and rejection, workmanship, permissible variation of 
gages, steel and iron castings, fittings, boilers for low-pressure heating 
and existing installations. The references (P. for page, p. for para- 
graph) refer to the report. These and the quotation points are 
required by the Society. The figure and table numbers are changed 
to suit present surroundings. 

Ultimate Strength of Material Used in Computing Joints 

P. 8, p. 14. The tensile strength used in the computations Sor steel 
plates shall be that stamped on the plates, which is the minimum of 
the stipulated range, or 55,000 lbs. per sq. in. for all steel plates, 
except for special grades having a lower tensile strength. 

p. 15. The resistance to crushing of steel plate shall be taken at 
95,000 lbs. per sq. in. of cross-sectional area. 

p. 16. In computing the ultimate strength of rivets in shear, the 
following values in lbs. per sq. in. of the cross-sectional area of the 
rivet shank shall be used : 

Iron rivets in single shear 38,000 

Iron rivets in double shear 76,000 

Steel rivets in single shear , 44,000 

Steel rivets in double shear 88,000 



The cross-sectional area used in the computations shall be that 
of the rivet shank after driving." 

Minimum Thicknesses of Plates and Tubes 

P. 9, p. 17. The minimum thickness of any boiler plate under 
pressure shall be H in. 

p. 18. The minimum thicknesses of shell plates, and dome plates 
after flanging, shall be as follows : 



36 ins. or under 



!4 in. 



When the Diameter of Shell is 

over 36 ins. over 54 ins. 

to 54 ins. to 72 ins. 

^6 in- %in. 



over 72 ins. 



ti m. 



p. 19. The minimum thicknesses of butt straps shall be as given in 
Table i. 

Table i. — Minimum Thicknesses of Butt Straps 



Thickness of 

shell plates, 

ins. 


Minimum 

thickness of 

butt straps, 

ins. 


Thickness of 

shell plates, 

ins. 


Minimum 

thickness of 

butt straps, 

ins. 


34 


M 


1^2 


Vie 


M2 


K 


He 


Vie 


He 


K 


H 


H 


1^2 


Vi 


y^ 


H 


M 


He 


K 


% 


1^2 


He 


I 


M 


Ke 


H 


iM 


M 


1^2 


Vs 


i^i 


Vs 


H 


He 







p. 20. "The minimum thicknesses of tube sheets for horizontal 
return tubular boilers, shall be as follows: 

When the Diameter of Tube Sheet is 

over 42 ins. over ^4 ins. 

42 ms. or under . ^ . over 72 ms. 

to 54 ms. to 72 ms. 

M in. He in- H in. ?lein. 

P. 10, p. 22. The minimum thicknesses of tubes used in fire-tube 
boilers measured by Birmingham wire gage, for maximum allowable 
working pressures not exceeding 175 lbs. per sq. in., shall be as 
follows: 

Diameters less than 2 3-^ ins No. 13 B.W.G. 

Diameter 2j^ ins. or over, but less than ^yi ins.. . . No. 12 B.W.G. 

Diameter 33<4 ins. or over, but less than 4 ins No. 11 B.W.G- 

Diameter 4 ins. or over, but less than 5 ins No. 11 B.W.G. 

Diameter 5 ins No. 9 B.W.G. 

"For higher maximum allowable working pressures than given 
above the thicknesses shall be increased one gage." 

Specifications for Boiler Plate Steel 

P. II, p. 23. "These specifications cover two grades of steel for 
boilers, namely : Flange and fire-box. 

pp. 24, 25. " The steel shall be made by the open-hearth process 



405 



406 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



and shall conform to the following requirements as to chemical 
composition : 



Carbon 

Manganese. 

Phosphorus 

Sulphur. . . . 
Copper 



Acid.. . . 
Basic. . . 



Flange 



Fire-box 



j Plates ?4 in. thick and under. .0. 12—0.25 per cent. 
\ Plates over % in. thick o. 12-0.30 per cent. 



0.30-0.60 per cent. 
Not over o . 05 per cent. 
Not over o . 04 per cent. 
Not over O.OS per cent. 



o . 30-0 . 50 per cent. 
Not over o . 04 per cent. 
Not over 0.035 P^r cent. 
Not over o . 04 per cent. 
Not over 0.05 per cent. 



P. 12, p. 28. a. "The material shall conform to the following 
requirements as to tensile properties: 



Tensile strength, lbs. per sq. in. . . . 
Yield point, min., lbs. per sq. in. . . 
Elongation in 8 ins., min., per cent. 



Flange 



55,000-65,000 

o . 5 tens. str. 

1,500,000 



Tens. str. 



Fire-box 



55,000-63,000 

0.5 tens. str. 

1,500,000 



Tens. str. 



h. "if desired steel of lower tensile strength than the above may be 
used in an entire boiler, or part thereof, the desired tensile limits to 
be specified, having a range of 10,000 lbs. per sq. in. for flange or 
8000 lbs. per sq. in. for fire-box, the steel to conform in all respects to 
the other corresponding requirements herein specified, and to be 
stamped with the minimum tensile strength of the stipulated range. 

c. "The yield point shall be determined by the drop of the beam of 
the testing machine. 

P. 13, p. 32. "Tension and bend test specimens shall be taken from 
the finished rolled material. They shall be of the full thickness of 






h-^-^r 



Parallel Section 



not less than 9 



^-i—L 



^. 






4— hS-t-H-H- 
* 



-18- 



FiG. I. — Standard test piece of 8 ins. gaged length, piece to be of 
same thickness as plate. 

material as rolled, and shall be machined to the form and dimensions 
shown in Fig. i ; except that bend test specimens may be machined 
with both edges parallel." 

Specifications for Boiler Rivet Steel 

A. Requirements for Rolled Bars 

P. 15, pp. 40, 41. "The steel shall be made by the open-hearth 
process and shall conform to the. following requirements as to chemical 
composition : 

Manganese 0.30-0.50 per cent. 

Phosphorus • not over 0.04 per cent. 

Sulphur not over o. 045 per cent. 

P. 16, p. 44. a. "The bars shall conform to the following require- 
ments as to tensile properties: 

Tensile strength, lbs. per sq. in 45,000-55,000 

Yield point, min., lbs. per sq. in 0.5 tens. str. 

Elongation in 8 ins., min., per cent 1,500,000 

but need not exceed 30 per cent. Tens. str. 

b. The yield point shall be determined by the drop of the beam of 
the testing machine." 



B. Requirements for Rivets 

P. 17, p. 55. The rivets, when tested, shall conform to the re- 
quirements as to tensile properties specified for welded bars above, 
except that the elongation shall be measured on a gaged length not 
less than four times the diameter of the rivet. 

p. 56. The rivet shank shall bend cold through 180 deg. flat 
on itself, as shown in Fig. 2, without cracking on the outside of the 
bent portion and 

P. 18, p. 57. Shall flatten, while hot, to a diameter two and one- 
half times the diameter of the shank, as shown in Fig. 3, without 
cracking at the edges." 





Fig. 2.— The bend 
test for rivets. 



Fig. 3. — The flatten- 
ing test for rivets. 



Specifications for Stay-bolt Steel 

Requirements for Rolled Bars 

P. 19, p. 63. Steel for stay-bolts shall conform to the require- 
ments for Boiler Rivet Steel specified above, except that the tensile 
properties shall be as follows : 

Tensile strength, lbs. per sq. in 50,000-60,000 

Yield point, min., lbs. per sq. in 0.5 tens. str. 

Elongation in 8 ins., min., per cent 1,500,000 

Tens. str. 

P. 19, pp. 64, 65. "The steel shall be made by the open-hearth 
process and shall conform to the following requirements as to chem- 
ical composition: 

_, , I Acid not over 0.06 per cent. 

Phosphorus \ -^ ■ ' 

[ Basic not over 0.04 per cent. 

Sulphur not over 0.05 per cent. 

P. 20, p. 67. a. The material shall conform to the following 
requirements as to tensile properties: 

Tensile strength, lbs. per sq. in 55,000-65,000 

Yield point, min., per sq. in 0.5 tens. str. 

Elongation in 8 ins., min., per cent 1,500,000 

Tens. str. 
Elongation in 2 ins., min., per cent 22 

h. 'The yield point shall be determined by the drop of the beam 
of the testing machine." 

Specifications for Boiler Rivet Iron 

A. Requirements for Rolled Bars 

P. 31, pp. 121, 122. "The iron shall be made wholly from puddled 
iron or knobbled charcoal iron, and shall be free from any admixture 
of iron scrap or steel. The term iron scrap applies only to foreign 
or bought scrap and does not include local mill products free from 
foreign or bought scrap. 

p. 123. a. The iron shall conform to the following requirements 
as to tensile properties: 



I 



STEAM BOILERS 



407 



Tensile strength, lbs. per sq. in 48,000-52,000 

Yield point, min., lbs. per sq. in 0.6 tens. str. 

Elongation in 8 ins., min., per cent 28 

Reduction of area, min., per cent 45 

b. "The yield point shall be determined by the drop of the beam 
of the testing machine. The speed of the cross-head of the machine 
shall not exceed i J^ ins. per minute." 

B. Requirements for Rivets 

P- 33) P- '^ZZ- "when specified, three rivets of each diameter 
shall be taken at random from each lot offered for inspection, and 
if they fail to stand the following tests the lot will be rejected. 

p. 134. a. "The rivet shank shall bend cold through 180 deg. fiat 
on itself, as shown in Fig. 2, without cracking on the outside of the 
bent portion, h. The heads must stand bending back, showing that 
they are firmly Joined, c. When nicked and broken gradually the 
fracture must show a clean, long and fibrous iron." 

Specifications for Stay-bolt Iron 

P. 34, p. 141. a. "The iron shall conform to the following require- 
ments as to tensile properties: 

TensUe strength, lbs. per sq. in 49,000-53,000 

Yield point, min., lbs. per sq. in 0.6 tens. str. 

Elongation in 8 ins., min., per cent 30 

Reduction of area, min., per cent 48 

b. "The yield point shall be determined by the drop of the beam 
of the testing machine. The speed of the cross-head of the machine 
shall not exceed i3^ ins. per minute." 

Specifications for Refined Wrought-iron Bars 

P. 37, p. 151. "Refined wrought-iron bars shall be made wholly 
from puddled iron, and may consist either of new muck-bar iron or a 
mixture of muck-bar iron and scrap, but shall be free from any admix- 
ture of steel. 

P. 152. a. "The iron shall conform to the following minimum 
requirements as to tensile properties. 

Tensile strength, lbs. per sq. in 48,000 

(See Pars. 153 and 154.) 

Yield point, lbs. per sq. in 25,000 

Elongation in 8 ins., per cent 22 

b. The yield point shall be determined by the drop of the beam 
of the testing machine. The speed of the cross-head of the machine 
shall not exceed i3^ ins. per minute." 

Specifications for Lapwelded and Seamless Boiler Tubes 

P. 40, p. 164. Lapwelded tubes shall be made of open-hearth 
steel or knobbled hammered charcoal iron. b. Seamless tubes shall 
be made of open-hearth steel. 

p. 165. a. " The steel shall conform to the following requirements 
as to chemical composition: 

Carbon 0.08-0.18 per cent. 

Manganese 0.30-0.50 per cent. 

Phosphorus not over 0.04 per cent. 

Sulphur not over 0.045 per cent. 

b. "chemical analyses will not be required for charcoal iron 
tubes. 

p. 167. a. A test specimen not less than 4 ins. in length shall 
have a flange turned over at right angles to the body of the tube with- 
out showing cracks or flaws. This flange as measured from the 



outside of the tube shall be % in. wide. b. In making the flange 
test, the flaring tool and die block as shown in Fig. 4, may be used. 

P. 41, p. 168. "a test specimen 3 ins. in length shall stand ham- 
mering flat until the inside walls are brought parallel and separated 
by a distance equal to three times the wall thickness, without showing 
cracks or flaws. In the case of lapwelded tubes, the test shall be 
made with the weld at the point of maximum bend. 

p. i6g. Tubes under 5 ins. in diameter shall stand an internal 
hydrostatic pressure of 1000 lbs. per sq. in. and tubes 5 ins. in diame- 
ter or over, an internal hydrostatic pressure of 800 lbs. per sq. in. 
Lapwelded tubes shall be struck near both ends, while under pressure, 
with a 2-lb. hand hammer or the equivalent." 



Position overusing Flaring Tool 
V Position after using 
^ / yj Flatter 




A 

Fig. 4. 



FLARING TOOL 
• O.S. Diam. of Tube less § 

, S." 

" •' " 8 




k--A --^'^Liners 
DIE BLOCK 
A= OS. Diam. of Tube * ^° 



-Details of flaring tool and die block for making flange 
test of boiler tubes. 



Construction and Maximum Allowable Working Pressures for 
Power Boilers 

P. 43, p. 179. The maximum allowable working pressure is that 
at which a boUer may be operated as determined by employing the 
factors of safety, stresses, and dimensions designated in these Rules. 
No boiler shall be operated at a higher pressure than the maximum 
allowable working pressure except when the safety valve or valves are 
blowing, at which time the maximum allowable working pressure 
shall not be exceeded by more than 6 per cent. Wherever the term 
maximum allowable working pressure is used herein, it refers to 
gage pressure, or the pressure above the atmosphere, in pounds per 
square inch. 

p. 180. ' The maximum allowable working pressure on the shell 
of a boiler or drum shall be determined by the strength of the weakest 
course, computed from the thickness of the plate, the tensile strength, 
stamped thereon, the efiiciency of the longitudinal joint, or of the 
ligament between the tube holes in shell or drum (whichever is the 
least), the inside diameter of the course, and the factor of safety. 

T'.^.X^X.E 
■n^rj r. — = maxlmum allowable working pressure, lbs. per sq. in. (a) 

in which r.5. = ultimate tensile strength stamped on shell plates, 
t = minimum thickness of shell plates in weakest course, 

in. 
£ = efficiency of longitudinal joint or of ligaments be- 
tween tube holes (whichever is the least). 
i? = inside radius of the weakest course of the shell or 
drum, ins. 
F,S. = factor of safety, or the ratio of the ultimate strength 
of the material to the allowable stress. For new 
constructions, F.S. in formula (a) = 5. 

P. 44, p. 181. The efficiency of a joint is the ratio which the 
strength of the joint bears to the strength of the solid plate. In the 
case of a riveted joint this is determined by calculating the breaking 
strength of a unit section of the joint, considering each possible mode 
of failure separately, and dividing the lowest result by the breaking 
strength of the solid plate of a length equal to that of the section con- 
sidered. Detailed methods are given below. 

p. 182. The distance between the center lines of any two adja- 
cent row of rivets, or the "back pitch' measured at right angles to the 



408 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



direction of the joint, shall be at least twice the diameter of the rivets 
and shall also meet the following requirements: 

a. "where a single rivet in the inner row comes midway between 
two rivets in the outer row, the sum of the two diagonal sections of 
the plate between the inner rivet and the two outer rivets shall be at 
least 20 per cent, greater than the section of the plate between the 
two rivets in the outer row. 

b. '"Where two rivets in the inner row come between two rivets in 
the outer row, the sum of the two diagonal sections of the plate 
between the two inner rivets and the two rivets in the outer row shall 
be at least 20 per cent, greater than the difference in the section 
of the plate between the two rivets in outer row and the two rivets 
in the inner row. 

p. 183. "On longitudinal joints, the distance from the centers of 
rivet holes to the edges of the plates, except rivet holes in the ends of 
butt straps, shall be not less than one and one-half times the diameter 
of the rivet holes. 

p. 184. a. "The strength of circumferential joints of boilers, the 
heads of which are not stayed by tubes or through braces shall be at 
least 50 per cent, that of the longitudinal joints of the same structure. 

b. " When 50 per cent, or more of the load which would act on an 
unstayed solid head of the same diameter as the shell, is relieved by 
the effect of tubes or through stays, in consequence of the reduction 
of the area acted on by the pressure and the holding power of the 




Fig. 5. — ^Circumferential joint for thick plates of horizontal 
return tabular boilers. 

tubes and stays, the strength of the circumferential joints in the shell 
'shall be at least 35 per cent, that of the longitudinal joints. 

p". 185. "When shell plates exceed ^{q in. in thickness in hori- 
zontal return tubular boilers, the portion of the plates forming the 
laps of the circumferential joints, where exposed to the fire or products 
of combustion, shall be planed or milled down as shown in Fig. 5, to J^ 
in. in thickness, provided the requirement above is complied with. 

P. 45, p. 186. "The ultimate tensile strength of a longitudinal 
joint which has been properly welded by the forging process, shall 
be taken as 28,500 lbs. per sq. in., with steel plates having a range in 
tensile strength of 47,000 to 55,000 lbs. per sq. in. 

p. 187. "The longitudinal joints of a shell or drum which exceeds 
36 in. in diameter, shall be of butt- and double-strap construction. 

p. 188. "The longitudinal joints of a shell or drum which does not 
exceed 36 ins. in diameter, may be of lap-riveted construction; but 
the maximum allowable working pressure shall not exceed 100 lbs. per 
sq. in. 

p. i8g. "The longitudinal joints of horizontal return tubular 
boilers shall be located above the fire-line of the setting. 

p. 190. "a horizontal return tubular boiler on which a longitu- 
dinal lap joint is permitted shall not have a course over 1 2 ft. in length. 
With butt- and double-strap construction, longitudinal joints of any 
length may be used provided the plates are tested transversely to 
the direction of rolling, which tests shall show the standards pre- 
scribed above under the Specifications of Boiler Plate Steel." 

Efficiency of Joints 

P. 95, p. 410. "The ratio which the strength of a unit length of a 
riveted joint has to the same unit length of the solid plate is known 
as the efficiency of the joint and shall be calculated by the general 
method illustrated in the following examples: 



T.S. = tensile strength stamped on plate, lbs. per sq. in., 
i = thickness of plate, ins., 
6 = thickness of butt strap, ins., 

P =pitch of rivets, ins., on row having greatest pitch, 
d = diameter of rivet after driving, ins. = diameter of rivet 

hole, 
a = cross-sectional area of rivet after driving, sq. ins., 
5 = shearing strength of rivet in single shear, lbs. per sq. in. 

as given in p. 16. 
5 = shearing strength of rivet in double shear, lbs. per sq. in. 

as given in p. 16. 
c = crushing strength of mild steel, lbs. per sq. in. as 

given in, p. 15. 
M = number of rivets in single shear in a unit of length of 

joint, 
iV = number of rivets in double shear in a unit length of 

joint. 

P. 96, p. 411. Lap joint, longihidinal or circumferential, single 
riveted, Fig. 6. 

A = strength of solid plate =PXtXT.S. 
B = strength of plate between rivet holes= (P— d)<X T.S. 
C = shearing strength of one rivet in single shear = wX.yXo. 
Z> = crushing strength of plate in front of one rivet = 
dXtXc. 

Divide B, C or D (whichever is the least) by A, and the quotient 
will be the efficiency of a single-riveted lap joint. 

p. 412. Lap joint, longitudinal or circumferential, double riveted, 
Fig. 7. 

A = strength of solid plate ^PXtXT.S. 
• 5 = strength of plate between rivet h.Qles = {P—d)tXT.S. 
C = shearing strength of two rivets in single shear = 

nXsXa. 
Z) = crushing strength of plate in front of two rivets = 
nXdXtXc. 

Divide B, C or D (whichever is the least) by A, and the quotient 
will be the efficiency of a double-riveted lap joint. 

P. 97, p. 413. "Butt and double strap joint, double riveted. Fig. 8 

4= strength of solid plate =PX<Xr.5. 

B = strength of plate between rivet holes in the outer row = 
iP-d)tXT.S. 

C = shearing strength of two rivets in double shear, 
plus the shearing strength of one rivet in single 
shear ^NXSXa+nXsXa. 

D = strength of plate between rivet holes in the second row, 
plus the shearing strength of one rivet in single 
shear in the outer row=(P— 2^);xr.5.-l-wX^Xo. 

E — strength of plate between rivet holes in the second row, 
plus the crushing strength of butt strap in front of one 
rivet in the outer row= {P—2d)tXT.S.-\-dXbXc. 

P = crushing strength of plate in front of two rivets, 
plus the crushing strength of butt strap in front o£ 
one rivet = iVX^XiXc-|-«X^X6Xc. 

G = crushing strength of plate in front of two rivets, 
plus the shearing strength of one rivet in single 
sheax^NXdXtXc+nXsXa. 

/? = strength of butt straps between rivet holes in the in- 
ner ro-w—{P — 2d)2bXT.S. This method of failure is 
not possible for thicknesses of butt straps required by 
these Rules and the computation need only be made 
for old boilers in which thin butt straps have been used. 
For this reason this method of failure will not be 
considered in other joints. 

Divide B,C, D, E, F,G or H (whichever is the least) by A , and the 



STEAM BOILERS 



409 



I 



quotient will be the efficiency of a butt and double strap joint, 
double riveted. 

P. 98, p. 414. "BhU and double strap joint, triple riveted, Fig. g. 
^ = strength of solid p\a.te=PXtXT.S. 
B = strength of plate between rivet holes in the outer row = 

(P-d)tXT.S. 
C = shearing strength of four rivets in double shear, 

plus the shearing strength of one rivet in single 

shear = NXSXa+nXsXa. 
Z) = strength of plate between rivet holes in the second 

row, plus the shearing strength of one rivet in single 

shear in the outer iow= {P — 2d)tXT.S.+nXsXa. 
E = strength of plate between rivet holes in the second row, 

plus the crushing strength of butt strap in front of 

one rivet in the outer Tow={P — 2d)XT.S.+dXh 

Xc 
i^ = crushing strength of plate in front of four rivets, 

plus the crushing strength of butt strap in front of 

onenvet = NXdXtXc-\-nXdXhXc. 
G = crushing strength of plate in front of four rivets, 



7? = strength of plate between rivet holes in the second row, 
plus the crushing strength of butt strap in front of 
one rivet in the outer roYr = {P — 2d)tXT.S.+dXb 
Xc. 

G = strength of plate between rivet holes in the third row, 
plus the crushing strength of butt strap in front of two 
rivets in the second row and one rivet in the outei 
royi = {P-/^d)tXT.S.+nXdXhXc. 

fl' = crushing strength of plate in front of eight rivets, 
plus the crushing strength of butt strap in front of 
three rivets = iVX(^X;Xc+MXrfX&Xc. 

/ = crushing strength of plate in front of eight rivets, 
plus the shearing strength of two rivets in the second 
row and one rivet in the outer row, in single shear = 
NXdXtXc+nXsXa. 

Divide B, C, D, E, F, G, H or I (whichever is the least) by A, and 
the quotient will be the efficiency of a butt and double strap joint, 
quadruple riveted. 

P. loi, p. 416. ' BiM and double strap joint, quintuple riveted, 
Figs. II and 12. 




Fig., 9. 



Fig. 10. 



Figs. 6 to 10. — Boiler joints. 



plus the shearing strength of one rivet in single 
sheax ^NXdXtXc+nXsXa. 

Divide B, C, D, E, F or G (whichever is the least) by A, and the 
quotient will be the efficiency of a butt and double strap joint, triple 
riveted. 

P. 99, p. 415. Butt and double strap joint, quadruple riveted. 
Fig. 10. 

A = strength of solid plate =PX/xr.>S. 

5 = strength of plate between rivet holes in the outer row— 

{P~d)tXT.S. 
C = shearing strength of eight rivets in double shear, 
plus the shearing strength of three rivets in single 
s\i&a.r = NXSXa-\-nXsXa. 
Z) = strength of plate between rivet holes in the second row, 
plus the shearing strength of one rivet in single 
shear in the outer ro^w = {P — 2d)tXT .S . + 's.XsXa. 
£ = strength of plate between rivet holes in the third row, 
plus the shearing strength of two rivets in the second 
row in single shear and one rivet in single shear in 
the outer row = (P-4(i)<Xr.5.+MX5Xa. 



4=strength of solid plate=PX/XT'-5. 

5= strength of plate between rivet holfs in the outer row = 
{P-d)tXT.S. 

C = shearing strength of 16 rivets in double shear, plus the 
shearing strength of seven rivets in single shear = 
NXSXa+nXsXa. 

D = strength of plate between rivet holes in the second row, 
plus the shearing strength of one rivet in single shear in 
the outer rovf={P — 2d)tXT.S. + iXsXa. 

£ = strength of plate between rivet holes in the third row, 
plus the shearing strength of two rivets in the second 
row in single shear and one rivet in single shear in the 
onttr ro^ = {P—Ad)tXT.S.~\riXsXa. 

F = strength of plate between rivet holes in the fourth row, 
plus the shearing strength of four rivets in the third 
row, two rivets in the second row and one rivet in the 
outer row in single shear = (P — 8i)/xr. 5. +«X.?Xa. 

G = strength of plate between rivet holes in the second row, 
plus the crushing strength of butt strap in front of one 
rivet in the outer rovf = {P — 2d)tXT.S.-\-dXbXc. 

fl' = strength of plate between rivet holes in the third row. 



410 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



plus the crushing strength of butt strap in front of two 
rivets in the second row and one rivet in the outer row 
= iP-4d)tXT.S.+3XdXbXc. 
7 = strength of plate between rivet holes in the fourth row, 
plus the crushing strength of butt strap in front of four 



P. 103, p. 417. "Figs. 13 and 14 illustrate other joints that may- 
be used. The butt and double strap joint with straps of equal width 
shown in Fig. 13 may be so designed that it will have an efficiency of 
from 82 to 84 per cent, and the saw-tooth joint shown in Fig. 14 so 
that it will have an efficiency of from 92 to 94 per cent." 




Fig. II. — Butt and double strap joint, 
quintuple riveted. 




Fig. 12. — Butt and double strap joint, 
quintuple riveted. 




Fig. 13.- — Butt and double strap joint with straps at 
equal width. 




Fig. 14. — Butt and double strap joint of the saw tooth 
type. 



rivets in the third row, two riv ets in the second row and 
one rivet in the outer iow = {P — 8d)tXT.S.-\-nXdX 
bXc. 
7 = crushing strength of plate in front of 16 rivets, plus the 
crushing strength of butt strap in front of seven rivets 
= NXdXtXc-\-nXdXhXc. 




'f>U-5li'->U-5k"->^-Sk''->U5k">^-Sk'-^ 




LoncjHudinal Line ~ ^ 

Fig. 15. 



(fe^'^-^^M^ 



■sr- 



'.5lt'-^ 



©-^-€>4>-<;>(^ 



LongifucHnal Lint - 

Fig. 16. 







Loncjifudinal Line ^ 

Fig. 17. 




Lonc^itudihal Line 
Fig. 18. 



Figs. 15 to 18. — Examples of tube spacing. 



X = crushing strength of plate in front of 16 rivets, plus the 
shearing strength of four rivets in the third row, two 
rivets in the second row and one rivet in the outer row 
in single s\it?Lr = NXdXtXc-\-nXsXa. 

Divide B, C, D, E, F, G, H, I,J or K (whichever is the least) by A , 
and the quotient will be the efficiency of a butt and double strap 
joint, quintuple riveted. 



Ligaments 

P. 46, p. 192. ''When a shell or drum is drilled for tubes in a line 
parallel to the axis of the shell or drum, the efficiency of the ligament 
between the tube holes shall be determined as follows: 

a. When the pitch of the tube holes on every row is equal (Fig. 
15), the formula is: 
p—d 

P 



■- efficiency of ligament, 



(6) 



I 



STEAM BOILERS 



411 



in which /» = pitch of tube holes, ins., 

d = diameter of tube holes, ins., 

b. "When the pitch of tube holes on any one row is unequal (as 
in Figs. i6 or 17), the formula is: 
p —nd 



M 



I 



- = eiEciency of ligament, 
r 
in which p= unit length of ligament, ins., 

n= number of tube holes in length, p, 
d= diameter of tube holes, ins. 

P. 47, p. 193. "When a shell or drum is drilled for tube holes in a 
line diagonal with the axis of the shell or drum as shown in Fig. 18, 
the efficiency of the ligament between the tube holes shall be deter- 
mined by the following methods and the lowest value used. 



- = efficiency of ligament 

- = efficiency of ligament 



id) 
(c) 



in which 



/)i=diagonal pitch of tube holes, ins., 
d = diameter of tube holes, ins., 
/»= longitudinal pitch of tube holes or distance 

between centers of tubes in a longitudinal 

row, ins. 

pi . 
The constant . 95 in formula {d) applies provided -7 is i-S or over. 

Dished Heads 

P." 49, p. 195. "The thickness required in an unstayed dished 
head with the pressure on the concave side, when it is a segment of a 
sphere, shall be calculated by the following formula: 

S-SXPXL 



i=- 



2XT.S. 



-+M 



(/) 



in which i = thickness of plate, ins., 



P = maximum allowable working pressure, lbs. per sq. in., 
r.5.= tensile strength, lbs. per sq. in., 

L = radius to which the head is dished, ins. 

"where the radius is less than 80 per cent, of the diameter of the 
shell or drum to which the head is attached the thickness shall be at 
least that found by the formula by making L equal to 80 per cent, of 
the diameter of the shell or drum. 

Dished heads with the pressure on the convex side shall have a 
maximum allowable working pressure equal to 60 per cent, of that 
for heads of the same dimensions with the pressure on the concave 
side. 

When a dished head has a manhole opening, the thickness as 
found by these rules shall be increased by not less than 3^ in. 

p. 196. 'When dished heads are of a less thickness than called for 
by formula (/), they shall be stayed as flat surfaces, no allowance 
being made in such staying for the holding power due to the spherical 
form. 

p. 197. The corner radius of an unstayed dished head meas- 
ured on the concave side of the head shall not be less than i ]4 ins. or 
more than 4 ins. and within these limits shall be not less than 3 per 
cent, of £ in formula (/) . 

p. 198. A manhole opening in a dished head shall be flanged to a 
depth of not less than three times the thickness of the head measured 
from the outside." 

Braced and Stayed Surfaces 

p. 199. The maximum allowable working pressures for various 
thicknesses of braced and stayed flat plates and those which by these 
rules require staying as flat surfaces with braces or staybolts of uni- 
form diameter symmetrically spaced, shall be calculated by the 
formula : 



in which P^ maximum allowable working pressure, lbs. per sq. in., 
i = thickness of plate in sixteenths of an inch, 
p = maximum pitch measured between straight lines passing 

through the centers of the staybolts in the different 

rows, which lines may be horizontal, vertical or inclined, 

ins., 
C= 112 for stays screwed through plates not over I/Iq m. 

thick with ends riveted over, 
C=i2o for stays screwed through plates over J-^e in. thick 

with ends riveted over, 
C=i35 for stays screwed through plates and fitted with 

single nuts outside of plate, 
C=i75 for stays fitted with inside and outside nuts and 

outside washers where the diameter of washers is not 

less than ./^p and thickness not less than t. 

If flat plates not less than % in. thick are strengthened with doubling 
plates securely riveted thereto and having a thickness of not less than 
%/, nor more than t, then the value of t in the formula shall be three- 
fourths of the combined thickness of the plates and the values of C 
given above may also be increased 15 per cent. 

P. 51, p. 202. "The ends of stays fitted with nuts shall not be 
exposed to the direct radiant heat of the fire. 

p. 203. "The maximum spacing between centers of rivets attach- 
ing the crowfeet of diagonal braces to the braced surface, shall be 
determined by formula (g) using 135 for value of C. 

The maximum spacing between the inner surface of the shell 
and lines parallel to the surface of the shell passing through the 
centers of the rivets attaching the crowfeet of diagonal braces to the 
head, shall be determined by formula (g) using 160 for the value 
of C. 

Table 2. — Maximum Allowable Pitch, in Inches, of Screwed 
Stay-bolts, Ends Riveted Over 





Thickness of plate, in. 


Pressure, 
lbs. per sq. in. 


Me 


% 1 


Me i Vi 1 Me 


% 


iHe 




Maximum pitch of stay-bolts 


, in. 




100 


SM 


6?i 


734 










no 


5 


6 


7 


83^^ 








120 


4% 


SM 


m 


8 








125 


4?i 


5=4 


(>H 


754 








130 


AH 


5H 


bVi 


7H 








140 


4H 


5?i 


6U 


7?^ 


&H 






ISO 


4H 


SH 


6 


iH 


8 






160 


4H 


5 


m 


6H 


7?4 






170 


4 


4% 


s=4 


63A 


7H 


8?^ 




180 




AYi 


5W 


6H 


7?^ 


8^^ 




190 




A% 


5% 


6H 


7\i 


7% 




200 




A\'i 


sK 


6H 


7 


7?4 


8H 


22s 




4K 


4% 


S% 


6H 


7M 


8 


2S0 




4 


m 


sH 


6M 


574 


7% 


300 






4H 


5 


5H 


6H 


7 











P = CX' 



is) 



p. 204. Formula (g) was used in computing Table 2. Where 
values for screwed stays with ends riveted over are required for 
conditions not given in Table 2, they may be computed from the 
formula and used, provided the pitch does not e.xceed 8Ji in." 

P. 104, p. 418. The allowable loads based on the net cross-sec- 
tional areas of stay-bolts with V-threads, are computed from the 
following formulas. The use of Whitworth threads with other pitches 
is permissible. 

The formula for the diameter of a stay-bolt at the bottom of a 
V-threadis: 

D-{PXi.732)=d 

in which Z) = diameter of stay-bolt over the threads, ins., 
P = pitch of threads, ins., 

c? = diameter of stay-bolt at bottom of threads, ins., 
1.732 = a constant. 



412 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



When U. S. threads are used, the formula becomes: 

Z)-(PXi.732X.75)='^ 
"Tables 3 and 4 give the allowable loads on net cross-sectional 
areas for stay-bolts with V-threads, having 12 and 10 threads per 
inch. 

Table 3. — Allowable Loads on Stay-bolts with V-theeads, 
12 Threads per Inch 



Table 6. — Net Areas of Segments of Heads 



Height 

from tubes 

to shell 



Diameter of boiler, ins. 



54 60 
ins. ins. 



66 
ins. 



72 
ins. 



78 
ins. 



84 



Outside diameter 


Diameter at 


Net cross-sectional 


Allowable load at 


of 


bottom of 


area (at bottom of 


7500 lb. stress, 


stay-bolts, in. 


thread, in. 


thread), sq. in. 


per sq. m. 


Vi 


.7500 


.6057 


.288 


2,160 


m^ 


.S12S 


.6682 


-351 


2,632 


H 


.8750 


.7307 


-419 


3,142 


■Me 


■ 9375 


.7932 


• 494 


3,705 


I 


1 . 0000 


.8557 


• 575 


4,312 


iMo 


1.062s 


.9 182 


.662 


4.965 


m 


I. 1250 


.9807 


■ 755 


5,662 


I?i6 


I. 187s 


1.0432 


.855 


6,412 


iM 


1.2500 


I. 1057 


.960 


7,200 


iMe 


I-3I2S 


I. 1682 


1.072 


8,040 


m 


1-3750 


1.2307 


I. 190 


8,925 


iMe 


1.4375 


1.2932 


1-3 13 


9,849 


iH 


I. 5000 


1-3557 


1.444 


10,830 



Table 4. — Allowable Loads on Sxay-bolts with V-threads, 
10 Threads per Inch 



Outside diameter 


Diameter at 


Net cross-sectional 


Allowable load at 


of 


bottom of 


area (at bottom of 


7500 lb. stress 


stay-bolts, in. 


thread, in. 


thread), sq. in. 


per sq. m. 


iW 


1.2500 


1.0768 


0.911 


6,832 


I«6 


1. 3125 


I. 1393 


1. 019 


7,642 


m 


1-3750 


1. 20 18 


I- 134 


8,50s 


iMe 


1.4375 


1.2643 


I. 255 


9,412 


iH 


1.5000 


1.3268 


1.382 


10,365 


I?i6 


1-5625 


13893 


1.515 


11,362 


iH 


1.6250 


1.45 18 


1.655 


12,412 



P. 105, p. 419. "Table 5 shows the allowable loads on net cross- 
sectional area of round stays or braces. 

Table 5. — Allowable Loads on Round Braces or Stay Rods 



Minimum 
diameter of 


Net cross- 
sectional 


Allowable stress, in lbs. per sq. in., net 
cross-sectional area 


6000 


8500 




circular 
ins 


stay. 


area of stay, 
in sq. ins. 


9500 


Allowable load, in lbs. on net cross- 








sectional area 




I 


I - 0000 


.7854 


4.712 


6,676 


7,462 


iMe 


1.062s 


.8866 


5,320 


7.536 


8,423 


m 


I. 1250 


• 9940 


5.964 


8.449 


9.443 


iMe 


I. 1875 


I. 1075 


6,645 


9.414 


10,521 


iM 


1.2500 


1.2272 


7,363 


10,431 


11,658 


iMe 


1. 3125 


1-3530 


8,n8 


11.501 


12,854 


m 


1-3750 


1.4849 


8,909 


12,622 


14. 107 


ilU 


1-4375 


1.6230 


9,738 


13,796 


15,419 


iH 


I. 5000 


I. 7671 


10,603 


15.020 


16,787 


I?16 


1.5625 


I-9175 


11,505 


16,298 


18,216 


i^A 


1.6250 


2.0739 


12,443 


17,628 


19,702 


I' Me 


1.6875 


2.2365 


13,419 


19,0 10 


21,247 


ifi 


1.7500 


2.4053 


14.432 


20,445 


22,852 


iHU 


1. 8125 


2.5802 


IS.481 


21,932 


24,512 


m 


1.8750 


2.7612 


16,567 


23,470 


26,231 


I'Me 


1.9375 


2 . 9483 


17,690 


25,061 


28,009 


2 


2.0000 


3.1416 


18,850 


26,704 


29,84s 


2\.i 


2. 1250 


3 . 5466 


21,280 


30. 147 


33,693 


2H 


2.2500 


3.9761 


23,857 


33.797 


37,773 


2H 


2.3750 


4-430 1 


26,580 


37,656 


42,086 


2]ri 


2.5000 


4-9087 


29,452 


41.724 


46,632 


2^A 


2.6250 


5 - 4 1 19 


32,471 


46,001 


5 1,4 13 


2?4 


2.7500 


5-9396 


35,638 


50,487 


56,426 


27^ 


2.8750 


6-4918 


38,951 


55.181 


61,673 


3 


3 . 0000 


7.0686 


42,412 


60,083 


67,152 



Area to be stayed, sq. ins. 



8 ms. 
8J.-^ ins. 

9 ins. 
9^ ins. 

10 ins. 

lo!.^ ins. 

1 1 ins. 
iiH ins. 

12 ins. 
12J4 ins. 

13 ins. 
13^2 ins. 

14 ins. 
14H ins. 

15 ins. 

isli ins. 

16 ins. 
16H ins. 

17 ins. 
17H ins. 

18 ins. 
i8J^ ins. 

19 ins. 
19H ins. 

20 ins. 

20^ ins. 

21 ins. 
2i>4 ins. 

22 ins. 
22j.^ ins. 

23 ins. 
23M ins. 

24 ins. 
24H ins. 

25 ins. 

2SH ins. 

26 ins. 
26H ins. 

27 ins. 
27 H ins. 

28 ins. 
28H ins. 

29 ins. 
29!^^ ins. 

30 ins. 

30^-5 ins. 

31 ins. 
31M ins. 

32 ins. 
32H ins. 

33 ins. 
33!^ ins. 

34 ins. 
34H ins. 

35 ins. 

35 H ins. 

36 ins. 
36J4 ins. 

37 ins. 



37 
46 
56 
66 

77 

89 
100 
112 
125 
138 

151 
164 
178 
192 
206 

220 
235 
249 
264 



40 
51 
62 
70 
85 

98 
III 
124 
139 
153 

168 
183 
199 
215 
231 

247 
263 
281 
297 
314 

331 
349 
366 
384 
401 



43 
55 
67 
80 
93 

107 
121 
137 
151 
167 

183 
200 
217 
235 
252 

271 
289 
308 
326 
345 

365 
384 
404 
424 
444 

464 
48s 
505 
526 



47 
59 
72 
86 
99 

114 
130 
146 
163 
180 

197 
216 
234 
254 
273 

291 
312 
332 
353 

374 

396 

417 
439 
461 
483 

505 
528 
551 
574 
597 



51 
63 
76 
91 
106 

123 
138 
156 
174 
193 

211 
230 
250 
271 
291 

312 
334 
357 
378 
400 

424 
448 
470 
496 
5 19 

543 
568 
594 
618 
643 



620 668 
642 69s 



667 
689 

714 

737 
761 



719 
745 
771 

798 
824 
850 
877 
904 

930 



53 
66 
82 
96 
112 

131 
147 
165 
184 
204 

224 
246 
266 
287 
309 

332 
355 
380 
402 
426 

450 
476 
500 
528 
552 

578 
604 
632 
658 
687 

713 
740 
768 
797 
825 

85s 
882 
909 
939 
968 

997 
1028 
1056 
1084 
U15 



55 
70 
86 

lOI 

117 

13s 
155 
173 
194 
216 

23s 
258 
280 
303 
326 

350 

374 
399 
425 
449 

476 
501 
529 
558 
583 

613 
640 
669 
697 
726 

754 
784 
814 
843 
875 

907 
936 
968 
998 
1030 

1060 
1092 
1123 
1155 
1187 

1218 
1252 
1286 
1317 



S8 

74 

90 

105 

123 

142 
161 
181 
203 
224 

247 
270 
294 
318 
343 

368 

394 
420 

447 
471 

Soo 
526 
555 
584 
613 

643 
673 
703 
734 
76s 

796 
827 
859 
892 
922 

956 
987 
1024 
10S3 
1089 

II20 
I157 
I187 
I22I 
I2SS 

1290 
1324 
1359 
1394 
1430 

1465 
1500 
1536 



60 


63 


76 


80 


92 


95 


III 


116 


129 


132 


147 


153 


169 


174 


189 


196 


213 


219 


234 


243 


256 


267 


282 


293 


305 


319 


333 


345 


357 


372 


382 


400 


411 


423 


436 


457 


467 


486 


494 


516 


520 


543 


552 


577 


580 


60^ 


613 


641 


642 


667 


675 


706 


705 


733 


739 


766 


769 


800 


800 


835 


830 


869 


866 


904 


897 


939 


934 


975 


966 


1010 


1003 


1047 


1035 


1083 


1073 


1120 


1106 


IIS7 


1145 


I195 


1177 


1232 


I2II 


1270 


1248 


1305 


1284 


1347 


I32I 


1382 


1358 


1424 


1394 


1459 


1433 


1496 


1467 


1538 


ISO8 


1575 


1542 


1617 


1578 


1655 


1617 


169s 


1654 


1735 


1692 


1775 




1810 




1857 





P. 53, p. 214. The area of a segment of a head to be stayed 
shall be the area enclosed by lines drawn 3 ins. from the shell and 2 
ins. from the tubes, as shown in Figs. 19 and 20. 

P. 105, p. 420. Table 6 gives the net areas of segments of heads 
for use in computing stays. 



STEAM BOILERS 



413 



Table 7. — Maximum Allowable Stresses for Stays and 
Stay-bolts 







Stresses, 


bs. per sq. in. 


Description of stays 


For lengths between 


For lengths between 




supports not exceed- 


supports exceeding 




ing 


120 diameters 


120 diameters 


Unwelded stays less than 20 




7500 


• . . • 


diam. long screwed through 








plates with ends riveted over . 








Unwelded stays and unwelded 




9500 


8500 


portions of welded stays, ex- 








cept as specified in line . 








Welded portions of stays. 




6000 


6000 




Fig. 19. 



OOOOOQiX^OOOOO 



Fig. 20. 

Figs. 19 and 20. — Areas to be braced in steam boiler heads. 

P. 54, p. 219. When stay rods are screwed through the sheets 
and riveted over, they shall be supported at intervals not exceeding 
6 ft. In boilers without manholes, stay rods over 6 ft. in length may 
be screwed through the sheets and fitted with nuts and washers on 
the outside. 



H-- 



i 



--H 



V.\V\ ' -.^ ' sVsVV~.^ VV\VVsVVV^Vs^<^'^'^^'^<^^'-V|j ^ 




Fig. 21. — Measurements for determining stresses in diagonal 
stays. 

p . 220. ' The maximum allowable stress per square inch net cross- 
sectional area of stays and stay-bolts shall be as given in Table 7. 
The length of the stays between supports shall be measured from the 
inner faces of the stayed plates. The stresses are based on tension 
only. 

p. 221. "To find the stresses in diagonal and gusset stays: 
multiply the area of a direct stay required to support the surface by 



the slant or diagonal length of the stay; divide this product by the 
length of a line drawn at right angles to surface supported to center 
of palm of diagonal stay. The quotient will be the required area of 
the diagonal stay. 

. aXL 



I 



(h) 



in which ^ = sectional area of diagonal stay, sq. ins. (see Table 7), 
= sectional area of direct stay, sq. ins., 
Z/ = length of diagonal stay, as indicated in Fig. 21, ins., 
^= length of line drawn at right angles to boiler head or 
surface supported to center of palm of diagonal stay, 
as indicated in Fig. 21, ins. 

P. 55, p. 222. For staying segments of tube sheets such as in 
horizontal return tubular boilers, where L is not more than 1.15 
times I for any brace, the stays may be calculated as direct stays, 
allowing 90 per cent, of the stress given in Table 7. 

p. 223. "The sectional area of pins to resist double shear and 
bending when secured in crowfoot, sling, and similar stays shall be 
at least equal to three-fourths of the required cross-sectional area of 
the brace. The combined cross-section of the eye at the sides of the 
pin shall be at least 25 per cent, greater than the required cross-sec- 
tional area of the brace. 

The cross-sectional area of the rivets attaching a brace to the 
shell shall be not less than one and one-quarter times the required 
sectional area of the brace. Each branch of the crowfoot shall be 
designed to carry two-thirds of the total load on the brace at the 
allowed stress. The net sectional areas through the sides of the 
crowfeet, tee irons or similar fastenings at the rivet holes shall be at 
least equal to the required rivet section. All rivet holes shall be 
drilled and sharp edges removed, and the pins shall be made a neat 
fit. 

Table 8. — Sizes or Angles Required for Staying Segments of 

Heads 

With the short legs of the angles attached to the head of the boiler 





30" Boiler 


34" Boiler 


36" Boiler 




Height of 
segment, 
dimension 




< S 




"m X 
< S 


C X 


B X 
< I, 


bo w 

c X 


S'x 


.2 3J 

bo ^ 

<^ 

<> 


Dimen- 
sion 
A in 


B in Pig. 
22, ins. 


i 


w 




in 


1 


.s 

Si m 
w 


c 

Si m 

M 


C 
Ji „- 


.5 


Fig. 22, 
ins. 




Vie 
Vie 


Vis 
H 
Vie 
Vie 


Me 
Vie 














6H 


1 1 


Vi6 


Vie 
Ve 


Me 
yie 
Me 
% 








7 
8 

8J.'2 


12 
13 
14 
15 
16 


Vie 
Vie 
% 
% 


Me 
■'Ae 


Ve 






9 












9H 





















P. 56, p. 224. "Gusset stays when constructed of triangular right- 
angled web plates secured to single or double angle bars along the two 
sides at right angles shall have a cross-sectional area (in a plane at 
right angles to the longest side and passing through the intersection 
of the two shorter sides) not less than 10 per cent, greater than would 
be required for a diagonal stay to support the same surface, figured 
by formula (h) assuming the diagonal stay is at the same angle as 
the longest side of the gusset plate. 

p. 225. "When the shell of a boiler does not exceed 36 ins. in 
diameter and is designed for a maximum allowable working pressure 
not exceeding 100 lbs. per sq. in., the segment of heads above the 
tubes may be stayed by steel angles as specified in Table 8 and Fig. 
22, except that angles of equal thickness and greater depth of out- 
standing leg, or of greater thickness and the same depth of outstand- 
ing leg, may be substituted for those specified. The legs attached 



414 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



to the heads may vary in depth J^ in. above or below the dimen- in which A = area of that portion of the tube plate containing the 



sions specified in Table 8. 

P. 57, p. 226. '"When this form of bracing is to be placed on a 
boiler, the diameter of which is intermediate to or below the diame- 
ters given in Table 8, the tabular values for the next higher diameter 
shall govern. Rivets of the same diameter as used in the longitudi- 
nal seams of the boiler shall be used to attach the angles to the head 
and to connect the outstanding legs. 

p. 227. "The rivets attaching angles to heads shall be spaced not 
over 4 ins. apart. The centers of the end rivets shall be not over 3 ins. 
from the ends of the angle. The rivets through the outstanding legs 
shall be spaced not over 8 ins. apart; the centers of the end rivets 
shall be not more than 4 ins. from the ends of the angles. The' ends 
of the angles shall be considered those of the outstanding legs and the 
lengths shall be such that their ends overlap a circle 3 ins. inside the 
inner surface of the shell as shown in Fig. 22. 

p. 2 28. "The distance from the center of the angles to the shell of 
the boiler, marked A in Fig. 22, shall not exceed the values given in 
Table 8, but in no case shall the leg attached to the head on the lower 
angle come closer than 2 ins. to the top of the tubes. 






/. 



I Not over 3 



'•Not, over 4" ■-■ 



Not /ess 
^than2"<l 



qogiooo 
o b d i o 00' 
ooolooo 

OOiOO 

Fig. 22. — Staying of head with steel angles in tubular boiler. 

P. 58, p. 229. When segments are beyond the range specified in 
Table 8, the heads shall be braced or stayed in accordance with the 
requirements in these Rules. 

p. 230. 'Crown bars and girder stays for tops of combustion 
chambers and back connections, or wherever used, shall be propor- 
tioned to conform to the following formula : 

CXd'^XT 



Maximum allowable working pressure = : 



H) 



iW-P)XDXW 
in which FF = extreme distance between supports, ins., 

P= pitch of supporting bolts, ins., 
D = distance between girders from center to center, ins., 

d= depth of girder, ins., 

T= thickness of girder, ins., 

C = 7ooo when the girder is fitted with one supporting bolt, 

C= 10,000 when the girder is fitted with two or three sup- 
porting bolts, 

C= 1 1,000 when the girder is fitted with four or five sup- 
porting bolts, 

C= 11,500 when the girder is fitted with six or seven sup- 
porting bolts, 

C= 12,000 when the girder is fitted with eight or more sup- 
porting bolts. 

p. 232. When stay tubes are used in multitubular boilers to give 
support to the tube plates, the sectional area of such stay tubes may 
be determined as follows: 

{A-a)P 
Total section of stay tubes, sq. ms. = Tf. (t) 



tubes, sq. ins., 
a = aggregate area of holes in the tube plate, sq. ins., 
P = maximum allowable working pressure, lbs. per sq. in., 
r = working tensile stress allowed in the tubes, not to exceed 
7000 lbs. per sq. in. 

P. 59, p. 233. The pitch of stay tubes shall conform to formula 
{g), using the values of C as given in Table 9. 



Table 9. — Values of C for Determining Pitch 


OF Stay Tubes 


Pitch of stay tubes in the bounding 
rows 


When tubes 

have no nuts 

outside of plates 


When tubes 

are fitted with 

nuts outside 

of plates 


Where there are two plain tubes be- 
tween each stay tube. 

Where there is one plain tube between 
each stay tube. 

Where every tube in the bounding rows 
is a stay tube and each alternate tube 
has a nut. 


120 

140 


130 
ISO 
170 







When the ends of tubes are not shielded from the action of flame 
or radiant heat, the values of C shall be reduced 20 per cent. The 
tubes shall project about J4 in. at each end and be slightly flared. 
Stay tubes when threaded shall not be less than ^e in. thick at 
bottom of thread; nuts on stay tubes are not advised. For a nest 
of tubes C shall be taken as 140 and S as the mean pitch of stay tubes. 
For spaces between nests of tubes .S shall be taken as the horizontal 
distance from center to center of the bounding rows of tubes and C 
as given in Table 9." 

Tube Sheets of Combustion Chambers 

p. 234. The maximum allowable working pressure on a tube 
sheet of a combustion chamber, where the crown sheet is not sus- 
pended from the shell of the boiler, shall be determined by the 
following formula: 

(D-d)T X27,ooo , 

^ WXD ^''^• 

in which P = maximum allowable working pressure, lbs. per sq. in., 
Z)= least horizontal distance between tube centers, ins., 
<^ = inside diameter of tubes, ins., 
7" = thickness of tube plate, ins., 

W = distance from tube sheet to opposite combustion cham- 
ber sheet, ins. 



Circular Furnaces and Flues 

P. 61, p. 239. The maximum allowable working pressure for 
unstayed, riveted, seamless or lapwelded furnaces, where the length 
does not exceed six times the diameter and where the thickness is at 
least 5^f 6 in. shall be determined by one or the other of the following 
formulas: 

a. Where the length does not exceed 120 times the thickness of 
the plate 

Si-S 



P = - 



D 



(i8.7sXr)- (1.03X1.) 



(0 



b. Where the length exceeds 120 times the thickness of the plate 

4250 xr2 



P = - 



LXD 



{m) 



in which P = maximum allowable working pressure, lbs. per sq. in., 
£) = outside diameter of furnace, ins., 
!. = length of furnace, ins., 
T = thickness of furnace walls, in sixteenths of an inch. 

"Where the furnaces have riveted longitudinal joints no deduction 



STEAM BOILERS 



415 



need be made for the joint provided the efficiency of the joint is 
greater thanPX-D divided by i25oXr. 

p. 240. "a plain cylindrical furnace exceeding 38 ins. in diameter 
shall be stayed in accordance with the rules governing flat surfaces. 

p. 241. "The maximum allowable working pressure for seamless 
or welded flues more than 5 ins. in diameter and up to and including 
18 ins. in diameter shall be determined by one or the other of the 
following formulas: 

a. "Where the thickness of the wall is less than .023 times the 
diameter 

10,000,000 X T 2 



Z)3 



(«) 



h. "Where the thickness of the wall is greater than .023 times the 
diameter 



17,300 xr 

P= 5 27S 



(0) 



in which P = maximum allowable working pressure, lbs. per sq. in., 
Z) = outside diameter of flue, ins., 
?■ = thickness of wall of flue, ins. 

c. "The above formulas may be applied to riveted flues of the 
sizes specified provided the sections are not over 3 ft. in length and 
provided the efi&ciency of the joint is greater than PXD divided by 
2o,oooXr. 

Adamson Type 

P. 62, p. 242. When plain horizontal flues are made in sections 
not less than 18 ins. in length, and not less than ^{q in. thick: 

a. They shall be flanged with a radius measured on the fire side, 
of not less than three times the thickness of the plate, and the flat 
portion of the flange outside of the radius shall be at least three times 
the diameter of the rivet holes. 

b. The distance from the edge of the rivet holes to the edge of the 
flange shall be not less than the diameter of the rivet hole, and the 
diameter of the rivets before driving shall be at least }4 in. larger than 
the thickness of the plate. 

c. 'The depth of the Adamson ring between the flanges shall be not 
less than three times the diameter of the rivet holes, and the ring 
shall be substantially riveted to the flanges. The fire edge of the 
ring shall terminate at or about the point of tangency to the curve of 
the flange, and the thickness of the ring shall be not less than }4 in. 

"The maximum allowable working pressure shall be determined by 
the following formula : 



^ = ^ I (i8.7SXr)-(i.03Xi) 



(P) 



in which P = maximum allowable working pressure, lbs. per sq. in., 
D= outside diameter of furnace, ins., 
i = length of furnace, ins., 
T = thickness of plate, in sixteenths of an inch. 

P. 63, p. 243. The maximum allowable working pressure on 
corrugated furnaces, such as the Leeds suspension bulb, Morison, 
Fox, Purves, or Brown, having plain portions at the ends not exceed- 
ing g ins. in length (except flues especially provided for) when new 
and practically circular, shall be computed as follows: 



P = 



CXT 
D 



(?) 



in which P = maximum allowable working pressure, lbs. per sq. in., 

r = thickness, ins. — not less than ^e in. for Leeds, Morison, 

Fox and Brown, and not less than %6 in. for Purves 

and other furnaces corrugated by sections not over 18 

ins. long, 

i) = mean diameter, ins., 

C= 17,300, a constant for Leeds furnaces, when corrugations 



are not more than 8 ins. from center to center and not 
less than 2j^ ins. deep, 

C= 15,600, a constant for Morison furnaces, when corruga- 
tions are not less than 8 ins. from center to center and 
the radius of the outer corrugations is not more than 
one-half that of the suspension curve, 

C = 14,000, a constant for Fox furnaces, when corrugations 
are not more than 8 ins. from center to center and not 
less than i3^ ins. deep, 

C = 14,000, a constant for Purves furnaces when rib pro- 
jections are not more than 9 ins. from center to center 
and not less than 1% ins. deep, 

C= 14,000, a constant for Brown furnaces, when corruga- 
tions are not more than 9 ins. from center to center and 
not less than i^ ins. deep, 

C = 10,000 a constant for furnaces corrugated by sections 
not more than 18 ins. from center to center and not less 
than 2}4 ins. deep, measured from the least inside to 
the greatest outside diameter of the corrugations, and 
having the ends fitted one into the other and sub- 
stantially riveted together, provided that the plain parts 
at the ends do not exceed 12 ins. in length. 

In calculating the mean diameter of the Morison furnace, the 
least inside diameter plus 2 ins., may be taken as the mean diameter." 

Manholes 

P. 65, p. 258. An elliptical manhole opening shall be not less 
than 11X15 ins. or 10X16 ins. in size. A circular manhole opening 
shall be not less than 15 ins. in diameter, p. 259. A manhole 
reinforcing ring when used, shall be of steel or wrought-iron, and shall 
be at least as thick as the shell plate. 




Fig. 23. — Riveting of man-hole frames. 



p. 260. Manhole frames on shells or drums when used, shall 
have the proper curvature, and on boilers over 48 ins. in diameter shall 
be riveted to the shell or drum with two rows of rivets, which may be 
pitched as shown in Fig. 23. The strength of the rivets in shear on 
manhole frames and reinforcing rings shall be at least equal to the 
tensile strength of that part of the shell plate removed, on a line par- 
allel to the axis of the shell, through the center of the manhole, or 
other opening. 

P. 66, p. 261. "The proportions of manhole frames and other 
reinforcing rings to conform to the above specifications may be deter- 
mined by the use of the following formulas, which are based on the 
assumption that the rings shall have the same tensile strength per 
square inch of section as, and be of not less thickness than, the shell 
plate removed. 

For a single-riveted ring : W = ^^^ -\- d (r) 



2Xt 

iXti 
For a double-riveted ring: W— ^ -\-2d 

2 Xt 



is) 



416 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 





Table io. — Discharge Capacities tor 


Direct 


Spring-loaded Pop Safety Valves with 


45 Deg. Bevel Seats 


Gage 

pressure, 




Diameter, i in. 


Diameter, iM in. 


Diameter, i}i in. 


Diameter, 2 in. 


Diameter, 2li in. 


I'bs. per 
sq. in. 


Min. 


Int. Max. 


Min. 


Int. 


Max. 


Min. 


Int. 


Max. 


Min. 


Int. Max. 


Min. Int. 





'Lift, in. . 


0.02 


0.04 


0.05 


0.03 


0.04 


0.05 


0.03 


0.05 


0.06 


0.04 


0.06 


0.07 


0.04 


0.06 


IS 


CH 


95,500 


191,000 


238,900 


179,200 


238,800 


293,500 


214,900 


358,300 


429,900 


382,200 


573,300 


668,900 


477,700 


716,600 




Lb. hr. . , 


65 


131 


163 


122 


163 


203 


146 


245 


293 


261 


391 


456 


326 


488 




Lift, in . . 


0.02 


0.04 


0.05 


0.03 


0.04 


0.05 


0.03 


O.OS 


0.06 


0.04 


0.06 


0.07 


0.04 


0.06 


25 


CH 


127,700 


255,400 


319.300 


239,500 


319,300 


399,100 


287,400 


478.900 


574.700 


510,900 


766,300 


894,000 


638,500 


957,900 




Lb. hr. . . 


87 


174 


218 


164 


218 


272 


196 


326 


392 


349 


523 


610 


435 


653 




Lift, in . . 


.02 


0.04 


0.0s 


0.03 


0.04 


O.OS 


0.03 


O.OS 


0.06 


0.04 


0.06 


0.07 


0.04 


0.06 


50 


CH 


208,200 


416,400 


520,400 


390,300 


520,400 


650,500 


468,300 


780.600 


936,600 


832,600 


1,249,000 


1,457,000 


1,041,000 


1,561,000 




Lb. hr. . . 


142 


284 


354 


266 


354 


444 


320 


532 


639 


S68 


851 


994 


710 


1,064 




Lift, in. . 


.02 


.04 


0.05 


0.03 


0.04 


O.OS 


0.03 


O.OS 


0.06 


0.04 


0.06 


0.07 


0.04 


0.06 


75 


CH 


288,600 


577,200 


721,400 


541,100 


721,400 


901,800 


649,300 


1,082,000 


1,299,000 


1,154,000 


1,731,000 


2,020,000 


1,443,000 


2,164,000 




Lb. hr. . . 


197 


393 


492 


369 


492 


615 


443 


738 


886 


787 


1,181 


1,377 


984 


1,475 




Lift, in. . 


0.02 


0.04 


0.05 


0.03 


0.04 


o.os 


0.03 


o.os 


0.06 


0.04 


0.06 


0.07 


0.04 


0.06 


100 


CH 


369,000 


738,000 


922,500 


691,900 


922,500 


1,153.000 


830,300 


1,384,000 


1,661,000 


1,476,000 


2,214,000 


2,583,000 


1,845,000 


2,768,000 




Lb. hr. . . 


252 


503 


629 


472 


629 


786 


S66 


944 


1,133 


1,007 


1,510 


1,761 


1,2S8 


1,887 




Lift, in. . 


0.02 


0.04 


o.os 


0.03 


0.04 


o.os 


0.03 


o.os 


0.06 


0.04 


0.06 j 0.07 


0.04 


0.06 


125 


CH 


449,400 


898,900 


1,124,000 


842,700 


1,124,000 


1,404,000 


1,011,000 


1,685,000 


2,022,000 


1,795,000 


2,693,000 


3,146,000 


2,247,000 


3,371,000 




Lb. hr. . . 


307 


613 


767 


575 


767 


957 


689 


1,149 


1,379 


1,224 


1.836 


2,14s 


1,532 


2,299 




Lift, in. . 


0.02 


0.04 


0.05 


0.03 


0.04 


O.OS 


0.03 


0.05 


0.06 


0.04 


0.06 


0.07 


0.04 


0.06 


150 


CH 


529,900 


i,c6o,ooo 


1,325,000 


993,500 


1,325,000 


1,656,000 


1,192,000 


1,987,000 


2,384,000 '2, 109,000 


3,179,000 


3,709,000 


2,649,000 


3,974.000 




Lb. hr. . . 


362 


723 


904 


677 


904 


1,129 


813 


1,355 


I,625[ 1,438 


2,158 


2,529 


1,806 


2,71 




Lift, in. . 


0.02 


0.04 


0.05 


0.03 


0.04 


0.05 


0.03 


0.05 


0.06 


0.04 


0.06 


0.07 


0.04 


0.06 


I7S 


CH 


610,300 


1,221,000 


1,526,000 


1,144,000 


1,526,000 


1,967,000 


1,373,000 


2,289,000 


2,746,000 


2,441,000 


3,662,000 


4,272,000 


3,051,000 


4.S77.000 




Lb. hr. . . 


416 


833 


1,040 


780 


1,040 


1,301 


936 


1,561 


1,872 


1,664 


2,497 


2,913 


2,081 


3. 121 




Lift, in. . 


0.02 


0.04' 


o.os 


0.03 


0.04 


0.05 


0.03 


O.OS 


0.06 


0.04 


0.06 


0.07 


0.04 


0.06 


200 


CH 


690,700 


1,381,000 


1,727,000 


1,295,000 


1,727,000 


2,158,000 


1,554,000 


2,590,000 


3,108,000 


2,763,000 


4,144,000 


4,835,000 


3,454,000 


5,180,000 




Lb. hr... 


471 


941 


1,178 


883 


1,178 


1,472 


1,060 


1,766 


2,119 


1,884 


2,826 


3,296 


2.3S4 


3.532 




Lift, in . . 


.02 


0.04 


0.05 


0.03 


0.04 


0.05 


0.03 


O.OS 


0.06 


0.04 


0.06 


.07 


0.04 


0.06 


225 


CH 


771,100 


1,542,000 


1,928,000 


1,446,000 


1,928,000 


2,410,000 


1,735.000 


2,892,000 


3,470,000 


3,085,000 


4,626,000 


5,398,000 


3,856,000 


5. 784.000 




Lb. hr... 


526 


1,052 


1,315 


986 


1,315 


1,643 


1,183 


1,972 


2,366 2,104 


3. 154 


3,680 


2,629 


3.944 




Lift, in.. 


0.02 


0.04 


o.os 


0.03 


0.04 


COS 


0.03 


0.05 


0.06 


0.04 


0.06 I 0.07 


0.04 


0.06 


250 


CH 


851,600 


1,703,000 


2,129,000 


l,S97,00O 


2,129,000 


2,661,000 


1,916,000 


3,193,000 


3,832,000 


3,406,000 


5,109,000 5,961,000 


4,258,000 


6,387,000 




Lb. hr. . . 


S81 


1,161 


1,451 


1,089 


1,451 


1,814 


1,307 


2,177 


2,613 


2,322 


3,484 4.064 


2,903 


4.3SS 




Lift, in.. 


,02 


0.04 


0.05 


0.03 


0.04 


0.05 


0.03 


0.05 


. 06 . 04 


.06 


0.07 


0.04 


0.06 


275 


CH 


932,000 


1,864,000 


2,330,000 


1,748,000 


2,330,000 


2,913,000 


2,097,000 


3,495,000 


4,194,000 


3,728,000 


5,592,000 


6,524,000 


4,660,000 


6,990,000 




Lb. hr. . . 


63s 


1,271 


1,589 


1,192 


1,589 


1,986 


1.430 


2,383 


2,860 


2,542 


3.813 


4,448 


3,177 


4.766 




Lift, in. . 


0.02 


0.04 


o.os 


0.03 


0.04 


0.05 


0.03 


O.OS 


0.06 


0.04 


0.06 


0.07 


0.04 


.06 


300 


CH 


1,024,000 


2,048,000 


2,531,000 


1,898,000 


2,S3i,ooo 


3,164,000 


2,278,000 


3,797,000 


4,556,000 


4,050,000 


6,075,000 


7,087,000 


5,062,000 


7,593.000 




Lb. hr... 


698 


1,397 


1,746 


1,294 


1,726 


2,157 


1,553 


2,589 


3,107 


2,762 


4.143 


4,832 


3,452 


5,177 



The discharge capacity of a fiat seat valve of a given diameter with a given lift may be obtained by multiplying the discharge capacity given in the Table 
for a 45 deg. bevel seat valve of same diameter and same lift, by 1.4. 



in which W = least width of reinforcing ring, ins., 
/i = thickness of shell plate, ins., 
rf = diameter of rivet when driven, ins., 
t = thickness of reinforcing ring — not less than thickness of 

the shell plate, ins., 
r = tensile strength of the ring, lbs., per sq. in. of section, 
a =net section of one side of the ring or rings, sq. ins., 
5 = shearing strength of rivet, lbs., per sq. in. of section, 
/ = length of opening in shell in direction parallel to axis of 

shell, ins., 
jV = number of rivets. 



Safety Valves 

P. 68, p. 269. " Each boiler shall have two or more safety valves, 
except a boiler for which one safety valve 3 inch size or smaller is 
required by these Rules. P. 68, p. 270. The safety valve capacity 
for each boiler shall be such that the safety valve or valves will dis- 
charge all the steam that can be generated by the boiler without 
allowing the pressure to rise more than 6 per cent, above the maxi- 
mum allowable working pressure, or more than 6 per cent, above the 
highest pressure to which any valve is set. 

p. 271. " One or more safety valves on every boiler shall be set at 



STEAM BOILERS 



417 



Table io.- — 


Discharge Capacities for Direct 


Spring-loaded Pop Safety Valves with 45 Deg. Bevel Seats — Continued 


Gage 

pres., 




Diameter, 
2^ in. 


Diameter, 3 in. 


Diameter, zYi in. 


Diameter, 4 


in. 


Diameter, 4\i in. 


lbs. per 
sq. in. 


Max. 


Min. 


Int. 


Max. 


Min. 


Int. 


Max. 


Min. 


Int. 


Max. 


Min. 


Int. 


Max. 




Lift, in. . 


o.o8 


0.05 


0.08 


O.IO 


.06 


.09 


O.II 


0.07 


O.IO 


. 12 


0.08 


O.II 


0.13 


IS 


CH 


955,500 


716,600 


1,147,000 


1,433,000 


1,003,000 


1,505,000 


1,839,000 


1,338,000 


1,911,000 


2,293,000 


1,720,000 


2,365,000 


2,795,000 




Lb. hr. . . 


651 


489 


782 


977 


684 


1,026 


1,254 


912 


1,303 


1,564 


1,173 


1,613 


1,906 




Lift, in, . 


o.o8 


0.05 


0.08 


O.IO 


0.06 


0.09 


O.II 


0.07 


. 10 


0. 12 


0.08 


0. 11 


. 13 


23 


CH 


1,277,000 


957,900 


1,533,000 


1,916,000 


1,341,000 


2,012,000 


2,459,000 


1,788,000 


2,554.000 


3,065,000 


2,299,000 


3,161,000 


3,736,000 




Lb. hr. . . 


871 


653 


1,046 


1,307 


914 


1,372 


1,676 


1,219 


1.742 


2,090 


1,568 


2,156 


2,547 




Lift, in. . 


o.o8 


0.05 


0.08 


. 10 


0.06 


0.09 


O.II 


0.07 


0. 10 


0. 12 


0.08 


0. II 


0.13 


SO 


CH 


2,081,000 


1,561,000 


2,498,000 


3,122,000 


2,186,000 


3,278,000 


4,007,000 


2,914,000 


4,163.000 


4,996,000 


3,747,000 


5,152,000 


6,088,000 




Lb. hr. . . 


1,419 


1,064 


1,703 


2,129 


1,490 


2,235 


2,732 


1,987 


2,839 


3.406 


2,555 


3,513 


4.151 




Lift, in. . 


0.08 


0.05 


0.08 


0. 10 


.06 


0.09 


0. II 


.07 


. 10 


0. 12 


0,08 


O.II 


0.13 


7S 


CH 


2,886,000 


2,164,000 


3,463,000 


4,329,000 


3,030,000 


4,545,000 


5,555.000 


4,040,000 


5,772,000 


6,926,000 


5,194.000 


7,142,000 


8,441,000 




Lb. hr. . . 


1,968 


1,475 


2,361 


2,951 


2,066 


3.099 


3.788 


2,754 


3,935 


4.722 


3.542 


4.870 


5,756 




Lift, in. . 


0.08 


0.05 


0.08 


0. 10 


0.06 


0.09 


0. II 


0.07 


0. 10 


0. 12 


0.08 


O.II 


0.13 


100 


CH 


3,690,000 


2,768,oooj 4,428,000 


5,535.000 


3,875,000 


S, 812, 000 


7.103.000 


5,166,000 


7,380,000 


8,856,000 


6,642,000 


9.133.000 


10,793,000 




Lb. hr. . . 


2,Si6 


1,887 


3,019 


3.774 


2,642 


3,963 


4,843 


3.522 


5,032 


6,038 


4.529 


6,227 


7.358 




Lift, in. . 


0.08 


0.05 


0.08 


O.IO 


0.06 


0.09 


O.II 


0.07 


0. 10 


0. 12 


0.08 


0. II 


0.13 


125 


CH 


4,494,000 


3,371,000 


5,393,000 


6,741,000 


4,719,000 


7,079,000 


8,652,000 


6,292,000 


8,988,000 


10,786,000 


8,989,000 


11,123,000 13,146,000 




Lb. hr. . . 


3,064 


2,299 3,677 


4.596 


3.218 


4,826 


S.899 


4.290 


6,128 


7,354 


5,516 


7.583 


8,963 




Lift, in. . 


0.08 


0.05 


0.08 


0. 10 


0.06 


0.09 


0. II 


0.07 


0. 10 


0. 12 


0.08 


0. II 


0.13 


ISO 


CH 


5,299,000 


3,974,000 


6,358,000 


7,948,000 


5,564.000 


8,345,000 


10,199,000 


7,418.000 


10,597,000 


12,717,000 


9,537.000 


13,114,000 


15,498,000 




Lb. hr. . . 


3,6i3 


2,710 


4,335 


5,419 


3.794 


5,690 


6,954 


5.058 


7,226 


8,670 


6,503 


8.940 


10,566 




Lift, in. . 


0.08 


0.05 


0.08 


O.IO 


0.06 


0.09 


0. II 


0.07 


0. 10 


0. 12 


0.08 j O.II 


0.13 


I7S 


CH 


6,103,000 


4,577.000 


7,323,000 


9,154.000 


6,408,000 


9,612,000 


11,748,000 


8,544,000 


12,206,000 


14,647,000 


10,985,000 15,105,000 


17,851,000 




Lb. hr... 


4,161 


3. 121 


4.993 


6,242 


4.369 


6,553 


8,010 


5.824 


8,320 


9.984 


7,490 


10,298 


12,173 




Lift, in. . 


0.08 


0.05 


0.08 


0. 10 


0.06 


0.09 


0. 11 


0.07 


0. 10 


0. 12 


0.08 


0. II 


0.13 


200 


CH 


6,907,000 


5,180,000 


8,289,000 


10,361,000 


7,253,000 


10,879,000 


13,296,000 


9,670,000 


13,814,000 


16,580,000 


12,433,000 17,095,000 


20,204,000 




Lb. hr... 


4.709 


3,532 


5.651 


7.064 


4,946 


7,418 


9,068 


6,593 


9,420 


11,305 


8,475 11,655 


13.773 




Lift, in. . 


0.08 


o.os 


0.08 


0. 10 


0.06 


0.09 


O.II 


0.07 


0. 10 


0. 12 


0.08 


0. II 


0.13 


225 


CH 


7,711,000 


5,784,000 


9,254,000 


11.567.000 


8,097,000 


12,146,000 


14,845,000 


10,796,000 


15,423,000 


18,507,000 


13,881,000 


19,086,000 


22,556,000 




Lb. hr. . . 


5,2S8 


3,944 


6,310 


7.890 


5,521 


8,280 


10,120 


7,361 


10,514 


12,616 


9,465 


13,013 


15,383 




Lift, in. . 


0.08 


0.05 


0.08 


0. 10 


0.06 


0.09 


0. 11 


0.07 


0. 10 


0. 12 


0.08 


0. II 


0.13 


250 


CH 


8,516,000 


6,387,000 


10,219,000 


12,774,000 


8,942,000 


13,412,000 


16,393,000 


11,922,000 


17,031,000 


20,438,000 


15,328,000 


21,076,000 


24,908,000 




Lb. hr. . . 


5,807 


4,355 


6,968 


8,708 


6,097 


9,143 


11.175 


8,130 


11,614 


13,938 


10,448 


14,366 


16,980 




Lift, in. . 


0.08 . 


0.05 


0.08 


0. 10 


0.06 


0.09 


0. II 


0.07 


0. 10 


0. 12 


0.08 


0. II 


0.13 


275 


CH 


9,320,000 


6,990,000 


11,180,000 


13,980,000 


9,786,000 


14,679,000 


17.941.000 


13,048,000 


18,640,000 


22,368,000 


16,776,000 


23,067,000 


27,261,000 




Lb. hr. . . 


6,355 


4,766 


7,620 


9,533 


6,672 


10,005 


12,233 


8,895 


12,707 


15,248 


11,438 


15,728 


18,58s 




Lift, in. . 


0.08 


0.05 


0.08 O.IO 


0.06 


0.09 1 O.II 


0.07 


0. 10 


0. 12 


0.08 


O.II 


0.13 


300 


CH 


10,124,000 


7,593,000 


1 2, 149, oooj 1 5, 1 86, 000 


10,630,000 


15,946,00019.489,000 


14,174,000 


20,249,000 


24,298,000 


18,224,000 


25,058,000 


29,614,000 




Lb. hr. . . 


6,903 


5,177 


8,280 


10,358 


7,248 


10,875 


13,290 


9,668 


13.807 


16,568 


12,428 


i7,o88| 20,195 



The discharge capacity of a flat seat valve of a given diameter with a given lift may be obtained by multiplying the discharge capacity given in the Table 
for a 45 deg. bevel seat valve of same diameter and same lift, by 1.4. 



or below the maximum allowable working pressure. The remaining 
valves may be set within a range of 3 per cent, above the maximum 
allowable working pressure, but the range of setting of all of the valves 
on a boiler shall not exceed 10 per cent, of the highest pressure to 
which any valve is set. 

P. 69, p. 272. ' Safety valves shall be of the direct spring loaded 
pop type with seat and bearing surface of the disk either inclined at an 
angle of about 45 deg. or flat at an angle of about 90 deg. to the center 
line of the spindle. The vertical lift of the valve disc measured 
immediately after the sudden lift due to the pop may be made any 
amount desired up to a maximum of .15 in. irrespective of the size 
27 



of the valve. The nominal diameter measured at the inner edge of 
the valve seat shall be not less than i in. or more than 4^^ ins. 

P. 274. The minimum capacity of a safety valve or valves to be 
placed on a boUer shall be determined on the basis of 6 lbs. of steam 
per hour per sq. ft. of boiler heating surface for water tube boilers, 
and 5 lbs. for all other types of power boilers, and upon the relieving 
capacity marked on the valves by the manufacturer, provided such 
marked relieving capacity does not exceed that given in Table 10. 
In case the relieving capacity marked on the valve or valves exceeds 
that given in Table 10, the minimum safety valve capacity shall be 
determined on the basis of the maximum relieving capacity given in 



418 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table lo for the particular size of valve and working pressure for 
which it was constructed. The heating surface shall be computed 
for that side of the boiler surface exposed to the products of combus- 
tion, exclusive of the superheating surface. In computing the heat- 
ing surface for this purpose only the tubes, shells, tube sheets and the 
projected area of headers need be considered." 

Table lo for the discharge capacity of safety valves was computed 



C = total weight or volume of fuel of any kind burned per 
hour at time of maximum forcing, lbs. or cu. ft., 

H = the heat of combustion, B.t.u. per lb. or cu. ft. of fuel 
used, 

£> = diameter of valve seat, ins., 

X = vertical lift of valve disk, ins., measured immediately 
after the sudden lift due to the pop, 



1 IH m V4 2 2»4 2H 254 



MaTriTinnn Pitch, Ins. 

3 3H ZVi 3*4 4 4'4 4I/2 454 8 EM BH 5'4 6 6'4 6H 634 7 iH IV2 1'A 8 




Traced downward from the maximum pitch to the curve for the diameter of the rivets and then to the left where read the per- 
centage of plate strength. 
Fig. 24. — Percentage of plate strength of boUer joints. 



Lloyds 




4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 
Maximum Pitch divided by number of rivets in Pitch and multiplied by Plate thickness in sixteenths 

Fig. 25. — Strength of boiler joint rivets compared with the solid plate. 



from formulas {11) and {v) wherein it is expressed as the product 
of C and H. 

p. 107, p. 421. The discharge capacities are given in Table 10 
for each valve size at the pressures shown and are calculated for vari- 
ous valve sizes, pressures and for three different lifts. The discharge 
capacities are proportional to the lifts, so that intermediate values . 
may be obtained from the table by interpolation. 



P = absolute boiler pressure or gage pressure plus 14.7 lbs. 
per sq. in., 
1 100 = the number of B.t.u. required to change a pound of feed 
water at 100 deg. Fahr. into a pound of steam. 
The boiler efficiency is assumed as 75 per cent. 
The coefficient of discharge, in Napier's formula, is taken as 96 
per cent. 



STEAM BOILERS 



419 




I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 

Per cent. Saved 
From the final temperature at the left trace horizontally to the diagonal leading from the initial temperature and then down 

where read the percentage of saving. 
Fig. 26. — The saving due to heating feed water for steam boilers. 



CXgX.75 _ 3-i4i6XZ)XXX.7o7XPX.96 for valve with 4S-deg. 
1100X3600 70' seat (0 

CH=i6o,8s6XPXDXL for valve with bevel seat at 45 deg. (u) 
CH = 22T,487XPXDXL for valve with ilat seat at 90 deg. (v) 




8000 2500 



Total Loss per Year of 320 Days 
Dollars 



1.00 1,50 2.00 
Price of Coal per Ton 
Dollars 



3.00 



From the price of coal per ton trace upward to the line for the 
thickness of scale, then horizontally to the line for the number 
of tons of coal consumed per day, then down, and read the loss 
in dollars per year of 320 days. 

Fig. 2 7 . — Loss of coal due to scale in boilers. 



The report of the A. S. M. E. Boiler Code Committee makes no 
provision for other than spring-loaded safety valves. The Massa- 
chusetts code includes Table 11. , 



Table ii. — Areas of Grate Sure aces in Sq. Fx. for Other 
THAN Spring-loaded Safety Valves 



Maximum pressure allowed per sq. in. j 


Zero to 


Over 25 to 


Over 50 to 


on the boiler 


1 


25 lbs. 


SO lbs. 


100 lbs. 


Diameter of valve 


Area of valve 








in ins. 


in sq. ins. 


Area 


of grate in 


sq. ft. 


I 


■ 7854 


1.50 


1-75 


2.00 


iH - 


1 .2272 


2.25 


2.50 


3.00 


iH 


I. 7671 


3 ■ 00 


3-75 


4.02 


2 


31416 


5.50 


6. 50 


7 .00 


2H 


4.9087 


8.25 


10. 00 


11.00 


3 


7 .0686 


11-75 


14.25 


16.00 


3H 


9-62II 


16.00 


19.50 


21.75 


4 


12. 5660 


21 . 00 


25.50 


28.2s 


4li 


15.9040 


26.75 


32.50 


36.00 


5 


19.6350 


32.75 


40. 00 


44.00 



The capacity of safety valves, according to the regulations of the 
Board of Supervising Inspectors of the Steamboat Inspection Service 
of the United States, is expressed by the formula: 

^=.2074 y 

in which A =area of valve disk, sq. ins., 

IF = weight of steam discharged per hour, lbs., 
P = absolute pressure, lbs. per sq. in. 
The above formula, due to L. D. Lovekin, chief engineer New 
York Shipbuilding Co., assumes the lift of the valve to be one- 
thirty-second of its diameter. Experiments by P. G. DarUng, 
mechanical engineer Manning Maxwell and Moore [Trajis. A. S. 
M. E., 1909) show that safety valves do not lift in proportion to their 
diameters; that the lift is practically the same for a large, as for a 
small valve, smaller for the larger valve if anything, and is around 
three-thirty-seconds of an inch for all valves in normal condition. 



420 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 13. — Proximate Analysis and Heating Value of American Coals 
From Steam, By Permission of the Babcock & Wilcox Co. 



Mois- 
ture 



Volatile 
matter 



Fixed 
carbon 



Ash 



Sul- 
phur 



Heating 

value per lb. 

coal, heat 

units 



Volatile 
matter per 
cent, of com- 
bustible 



Heating 

value per lb. 

combustible, 

heat units 



Theoretical 
evaporation lbs. 
water from and at 
212° per lb. com- 
bustible 



A nthracile. 

Northern coal field 

East Middle coal field 

West Middle coal field 

Southern coal field 

A nthracile from one mine. 

Egg, screen 2\-i\ ins ... 

Stove, screen ij-ii ins.... 

Chestnut, screen i\-l in 

Pea, screen \~\ in 

Buckwheat, screen \-\ in 

Semi- A nthracile. 

Loyalsock field 

Bernice basin 

Semi-Bituminous. 

Broad Top, Pa 

Clearfield County, Pa 

Cambria County, Pa 

Somerset County, Pa 

Cumberland, Md 

Pocahontas, Va 

New River, W. Va 

Bituminous. 

Connellsville, Pa 

Youghiogheny, Pa 

Pittsburg, Pa 

Jefferson County, Pa 

Middle Kittanning seam. Pa. ... 

Upper Freeport seam. Pa. and O. 

Thacker, W. Va 

Jackson County, O 

Brier Hill, O 

Hocking Valley, O 

Vanderpool, Ky 

Muhlenberg County, Ky 

Scott County, Tenn 

Jefferson County, Ala 

Big Muddy, 111 

Mt. Olive, 111 

Streator, 111 

Missouri 

Lignites and Lignitic Coals. 

Iowa 

Wyoming 

Utah 

Oregon lignite 



3.42 
3-71 
3.16 
3.09 



4-38 
3.08 
3.72 
4.28 



1.30 
.65 

.79 

.76 

• 94 

1.58 

1.09 

1 .00 

.85 

1 . 26 
1.03 
1.37 
1 . 21 
I. 81 
1-93 
1.38 
3.83 
4.80 
'6.59 
4.00 
4-33 
1 . 26 
I-SS 
7.50 

II ,00 

12.00 

6.44' 

8.4s 

8, 19 

9.29 

15.25 



8.10 
9.40 

15.61 
22.52 
ig. 20 
16.42 
17.30 
21 . 00 
17.88 

30. 12 
36.50 
35-90 
32.53 
35-33 
35.90 
35-04 
32.07 
34.60 
34-97 
34-10 
33 65 
35-76 
34-44 
30.70 
35.65 
33-30 
37.57 

37.09 
38.72 
41-97 
42 .98 



83-27 


8.20 


86.40 


6.22 


81.59 


10. 6s 


83.81 


8.18 


88-49 


5.66 


83.67 


10.17 


80.72 


12.67 


79-05 


14.66 


76.92 


16.62 


83.34 


6.23 


83.69 


5. 34 


77.30 


5.40 


71 .82 


3.99 


71. 12 


7.04 


71.51 


8.62 


73.12 


7.75 


74.39 


3.03 


77.64 


3.36 


59.61 


8.23 


59-05 


2.61 


52-21 


8.02 


60-99 


4.27 


S3 -70 


7.18 


50-19 


9. 10 


56.03 


6.27 


57.60 


6.50 


56.30 


4.30 


48.85 


8.00 


54-60 


7.30 


55-50 


4.95 


53-14 


8.02 


59.77 


2.62 


53-80 


8.00 


37-10 


13.00 


40.70 


14.00 


47.94 


8.05 


35.60 


18.86 


41.83 


II. 26 


44.37 


3.20 


33.32 


7. II 



.73 
.58 

.50 
.64 



13,160 
13,420 
12,840 
13,220 



5.00 
3.44 
4 36 
4.85 



14,900 
14,900 
14,900 
14,900 



15.42 
15.42 
15.42 
15.42 



1.63 
.91 

.90 

.91 

1 . 70 

1.87 

■74 
.58 



.78 

.81 

1.80 

I .00 

1.98 
2.89 
1.28 



I. 57 
1.80 

1 .42 



.18 
.66 



13.920 
13.700 

14,820 
14.950 
14.450 
14,200 
14,400 
15.070 
15,220 

14,050 
14.450 
13.410 
14.370 
13,200 
13.170 
14,040 
13,090 
13.010 
12,130 
12,770 
13,060 
13,700 
13,770 
12,420 
10,490 
10.580 
12,230 

8,720 
10,390 
11,030 

8,540 



8.86 
10.98 

17.60 
24.60 
22.71 
20.37 
19.79 
22.50 
18.95 

34.03 
38.73 
41 .61 
35.47 
40.27 
43.59 
39.33 
35.76 
38. 20 
42.81 
38.50 
38.86 
34.17 
37-63 
36.30 
47.00 
45-00 
43-94 

51 03 
48-07 
48.60 
54-95 



15.500 
15.500 

15.800 
15.700 
15.700 
IS. 800 
15.800 
IS. 700 
15.800 

15.300 

IS. 000 

14,800 
15,200 
14,500 
14.800 

IS, 200 
14.600 
14.300 
14,200 
14,400 

I4.400(?) 

I5,I00(?) 

I4,400(?) 

14,700 

13,800 

14,300 

i4,3O0(?) 

I2,000(?) 
I2,900(?) 
I2,600(?) 
II,000(?) 



16. OS 
16. OS 



15.84 

15-53 
15-32 
15-74 
1501 
15.32 
15-74 
15. II 
14.80 
14.70 
14.91 
14.91 
15.63 
14.91 
15.22 
14.29 
14.80 
14. 80 

12.42 
13.3s 
13.04 
11.39 



From this fact and Napier's formula for the discharge of steam through 
an orifice, Power {Mar. 9, 1909) deduces the very simple formula: 

d = .i-p 

in which d = diameter of disk, ins. 
the remaining notation being as before. 

Mr. Lovekin's formula, which has proven sufficient, gives the 
same results as Power's formula for valves of 2.64 ins. diameter. 

The Massachusetts formula is: 

^=770 ~p 

in which A = total area of safety valve or valves, sq. ins., 

W = lbs. of water evaporation per sq. ft. of grate surface 

per sec, 
P = boiler pressure (absolute). 

The Philadelphia formula is: 

22.5 G 



A-- 



PX8.62 



in which A = area of safety valve, sq. ins. per sq. ft. of grate, 
G= grate area, sq. ft., 
P = boiler pressure (gage). 



The saving due to heating feed water for boilers may be determined 
from Fig. "26 by W. M. Wright {Power, June 25, 1912). The use 
of the chart is explained below it. The chart is calculated for 100 
lbs. boiler pressure. For 50 lbs. pressure, percentages are less than 
.15 higher. For 200 lbs. pressure, percentages are less than .2 
lower. 

The loss due to scale in boilers may be estimated by the use of' 
Fig. 27 by Chas. Brossmann {Power, Apr. 16, 1912). The use of 
the chart is explained below it. 

The horse-power of chimneys is given in Table 14 from Wm. Kent's 
well-known formula, the figures for the horsepower being, however, 
increased for the larger sizes by unimportant amounts by the Bab- 
cock & Wilcox Company. The table is based on the assumption 
that a commercial horsepower requires an average consumption 
of s lbs. of coal per hour. 

Hanging or Supporting the Boiler 

The boiler should be supported on points where there is the greatest 
excess of strength. Excessive local stresses from weight of boiler 
and contents must be avoided and distortion of parts prevented by 
using long lugs or brackets, and only half the stress which they may 
carry in the seams, to be allowed on rivets. 



nan 



STEAM BOILERS 



421 



Table 14. — Horse-power of Chimneys for Steam Boilers 

From Steam, By Permission of the Babcock & Wilcox Co. 



Diameter 
in ins. 



Height of Chimneys and Commercial Horse-power 



50 ft. 



60 ft. I 70 ft. I Soft. I 90 ft. I 100 ft. I no ft. I I2S ft. I ISO ft. I 17s ft. I 200 ft. 



Side of 
square. 



Effective 

area, sq. 

ft. 



Actual area, 
sq ft. 



18 
21 
24 
27 
30 

33 
36 

39 

42 



54 
60 
66 

72 
78 

84 

90 

96 

102 

108 

114 
120 
126 
132 
138 

144 



23 

35 
49 
65 
84 

: ■ 


25 
38 

54 
72 
92 

IIS 

141 


27 
41 
58 
78 
100 

I2S 
152 
183 
216 


































62 

83 

107 

133 

163 
196 
231 
311 

363 

SOS 






























113 

141 
173 
208 
24s 
330 

427 
536 
6s8 
792 


























182 
219 
258 
348 

449 
56s 
694 
83s 
995 

1 163 

1344 
1337 


























271 
365 

472 
593 
728 
876 
1038 

1214 
1415 
1616 














389 

503 
632 
776 
934 
II07 

1294 
1496 
1720 
1946 
2192 

2459 














551 

692 

849 

1023 

1212 

1418 
1639 
1876 
2133 
2402 

2687 
2990 
3308 
3642 
3991 

4357 












748 
918 
IIOS 
1310 

1531 
1770 
2027 
2303 
2594 

2903 
3230 
3573 
3935 
4311 

4707 


981 
I181 
1400 

1637 
1893 
2167 
2462 
2773 

3003 
3452 
3820 
420S 
4608 

5031 











































































































































































16 
19 

22 



30 
32 
35 

38 
43 

48 
54 
59 
64 
70 

75 
80 
86 
90 
96 

lOI 

106 
112 
117 
122 



■ 97 
1.47 
2.08 
2.78 
3.58 

4.48 
S-47 
6. 57 
7.76 
10.44 

13.51 
16.98 
20.83 
25.08 
29.73 

34-76 
40.19 
46.01 
52.23 
58.83 

65.83 

73-22 

81.00 
89-19 

97-75 
106.72 



1.77 
2.41 
3.14 
3-98 
4.91 

5. 94 
7.07 
8.30 
9. 62 

12-57 

15-90 
19.64 
23.76 
28. 27 
33-18 

38.48 
44-18 
50-27 
56.75 
63.62 

70.88 
78.54 
86. S9 
95-03 
103.86 

113. 10 



The supports must permit rebuilding the furnace without disturb- 
ing the proper suspension of the boiler. The boiler should be slightly 
inclined so that a little less water shows at the gage cocks than at 
the opposite end. 

The percentage of plate strength of boiler joints may be obtained 
from Fig. 24 {Anier. Mach., June 16, 1892). The use of the chart is 
explained below it. 

The strength of the rivets of boiler joints compared with the strength 
of the solid plate may be obtained from Fig. 25 {Amer. Mach., Apr. 
14, 1892) which is drawn for Lloyd's and the British Board of Trade 
rules. The use of the chart is best shown by an example: 

Find the percentage of strength of rivets in single shear com- 
pared with the tensile strength of the solid plate in the case of a 
double-riveted lap joint; plate thickness | in., rivets | in. diameter, 
pitch 2^ ins., iron plate and rivets. Divide the maximum pitch by 
the number of rivets in one pitch length and multiply the quotient 



Serrating the tube seat in a straight machined hole by rolling or 
cutting square edged grooves .01 in. deep and ten pitch will raise 
the slipping point to three or four times that in a smooth hole. 

It is possible to make a rolled joint that will offer a resistance 
beyond the elastic limit of the tube and remain tight. 

Table 12. — Properties of Standard Boiler Tubes and Flues 

Weights and Dimensions are Nominal 

From the National Tube Company's Book of Standards 



Find 



2.5 
by the plate thickness in sixteenths of an inch; i.e., — X 8 = 10. 

the intersection of the ordinate 10 on the chart with the curve for 
j-in. rivets and read 71 per cent, for punched plates. If rivets 
in double shear are considered to be 75 per cent, stronger than in 
single shear, multiply the result given by the chart by 1.75. 

By a reverse reading the rivet diameter can be obtained if the 
pitch, number of rows, plate thickness and percentage of strength 
are given. 

The resistance offered by the expanded tubes in tube sheets formed the 
subject of experiments by Profs. O. P. Hood and G. L. Christensen 
{Trans. A. S. M. E., 1908) of which the following are the conclusions: 

The slipping point of a 3-in. twelve-gage Shelby cold drawn tube 
rolled into a straight smooth machined hole in a lin. sheet occurs 
with a pull of about 7000 lbs. 

Various degrees of rolling do not greatly affect the point of initial 
slip. 

The frictional resistance of such tubes is about 750 lbs. per sq. 
in. of tube-bearing area in sheets -| in. and i in. thick. 

For a higher resistance to initial slip other resistance than friction 
must be depended upon. 



Diameters 


Thickness 


Weight 
per 

ft. 


Length of tube 
per sq. ft. 


Sq. 
surfa 
linea 


Et. of 
:e per 
1 ft. 


Exter- 


Inter- 


Ins. 


B.W.G 


External 


Internal 


External 


Internal 


nal 


nal 




surface 


surface 


surface 


surface 


li 


1.560 


.095 


13 


1-679 


2.182 


2.448 


-458 


.408 


2 


1. 810 


• 095 


13 


1.932 


1.909 


2. no 


-523 


■473 


2i 


2.060 


.095 


13 


2.186 


1.697 


1.854 


-589 


-539 


2^ 


2-282 


. 109 


12 


2.783 


1-527 


1.673 


-654 


-597 


2| 


2-532 


. 109 


12 


3-074 


1.388 


1.508 


■ 719 


.662 


3 


2.782 


. 109 


12 


3365 


1-273 


1-373 


.785 


.728 


3i 


3-010 


. 120 


II 


4. on 


1-175 


1 . 269 


.850 


.788 


3i 


3-260 


. 120 


II 


4-331 


I -091 


1. 171 


.916 


.853 


3i 


3.510 


. 120 


II 


4-652 


I -018 


1.088 


.981 


.918 


4 


3-732 


.134 


10 


5-532 


-954 


1.023 


1.047 


.977 


4i 


4.232 


.134 


10 


6.248 


-848 


.902 


1.178 


1. 107 


5 


4-704 


.148 


9 


7.669 


-763 


.812 


1.308 


I. 231 


6 


5-670 


.I6s 


8 


10.282 


-636 


.673 


1-570 


1.484 


7 


6.670 


.165 


8 


12.044 


-545 


.572 


1-832 


1-746 


8 


7.670 


.i6s 


8 


13-807 


-477 


.498 


2.094 


2 .008 


9 


8.640 


. 180 


7 


16.955 


-424 


.442 


2-356 


2.261 


10 


9-594 


.203 


6 


21 .240 


-381 


.398 


2.617 


2.511 


II 


10-560 


.220 


5 


25 329 


.347 


.361 


2.879 


2.764 


12 


11.542 


.229 




28.788 


.318 


.330 


3-141 


3.021 


13 


12.524 


.238 


4 


32.439 


.293 


-304 


3-403 


3.278 


14 


13-504 


.248 




36.424 


.272 


.282 


3 -66s 


3-535 


15 


14.482 


• 259 


3 


40.775 


.254 


.263 


3926 


3-791 


16 


15 460 


.270 




45.359 


.238 


.247 


4.188 


4-047 



THE STEAM ENGINE 



Table i. — Properties of Saturated Steam 
From Steam, By Permission of the Babcock & Wilcox Co. 



Pressure in 

lbs, per sq. 

in. above 

vacuum 


Temperature in 
deg. 
Fahr. 


Total heat in 
heat units from 
water at 32 deg. 


Heat in liquid 

from 32 deg. in 

units 


Heat of vapor- 
ization, or latent 
heat in heat units 


Density or 

weight of cu. 

ft. in lbs. 


Volume of 1 
lb. in cu. ft. 


Factor of equiv- 
alent evapora- 
tion at 212 deg. 


Total pressure 
above vacuum 


I 


loi .99 


1113. I 


70.0 


1043.0 


.00299 


334-5 


.9661 


I 


2 


126.27 


1120.5 


94.4 


1026. I 


.00576 


173.6 


.9738 


2 


3 


141 .62 


1125. 1 


109.8 


1015.3 


.00844 


118. S 


.9786 


3 


4 


1S309 


1128.6 


121. 4 


1007.2 


.01107 


90.33 


.9822 


4 


S 


162.34 


1131.S 


130.7 


1000.8 


.01366 


73.21 


.9852 


5 


6 


170.14 


1133.8 


138.6 


995.2 


.01622 


61. 6S 


.9876 


6 


7 


176.90 


1135-9 


145.4 


990.5 


.01874 


53-39 


.9897 


7 


8 


182 .92 


1137.7 


151. 5 


986.2 


.02125 


47.06 


.9916 


8 


9 


188.33 


1139.4 


156.9 


982. 5 


.02374 


42. 12 


.9934 


9 


10 


193.25 


1140.9 


161.9 


979.0 


.02621 


38.15 


.9949 


10 


IS 


213.03 


1146.9 


181.8 


965.1 


.03826 


26.14 


I . 0003 


IS 


20 


227.95 


1151.S 


196.9 


954.6 


.05023 


19.91 


1.0051 


20 


25 


240 . 04 


IISS.I 


209.1 


946.0 


.06199 


16. 13 


1.0099 


2S 


30 


250.27 


1158.3 


219.4 


938.9 


.07360 


13.59 


1.0129 


30 


3S 


259.19 


1x61.0 


228.4 


932.6 


.08508 


11.75 


1.0157 


35 


40 


267.13 


1163.4 


236.4 


927.0 


. 09644 


10.37 


1.0182 


40 


AS 


274.29 


1165.6 


243.6 


922.0 


.1077 


9.285 


X.0205 


45 


SO 


280.8s 


1167.6 


250.2 


917.4 


.1188 


8.418 


1.0225 


50 


ss 


286.89 


1169.4 


256.3 


913.1 


.1299 


7.698 


1.0245 


SS 


6o 


292.51 


1171.2 


261.9 


909.3 


. 1409 


7.097 


1.0263 


60 


6S 


297.77 


1172.7 


267.2 


905.5 


.1519 


6.S83 


1.0280 


6S 


70 


302.71 


1174.3 


272.2 


902. 1 


.1628 


6.143 


1.029s 


70 


75 


307.38 


II7S.7 


276.9 


898.8 


-1736 


S.760 


1.0309 


75 


80 


311.80 


1177.0 


281.4 


895.6 


.1843 


5.426 


1.0323 


80 


8S 


316.02 


1178.3 


285.8 


892. S 


-1951 


S.126 


1.0337 


85 


90 


320.04 


1179.6 


290.0 


889.6 


.2058 


4.859 


1.0350 


90 


95 


323.89 


1180.7 


294.0 


886.7 


.2165 


4.619 


1.0362 


95 


100 


327.58 


1181.9 


297.9 


884.0 


.2271 


4-403 


1.0374 


100 


los 


331.13 


1182.9 


301.6 


881.3 


.2378 


4-205 


1.038s 


los 


no 


334.56 


1184.0 


30s. 2 


878.8 


.2484 


4.026 


1.0396 


no 


IIS 


337.86 


118S.0 


308.7 


876.3 


-2589 


3-862 


1.0406 


lis 


120 


341 05 


1186.0 


312.0 


874-0 


.269s 


3-711 


1.0416 


120 


125 


344.13 


1186.9 


3IS.2 


871.7 


.2800 


3-571 


1.0426 


I2S 


130 


347.12 


1187.8 


318.4 


869.4 


.2904 


3-444 


1.0435 


130 


140 


352.8s 


1189.5 


324.4 


865.1 


-3113 


3-212 


I.04S3 


140 


ISO 


358.26 


1191.2 


330.0 


861.2 


.3321 


3-0II 


1.0470 


ISO 


160 


363 . 40 


1192.8 


335.4 


857.4 


-3530 


2-833 


1.0486 


160 


170 


368.29 


1194.3 


340. s 


8S3-8 


-3737 


2.676 


1.0502 


170 


180 


372.97 


1195.7 


345.4 


850.3 


0-3945 


2.535 


1.0517 


180 


190 


377.44 


1197.1 


3S0.I 


847.0 


-4153 


2.408 


1.0S31 


190 


200 


381.73 


1198.4 


354-6 


843.8 


-4359 


2.294 


1.0S4S 


200 


225 


391.79 


I20I.4 


36s. I 


836.3 


.4876 


2.0SI 


1.0576 


225 


250 


400 . 99 


1204.2 


374.7 


829. S 


-5393 


1.854 


1.0605 


250 


275 


409 . SO 


1206.8 


383.6 


823.2 


-5913 


1. 691 


1.0632 


275 


300 


417.42 


1209.3 


391.9 


817.4 


-644 


1.S53 


1.0657 


300 


325 


424.82 


1211. 5 


399.6 


811.9 


.696 


1.437 


1.0680 


32s 


350 


431.90 


1213.7 


406.9 


806.8 


.748 


1.337 


1.0703 


350 


375 


438.40 


1215.7 


414.2 


801. S 


.800 


1.250 


1.0724 


375 


400 


445.15 


1217.7 


421.4 


796.3 


.853 


1 . 172 


1.0745 


400 


500 


466.57 


1224.2 


444.3 


779.9 


1.065 


.939 


1.0812 


500 



condensing engines use 18 to 22 lbs.; while the best types of multi- 
expansion condensing engines utilizing superheated steam have re- 
duced the steam consumption below 12 lbs. per h.p. hour. Table 2 
cylinder non-condensing engines use 28 to 50 lbs.; ordinary compound gives the relative efiaciency of various types of pumping engines. 

422 



Steam and Coal Consumptioii 

The steam consumption of steam engines varies with the type. Single- 



THE STEAM ENGINE 



423 



Table iA. — Properties of Superheated Steam 

Abstracted from Marks and Davis's Steam Tables and Diagrams by permission of the publishers, Messrs. Longmans, Green & Company. 
( = temperature Fahr.; D = specific volume, cu. ft. per lb.; ft = total heat, B.t.u. from water at 32 deg. 



Press., 


Water J^*;^ 


Degrees of superheat 


lbs. 
abs. 


10° 


20° 


30° 


40° 


50° 


60° 


70° 


80° 


90° 


100° 


110° 


120° 


130° 


1 
140° , 150° 1 160° 


170° 


180° 


190° 


200° 


20 t 

V 

h 


228.0 
.02 20.08 
196. I 1156.2 


238.0 

20.41 

1160.9 


248 .0 

20.73 
II65.7 


258.0 

21 .05 

II70.4 


268.0 
21 .37 
II7S-2 


278.0 

21 .69 
1179.9 


288.0 

22 .01 

I184.6 


298.0 

22 .32 
II89-3 


308.0 

22.63 

I194-I 


318.0 
22.94 
1198.8 


328.0 

23.25 

1203.5 


338.0 

23-56 

1208 .3 


348.0 
23.87 
1213 .0 


358.0 

24. 18 

1217.7 


368.0 

24-49 
1222 .4 


378.0 

24.80 

1227 . I 


388,0 

25-11 

1231,8 


398.0 
25,41 
1236,5 


408.0 

25-72 

1241 .2 


418 .0 

26,02 

1245-9 


428 

26.33 
1250,6 


25 t 

V 

h 


240.1 
.02 16,30 
208.4 1160.4 


250.1 

16.57 

1165.2 


260.1 

16.84 

II7O.O 


270.1 

17.10 

1174.8 


280.1 

17-35 
II79-6 


290.1 

17.60 

II84.4 


300.1 

17.86 

II89-2 


310.1 

18.11 

II93.9 


320.1 

18.36 

II98.7 


330.1 

18.61 

1203.4 


340.1 
18.86 

1208 .2 


350.1 

19.11 

1213 .0 


360. 1 

19.36 

I2I7.7 


370.1 

19-6l 
1222 .4 


380.1 

19.86 

1227 . 1 


390.1 

20. 10 
I23I .9 


400, 1 

20,35 

1236,6 


410,1 

20.60 

1241,3 


420.1 

20.84 

1246 .0 


430 , 1 

21 ,08 

1250,8 


440,1 

21 ,32 
12556 


30 < 

V 

h 


250.4 
.02 13.74 
218.8 1163.9 


260.4 

13.97 

1168.8 


270.4 

14-19 
1173.6 


280.4 
14.41 

1178. 5 


290.4 

14.62 

1183.3 


300.4 

14-83 

1188. I 


310.4 

15.05 

II92 .9 


320.4 

15.26 

II97.7 


330.4 

15-47 
1202 .5 


340.4 

15.68 

1207.3 


350.4 

IS. 89 

I2I2 . 1 


360.4 

16.10 

1216.9 


370.4 
16.31 

I22I .7 


380.4 

16.52 

1226 .4 


390.4 

16.73 

1231 .2 


400.4 
16.93 

1236 .0 


410,4 

17.14 

1240,7 


420,4 

17-35 

1245-4 


430.4 

17.55 

1250.2 


440,4 

17-76 

1255,0 


450.4 

17.97 

1259-7 


35 < 

V 

h 


259.3 
.02 11.89 
227.9 1166.8 


269.3 

12 .09 

1171.7 


279-3 

12.29 

1176.6 


289.3 

12 .48 
I18I.S 


299.3 

12 .67 

ri86.4 


309.3 

12.8s 
1191.3 


319.3 

13 03 

1196. I 


329.3 

13 .21 

I20I .0 


339.3 

13.39 

1205 .8 


349.3 

13-57 

1210 .6 


359.3 

13.75 

1215-4 


369.3 

13-93 

1220 .2 


379.3 

14.11 

1225.0 


389.3 

14.29 

1229.8 


399.3 

14-47 
12346 


409.3 

14-65 

1239.4 


419.3 

14.83 

1244.2 


429.3 

15 01 

1248 ,9 


439.3 

15 . 18 

1253.7 


449.3 

15,36 

1258,5 


459,3 

15-54 
1263,3 


40 < 

V 

h 


267.3 
.02 10.49 
236.0 1169.4 


277.3 

10.66 

1174.3 


287.3 

10.83 

1179.3 


297.3 
11 .00 

II84.2 


307.3 

11 .16 

1189.1 


317.3 

11.33 

I 194.0 


327.3 

11.50 

1198.9 


337.3 

11.66 
1203.8 


347.3 

11 .82 

1208.7 


357-3 

11.98 

1213.5 


367.3 

12.13 

1218.4 


377.3 

12 .29 

1223.2 


387.3 

12.45 

1228.0 


397.3 

12 .61 
1232.9 


407.3 

12.77 

1237.7 


417.3 

12.93 

1242.4 


427.3 

13 -08 

1247.2 


437.3 

13-23 

1252 ,0 


447.3 

13 39 

1256.8 


457.3 

13,54 

1261 ,6 


467.3 

13-70 

1266,4 


45 < 

V 

h 


274-S 
.02 9-39 
243.4 I171.6 


284.5 

9-55 

1176.6 


294-5 

9-70 

I181 .6 


304 -5 

9-85 

1186.6 


314-5 

10.00 

1191.6 


324.5 

10.14 

1196.6 


334-S 

10.29 

1201 .5 


344-5 

10.44 

1206.4 


3S4-5 

10.58 

1211 .3 


364.5 

10.73 

1216 .2 


374-5 

10.86 

1221 .0 


384.5 

11 .00 

1225.9 


394-5 

II. IS 

1230.7 


404.5 
11 .29 

1235 6 


414-5 

11.43 

1240 .4 


424,5 

11-57 
1245.2 


434-5 

11.71 

1250.0 


444 5 

11.85 

I2S4-8 


454-4 

11.99 

1259.7 


464.5 

12 , 13 

1264,5 


474-5 

12 .27 

1269.3 


sot 

V 

h 


281.0 
.02 8.51 
250.1 II73-6 


291 .0 

8.65 

I178.7 


301 .0 

8.78 

1183.7 


311. 

8.92 

I188.8 


321 .0 

9.06 

1193.8 


331.0 

9.19 

1198.8 


341-0 

9-32 

1203.8 


351.0 

9-45 

1208.7 


361 .0 

9-58 

1213.6 


371.0 

9.71 

1218.S 


381.0 

9-84 

1223.4 


391.0 

9-97 

1228.3 


401 .0 

10.10 

1233.2 


411 .0 

10.23 

1238. 1 


421 .0 

10.35 
1242.9 


43I-O 

10,48 

1247.7 


441 -0 

10.61 

1252.5 


451.0 

10.73 

1257.3 


461 .0 

10.86 

1262 .2 


471.0 
10.99 

1267 .0 


481 .0 

II . 11 

1271.8 


60 t 

V 

h 


292.7 
.02 7-17 
262.1 I177.O 


302.7 
7.29 

1182 .2 


312.7 

7.40 

1187.3 


322.7 

7-52 
1192.4 


332.7 

7-63 

1197-5 


342.7 
7-75 

1202 .6 


352.7 

7.86 

1207.7 


362.7 
7-97 

1212 .7 


372.7 

8.08 

1217.7 


382.7 

8.19 

1222 .6 


392.7 

8.30 

1227 .6 


402.7 

8.41 

1232.5 


412.7 

8.52 

1237.4 


422.7 

8.62 

1242.3 


432.7 
8.73 

1247.2 


442.7 

8.84 

1252 .1 


452.7 

8.94 

1256.9 


462,7 

9-05 

1261,8 


472.7 

915 

1266 .6 


482.7 

9.26 

1271.5 


492.7 

9,36 

1276.4 


lot 

V 

h 


302.9 
.02 6.20 
272.6 II79.8 


312.9 

6.30 

1185. I 


322.9 

6.40 

1190.4 


332.9 

6.51 

II9S-6 


342.9 

6.61 

1200.8 


352.9 

6.71 

1205.9 


362.9 

6.81 

1211 .0 


372.9 

6.90 

I216. I 


382.9 

7.00 

1221 .2 


392.9 

7-09 

1226.2 


402.9 

7.18 

1231 .2 


412.9 

7.28 

1236.2 


422.9 

7-37 

1241 . I 


432.9 

7.46 

1246.0 


442.9 
7-56 

1250.9 


452.9 

7.65 

125s. 8 


462.9 

7-74 
1260.7 


472,9 

7.83 

1265,5 


482.9 

7.92 

1270.4 


492.9 
8.02 

1275,3 


502.9 

8.11 

1280.2 


iot 

V 

h 


312.0 
.02 5-47 
282.0 1182.3 


322.0 

5.56 

1187.7 


■632.0 

5-65 
11930 


342.0 

5-74 

1198.3 


352.0 

S.83 

1203.6 


362.0 

5-92 

1208.8 


372.0 

6.00 
1214.0 


382.0 

6.09 

I2I9 . I 


392.0 
6.18 

1224.2 


402 .0 

6.26 

1229.3 


412 .0 

6.34 

1234-3 


422.0 

6.42 

1239.3 


432.0 
6.50 

1244.2 


442.0 

6.58 

1249.2 


452.0 

6.67 

I254-I 


462 .0 

6-75 

1259.0 


472.0 

6.83 

1264 


482 ,0 

6,91 

1268,9 


492.0 

7 .00 

1273.8 


502 ,0 

7 .08 

1278.7 


512 .0 

7-17 

1283.6 


90 ( 

V 

h 


320.3 
.02 4.89 
290. s 1184.4 


330.3 

4-97 
I 190.0 


340.3 

5-05 

1195.4 


350.3 

5 13 

1200.8 


360.3 

5.21 

1206. I 


370.3 

5 -29 
1211 .4 


380.3 

5-37 

1216.7 


390.3 

5-44 
1221 .9 


400.3 

5-52 

1227 .0 


410.3 

5 .60 

1232. I 


420.3 

5.67 

1237.2 


430.3 

5-74 

1242 . 2 


440.3 

5. 82 

1247.2 


450.3 

5-90 

1252 .2 


460.3 

5.97 

1257,1 


470.3 

6.04 

1262.0 


480.3 

6.12 

1266.9 


490.3 

6 . 19 

1271.9 


500.3 

6.26 

1276.8 


510.3 

6.34 

1281.7 


520.3 

6 .40 

1286.6 


100 t 

V 

h 


327.8 
.02 4.43 
298.3 1186.3 


337-8 

4-51 

1192 .0 


347-8 

4-58 

1197-5 


357.8 

4-65 

1203.0 


367.8 

4.72 

1208.4 


377.8 

4-79 

1213.8 


387.8 

4.86 

1219.1 


397.8 

4-93 

1224.3 


407.8 

5 00 

1229. S 


417.8 

5.07 

1234.6 


427.8 

5-14 

1239.7 


437.8 

5-21 

1244.7 


447.8 

5. 27 

1249.7 


457.8 

5-34 

1254.7 


467.8 

5-41 
1259.7 


477.8 

5-47 

1264.7 


487.8 

5-54 
1269.6 


497.8 

5-61 

1274-6 


507,8 

5.67 

1279,5 


517,8 

5-74 

1284.5 


527.8 

5- 80 

1289.4 


no t 

V 

h 


334-8 
.02 4.05 
305. 5 1188.0 


344-8 

4.12 

1193.8 


354-8 

4.18 

1199.4 


364.8 

4-25 

1205.0 


374-8 

4.32 

1210.5 


384.8 

438 

121S.9 


394-8 

4.45 

1221.2 


404.8 

4-51 

1226.5 


414-8 

4-57 

1231.7 


424.8 

4.64 

1236.9 


434.8 

4.70 

1242.0 


444-8 

4.76 

1247. I 


454.8 

4 83 

1252. I 


464.8 

4-89 

1257. I 


474-8 

4-95 

1262. I 


484.8 

501 

1267.1 


494.8 

5.07 

1272.1 


504-8 

5 -13 

1277,0 


514-8 

5 19 

1282,0 


524.8 

5-25 

1287.0 


534-8 

5 31 

1291.9 


120 t 

V 

h 


341 -3 
.02 3.73 

312.3 1189.6 


351.3 

3-79 

1195-4 


361.3 
3.8s 

1201 .1 


371.3 

3-92 

1206.8 


381.3 

3.98 

1212.4 


391.3 

4.04 
1217.9 


401.3 

4. 10 

1223.3 


411-3 

4.16 

1228.6 


421.3 

4.22 

1233.8 


431.3 

428 

1238.9 


441.3 

4-33 

1244.1 


451.3 

4-39 

1249.2 


461.3 

4-45 
1254.2 


471.3 

4-50 

1259.3 


481.3 

4.56 

1264,3 


491.3 

4-62 

1269.3 


501. 3 

4.68 

1274.3 


5II-3 

4-73 

1279 .2 


521,3 

4-78 

1284,2 


531.3 

4-83 

1289.2 


54I-3 

4-89 

1294 -I 


130 t 

V 

h 


347.4 

.02 3-45 
318 .6 1191 .0 


357.4 

3 51 

1197.0 


367.4 

3-57 

1202.8 


377-4 

3.63 

1208. s 


387.4 
3.69 

1214. I 


397.4 

3-74 

1219.7 


407.4 

3 -80 

1225.1 


417.4 

3.85 

1230.4 


427.4 

3-91 

1235.7 


437.4 

3 96 

1240 .9 


447.4 

4.02 

1246.1 


457.4 

4 07 

1251 .2 


467.4 

413 

I2S6.3 


477.4 

4.18 

1261.4 


487.4 

4-23 

1266.4 


497-4 

4,28 

1271.4 


S07.4 

4-34 

1276.4 


S17.4 

4-39 

1281 ,4 


527.4 

4-44 

1286,3 


537.4 

4-49 

1291.3 


547.4 

4 54 

1296.2 


140 I 

V 

h 


353-1 
.02 3.22 
324.6 1192.2 


363.1 

3-27 

1198.3 


373.1 
3.32 

1204.3 


383.1 

3.38 

1210.I 


393.1 

3.44 
1215.8 


403.1 

3-49 

1221 .4 


413.1 

3.54 

1226.8 


423.1 

3 -60 

1232.2 


433-1 

3-65 

1237-S 


443.1 

3-70 

1242.8 


453. 1 

3-75 

1248.0 


463.1 

3.80 

1253.1 


473.1 

3.85 

1258.2 


483-1 

3-90 

1263.3 


493.1 

3-95 

1268.3 


S03.1 

4-00 

1273.3 


513,1 

4 05 

1278.3 


523,1 

4-09 

1283-3 


533.1 

4-14 
1288.3 


543-1 

4-19 

1293.2 


553.1 

4.24 

1298.2 


ISO t 

V 

h 


358. 5 

.02 3,01 

330.2 II93-4 


368. 5 

3 06 

I 199. 6 


378.5 

3-II 
120S.7 


388. s 

3-17 

I2I1 .6 


398. 5 

3 .22 

1217.3 


408.5 

3-27 

1223.0 


418. s 

3.32 

1228.5 


428.5 

3 37 

1233.9 


438.5 

3-42 

1239.2 


448. S 

3-46 
1244.4 


458.5 

3-51 

1249.6 


468. 5 

3-S6 

1254-8 


478.5 
3-61 

1259. S 


488.5 

3.66 

1265 .0 


498,5 

3-70 

1270, I 


508,5 

3-75 

1275,1 


518,5 

3.79 

1280. I 


528,5 

3.84 

1285,1 


538.5 

3-88 

1290. I 


548.5 

3 92 

1295,0 


558.5 

3-97 

1300.0 


160 t 

V 

h 


363.6 

.02 2.83 

335-6 11945 


373.6 

2.88 

1200.8 


383.6 

2.93 

1207.0 


393.6 

2.98 

1213 .0 


403.6 

3.03 

1218.8 


413.6 

3-07 

1224. s 


423.6 

3.12 

1230.0 


433-6 

3-17 

1235-S 


443.6 

3.21 

1240.8 


4S3.6 

3-20 

1246. I 


463.6 

3.30 

1251.3 


473-6 

3-35 

1256. 5 


483.6 

3-40 

1261 .6 


493.6 

3-44 

1266.7 


503.6 

3,48 

1271 ,8 


513,6 

3,53 

1276,8 


523,6 

3-57 
I28l,8 


533,6 

3-61 

1286,8 


543-6 

3.66 

1291 .8 


553.6 

3.70 

1296.8 


563.6 

3-74 
1301.7 


170 t 

V 

h 


368. 5 
.02 2.68 

340.7 1195.4 


378. 5 

2.73 

1202.0 


388.5 

2.78 

1208.2 


398. 5 

2.82 

1214.3 


408.5 

2.86 

1220.2 


418. 5 

2.91 

1225.9 


428. 5 

2.95 

1231.5 


438.5 

3-00 

1237-0 


448.5 

3.04 

1242.3 


458. 5 

3 08 

1247 .6 


468. 5 

3-12 

1252.8 


478.5 

3.17 

1258.0 


488.5 

3.21 

1263 .1 


498,5 

3-25 

1268.2 


SO8.5 

3.29 

1273.3 


Sl8,5 

3-34 

1278,4 


528,5 

3.38 

1283,4 


538,5 

3-42 

1288,4 


548.5 

3-46 

1293-4 


558. 5 

3 50 

1298 .4 


568. S 

3-54 

1303.3 


180 / 

V 

h 


373-1 
.02 2.53 
345-6 II96.4 


383.1 

2.58 

1203 


393.1 

2.62 

1209.4 


403.1 

2.67 

12155 


413. 1 

2.71 

1221 .5 


423.1 
2.75 

1227 .2 


433-1 

2.80 

1232.8 


443.1 

2.84 

1238.4 


453-1 

2.88 

1243.8 


463.1 

2.92 

1249.1 


473-1 

2 .96 

I2S4-3 


483.1 

3-00 

1259.5 


493 I 

3.04 

1264.6 


503,1 
3 08 

1269,7 


S13.I 

3-12 

1274.8 


523,1 

3.16 

1279,9 


533,1 

3-20 

1284,9 


543 -I 

3,24 

1289-9 


553.1 

3-28 

1294-9 


563.1 

3 32 

1299,9 


573,1 

3 35 

1304,8 


190 t 

V 

h 


377-6 
.02 2.41 
350.4 II97-3 


387-6 

2.45 

1204.0 


397.6 

2.49 

1210. 5 


407.6 

2.53 

1216.7 


417.6 

2.58 

1222 .7 


427.6 

2.62 

1228.6 


437.6 

2.66 

1234-3 


447.6 

2.70 

1239.8 


457-6 

2.74 

1245.1 


467.6 

2.78 

1250.4 


477-6 

2.81 

1255-7 


487.6 

2.85 

1260 .9 


497.6 

2.8g 

1266. I 


507,6 
2.93 

I27I .2 


S17.6 

2.97 

1276,3 


527.6 

3 00 

1281 ,3 


537.6 

3 04 

1286,4 


547.6 

3.08 

1291,4 


557,6 

3.12 

1296 ,4 


567,6 

3.15 

1301,3 


577,6 

3 19 

1306,3 


200 t 

V 

h 


381.9 
.02 2.29 
354-9 1198.1 


391.9 

2.33 

1205.0 


401.9 

2.37 

I2II .6 


411. 9 

2.41 

1217 .8 


421.9 

2.45 

1223.9 


431.9 

2.49 

1229.8 


441.9 

2.53 

1235-S 


451.9 

2.57 

1241 .1 


461.9 

2 .61 

1246.5 


471.9 

2 .64 

I2S1.8 


481.9 

2.68 

1257. I 


491.9 

2.72 

1262 .3 


501. S 

2 .76 
1267.4 


SII.9 
2.79 

1272. s 


521 .9 

2.83 

1277.6 


531,9 

2,86 

1282 ,6 


541-9 

2 .90 

1287.7 


551-9 

2.94 

1292 .8 


561 .9 

2.97 

1297,8 


571,9 581. 9 

3 ■ 00 3 ■ 04 

1302.7 1307.7 



424 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 2. 



-Efficiencies of Various Types of Pumping Engines Using Saturated Steam 

From Steam, By Permission of the Babcock & Wilcox Co. 



Type 



Duty. 
Million ft.-lbs. work done per 1000 lbs. 
steam consumed, with varying con- 
ditions of service 



Lbs. of steam 
per pump h.p. hour 



Condensins. 
Direct acting and crank and fly-wheel. Triple expansion. 

Direct acting and crank and fly-wheel. Compound 

Direct acting low duty ; Triple expansion. 

Direct acting low duty Compound 

Non-condensing. 

Direct acting low duty Triple expansion. 

Direct acting low duty Compound 

Direct acting small sizes Non-compound.. . 

Vacuum pumps, direct acting, independent 

Vacuum pumps, fly-wheel, independent 

Injectors 



I2S to 140 


100 to 120 


75 to 


90 


40 to 


60 


50 to 


70 


30 to 


40 


8 to 


20 


8 to 


20 


45 to 


80 


2 to 


5 



16 to 


13 


20 to 


16 


26 to 


20 


50 to 


33 


40 to 


28 


66 to 


50 


250 to 


100 


250 to 


100 


45 to 


25 


000 to 


400 



Table 3. — Useful Steam per I.h.p.-hr., 5„ in Lbs. with Single- 
Cylinder Non-condensing Engine Steam Throttled 

With new engines Su may be taken approximately 1.75 lbs. less. 



Table 4. — Useful Steam per I.h.p.-hr., Su in Lbs. with Single- 
cylinder, Non-condensing Engines, Automatic Cut-off 

With new engines these values may be taken approximately 1.5 lbs. less. 



Avg. abs. 






Cut-off in per cent. 


of full stroke 






i -2 <" 




admission 




















Avg. a 
admiss 
pressui 


Cut-off in per cent, of full stroke 


pressure 


70 


60 


50 


40 


33.3 


30 


2S 


20 


IS 




30 


SI. SO 


51.00 
















70 


60 


50 1 .10 1 33.3 


30 


25 


20 


IS 


12. si 10 


35 


3S 


SO-SO 






















40 


46.00 


44-50 


44-50 














40 


44 so 


40.50 


39.50 


3S.30 


39 . 50 














45 


42.50 


40.50 


39.50 


39.00 












45 


40-50 


36.80 


35.50 


34-00 


33-50 


34-00 


35-50 










SO 


39.50 


37-75 


35.75 


35.00 


35.00 










50 
55 


38.00 
36.00 


34.80 
33.00 


32.50 
30.75 


30.80 
29.00 


30-25 
27.90 


30-50 
27-80 


31-60 

28.55 


30.50 








55 


37.8s 


35-55 


33-50 


32.2s 


32.00 


































60 


36.50 


34-00 


31.80 


30.40 


29.95 


29. 80 








60 


34-50 


31.7s 


29.50 


27.50 


26.40 


25-90 


26, IS 


26.50 


28.00 






65 


35-25 


33.00 


30.75 


28.80 


28.20 


27-75 








65 


33-45 


30.75 


28.40 


26.25 


25.00 


24.50 


24-45 


24.40 


25.50 


27.50 




70 


34-25 


32.00 


29.80 


27.80 


26.85 


26.30 


25.95 






70 


32-50 


29.90 


27.50 


25.20 


24.00 


23.4s 


23-lQ 


22.75 


23.55 


25.00 


27.2s 


75 


33-50 


3 1 - 00 


29.00 


27.00 


25-85 


25.20 


24-75 






75 


31-55 


29.20 


26.75 


24-45 


23-15 


22.60 


22.25 


21.60 


22. 10 


23.20 


25.00 






















80 


31 -00 


28.50 


26.00 


23.75 


22, 50 


22.00 


21.50 


20.80 


21.00 


21.95 


23.00 


80 


32-65 


30.40 


28.20 


26.20 


25.00 


24.30 


23-75 






























85 


32.00 


29.80 


27.5s 


25.50 


24.4s 


23.52 


23-00 


22. 10 




85 


30-50 


28.00 


25.45 


23.20 


22 .00 


21.4s 


21 .00 


20.00 


20.00 


20.80 


21.50 


90 


31.50 


29.3s 


27.1s 


25.00 


23.8s 


23.0s 


22.50 


21.4s 




90 


30.00 


27 -SO 


25.00 


22.75 


21.50 


20.95 


20.50 


19.50 


19.50 


19.90 


20.50 


95 


31 .00 


28.95 


26.60 


24.60 


23.35 


22.75 


22.00 


20.90 




95 


29.50 


27 .00 


24.60 


22.40 


21 . 10 


20.55 


20.00 


19.00 


19.00 


19. 10 


19.50 


100 


30.60 


28,50 


26.30 


24.25 


22.90 


22.25 


21.50 


20.50 


19.60 


100 


29. 10 


26.50 


24. 20 


22.00 


20.85 


20. 10 


19.50 


18.60 


18.45 


18.50 


19.00 






















105 


28.80 


26. 10 


23.90 


21.6s 


20.45 


19.80 


19.00 


18.40 


17.8s 


18.00 


18. so 


105 


30.30 


28.05 


26.00 


24.00 


22. so 


21.95 


21.00 


20. 10 


19.4s 


























no 


30.00 


27.80 


25-75 


23.6s 


22.25 


21.55 


20.75 


19.90 


19- 10 


no 


28.50 


25-75 


23-65 


21.45 


20. 10 


19.50 


18.80 


18. IS 


17-45 


17.50 


18.00 


115 


29-75 


27.50 


25-50 


23.40 


22.00 


21.30 


20.50 


19.50 


18.96 


•JiS 


28.2s 


25-55 


23-40 


21. 20 


19.85 


19. 20 


18. 55 


17.80 


17.00 


17. 10 


17.5s 


120 


29.50 


27.30 


25-25 


23-05 


21.65 


21 . 10 


20.25 


19-25 


18.52 


120 


28.00 


25-40 


23-15 


21 .00 


19. 55 


19.00 


18.35 


17.50 


16.60 


16.80 


17.10 


125 


29. IS 


27.05 


24-95 


22.80 


21.50 


20.96 


20.00 


19.00 


18.40 


125 


27.75 


25.25 


22.95 


20.75 


19.40 


18.80 


18.10 


17.20 


16.40 


16. so 


16.6s 






















130 


27.50 


25.10 


22.75 


20.50 


19.25 


18.60 


17.90 


17.00 


16.2s 


16. 25 


16.3s 


130 


28.8s 


26.80 


24-75 


22.60 


21.35 


20.75 


19-75 


18.8s 


18.2s 


























135 


28.60 


26.55 


24 -SO 


22.45 


21.15 


20.50 


19.50 


18.60 


18.00 


135 


27-35 


24.90 


22.55 


20.30 


19. OS 


18.45 


17 .60 


16.80 


16. 10 


16.00 


16.00 


140 


28.45 


26.30 


24-30 


22.30 


21.00 


20.30 


19.30 


18.50 


17-82 


140 


27.10 


24.65 


22.40 


20.15 


18.8s 


18.30 


17.35 


16.55 


15.95 


15.7s 


IS-SS 


14s 


28. 22 


26.15 


24-iS 


22.25 


20.80 


20. 15 


19. IS 


18.2s 


17-75 


14s 


26.90 


24.4s 


22.30 


20.00 


18.70 


18.15 


17 . 10 


16.4s 


15-70 


15.55 


IS-2S 


ISO 


28.05 


26.00 


24-05 


22. 15 


20.60 


20.00 


19.00 


17.80 


17.55 


ISO 


26.7s 


24.25 


22.15 


19.88 


18. 55 


18.00 


16.9s 


16.35 


15-50 


15.40 


15 00 



The approximate steam consttmption oj various types of steam engines 
may be obtained from Tables 3-8 and the following formulas by J. 
A. Knesche {Power, Nov. 12, 1912). The useful steam is to be 
taken from Tables 3-6 in accordance with the class of engine under 
consideration and to the quantity thus obtained additions are to 
be made as follows: 

The greater part of the steam loss within the cylinder is due to 
condensation and the smaller part to leakage past the piston and 
valves. The condensation losses Sc are determined from the formula 

Sc = ^— {a) 

in which 50 = steam losses through condensation, 
P = piston speed in ft. per sec, 
X = coefficient as given in Table 7, 



If the admission steam is superheated sufficiently, cylinder con- 
densation may be entirely avoided. With cutoff in the high-pressure 
cylinder ranging from 40 to 25 per cent, a superheat of from 175 
to 250 deg. Fahr. is sufficient to prevent condensation. But even in 
such a case, Sc must not be taken as zero, because superheated 
steam, compared with saturated steam, does less work in the engine 
cylinder on account of the more rapid fall of its expansion curve and 
also, because heat is required to superheat the steam. If Sc is de- 

K 
termined from the formula Sc = ~y= when superheated steam is 

used, then K will be from j to 5 the value for saturated steam, as 
given in Table 7. 

For single-cylinder engines the leakage past the piston Si may be 
determined according to the formula 



when the ratio of stroke to diameter, ("5) is approximately 2. 

The smaller figures are to be applied to engines that are new or in 
very good condition. 

When -^ differs considerably from 2, the values in Table 7 are to 

be multiplied by the coefficients which are given in Table 8. 



Si = 



35-14 



+ 



3.62 



Vi-hp.XP P 



{h) 



in which i.A^. = indicated h.p., 

P = piston speed in ft. per sec. 
For compound engines the leakage loss is 80 per cent, and for 
triple-expansion engines 64 per cent, of the value given by this 



THE STEAM ENGINE 



425 



Table 5. — Useful Steam per I.h.p.-hr., Su in Lbs. with Single- 
cylinder Condensing Engines, Automatic Cut-off 
With new engines these values maybe taken from i to 1.5 lbs. less in 
the smaller cut-offs. 



g. abs. 

mission 

essure 








Cut-off in 


per cent of full stroke 








^ a 0. 


50 


40 i33 


.3 1 30 1 25 


20 1 IS 1 12 


.5 1 10 1 7 


S 


35 


24 


IS 










































40 


23 


55 


21 


50 


20 


20 


19 


40 


18 


60 


17 


80 


16 


9S 


16 


95 


16 


60 










45 


23 


25 


21 


10 


19 


85 


19 


00 


18 


35 


17 


45 


16 


70 


16 


50 


16 


30 


x6 


^5 






50 


23 


00 


20 


80 


19 


55 


18 


75 


18 


OS 


17 


10 


16 


45 


16 


10 


16 


00 


15 


75 


16 


10 


55 


22 


75 


20 


60 


19 


30 


18 


50 


17 


80 


16 


90 


16 


15 


IS 


75 


IS 


75 


15 


45 


IS 


6s 


60 


22 


50 


20 


45 


19 


10 


18 


25 


17 


55 


16 


70 


IS 


90 


IS 


50 


15 


50 


IS 


15 


IS 


45 


65 


22 


30 


20 


30 


18 


90 


18 


05 


17 


40 


16 


50 


IS 


70 


IS 


30 


IS 


35 


14 


95 


IS 


2S 


70 


22 


10 


20 


10 


18 


70 


17 


95 


17 


25 


16 


35 


IS 


50 


IS 


10 


IS 


20 


14 


75 


IS 


OS 


75 


21 


95 


19 


97 


18 


55 


17 


80 


17 


10 


16 


18 


IS 


35 


15 


00 


IS 


OS 


14 


60 


14 


85 


80 


21 


80 


19 


81 


18 


40 


17 


72 


16 


95 


16 


OS 


IS 


20 


14 


90 


14 


90 


14 


SO 


14 


65 


85 


21 


70 


19 


70 


18 


25 


17 


60 


16 


85 


IS 


97 


IS 


OS 


14 


80 


14 


75 


14 


40 


14 


45 


90 


21 


60 


19 


60 


18 


10 


17 


50 


16 


75 


IS 


85 


14 


95 


14 


70 


14 


60 


14 


30 


14 


25 


95 


21 


50 


19 


SO 


18 


00 


17 


42 


16 


6s 


15 


75 


14 


85 


14 


67 


14 


SO 


14 


20 


14 


05 


100 


21 


40 


19 


40 


17 


93 


17 


35 


16 


55 


15 


65 


14 


75 


14 


60 


14 


40 


14 


10 


13 


90 


105 


21 


30 


19 


30 


17 


85 


17 


27 


16 


45 


IS 


55 


14 


67 


14 


53 


14 


30 


14 


00 


13 


80 


no 


21 


20 


19 


20 


17 


76 


17 


IS 


16 


40 


IS 


45 


14 


60 


14 


45 


14 


20 


13 


95 


13 


75 


115 


21 


10 


19 


10 


17 


67 


17 


07 


16 


35 


IS 


37 


14 


55 


14 


40 


14 


10 


13 


90 


13 


70 


120 


21 


00 


19 


00 


17 


60 


17 


00 


16 


30 


15 


30 


14 


50 


14 


35 


14 


00 


13 


85 


13 


65 


125 


20 


90 


18 


95 


17 


55 


16 


95 


16 


25 


IS 


23 


14 


45 


14 


30 


13 


95 


13 


80 


13 


60 


130 


20 


80 


18 


90 


17 


SO 


16 


90 


16 


20 


IS 


IS 


14 


40 


14 


25 


13 


90 


13 


75 


13 


55 


I3S 


20 


75 


18 


85 


17 


45 


16 


85 


16 


IS 


IS 


08 


14 


37 


14 


20 


13 


85 


13 


70 


13 


50 


140 


20 


70 


18 


80 


17 


40 


16 


80 


16 


10 


IS 


00 


14 


35 


14 


15 


13 


77 


13 


65 


13 


45 


14s 


20 


65 


18 


75 


17 


35 


16 


75 


16 


05 


14 


95 


14 


32 


14 


10 


13 


75 


13 


60 


13 


40 


ISO 


20 


60 


18 


70 


17 


30 


16 


70 


16 


00 


14 


92 


14 


30 


14 


05 


13 


73 


13 


55 


13 


35 



formula. With engines in very good condition Si may be only one- 
half the foregoing values while, with pistons in visibly leaky con- 
dition, the leakage loss may be twice this or even more. 

The condensation losses in the steam lines plus any water carried 
over with the steam when the boilers prime may be taken from 4 
to 10 per cent., depending upon the size and length of the steam line, 
its covering and the frequency with which the boilers prime. 

The method of procedure is best shown by an example: Required 
the steam consumption of a 42X6o-in. vertical, single-cylinder, 
piston-valve throttling engine, the diagrams from which are shown 
in Fig. I. Taking first the top end: 

Useful steam per indicated h.p.-hr. from Table 3, Su = 34-25 lbs. 

Average admission pressure = 67.8 lbs., absolute. 

Cut-off = 73.45 per cent. 

Ratio of stroke to diameter I -j I =1.43. 

Piston speed {P) = 5.36 ft. per sec. 
Then 

\/P = V5^6 = 2.315 

From Tables 7 and 8, -ff = 27.93 X.91 = 25.4, and from equation 

(a) 

25-4 



Sc = - 



2-315 



= II lbs. 



The leakage losses Si from equation (6) are 

3S-I4 , 3-62 

/ — 1 2 = 1-57 lbs. 

\/289XS-36 S-36 

the h.p. being computed from the indicator diagram, Fig. i. 

Therefore, 

5 = 34.25-1-11-^1-57=46.82 lbs. 

The steam-line losses are taken at 4 per cent.; hence the total 
steam consumption is 46.82X1.04 = 48.7 lbs. per h.p.-hr. 
Taking next the bottom end: 

Useful steam per indicated h.p.-hr. from Table 3, 5„=33.5 lbs. 

Average admission pressure = 57.8 lbs., absolute. 



Table 6. — Useful Steam per I.h.p.-hr., Su in Lbs. with 
Compound Condensing Engines 

These values are for engines in good condition and well defined cut-off 
and without preheating in the receiver. 

With new engines these values may be taken from i to 1.5 lbs. less in the 
smaller cut-offs. 



Avg. abs. ad- 


Cut 


■off 


in per cent. 


of full Stroke reduced to low pressure c 


yl. 


mission pressure 


25 


20 


IS 


12 


S 


10 


7 


S 


4 


40 


































45 


17 


50 


16 


45 


15 


25 


14 


80 


14 


50 


14 


75 


15 


30 






50 


17 


25 


16, 


10 


IS 


00 


14 


SO 


14 


10 


14 


25 


14 


SO 






55 


16 


95 


IS 


8s 


14 


80 


14 


IS 


13 


65 


13 


75 


14 


00 






60 


16 


70 


IS 


55 


14 


60 


13 


85 


13 


25 


13 


40 


13 


50 


13 


50 


6s 


16 


SO 


15 


35 


14 


40 


13 


6s 


13 


10 


13 


05 


13 


00 


13 


00 


70 


16 


35 


IS 


IS 


14 


20 


13 


50 


12 


95 


12 


70 


12 


60 


12 


75 


75 


16 


25 


15 


00 


14 


00 


13 


35 


12 


85 


12 


50 


12 


25 


12 


SO 


80 


16 


15 


14 


90 


13 


85 


13 


25 


12 


75 


12 


30 


12 


00 


12 


25 


85 


16 


05 


14 


80 


13 


70 


13 


IS 


12 


65 


12 


10 


II 


80 


12 


00 


90 


15 


95 


14 


70 


13 


55 


13 


OS 


12 


55 


II 


95 


II 


60 


II 


75 


95 


IS 


90 


14 


65 


13 


45 


12 


95 


12 


45 


II 


85 


II 


45 


11 


50 


100 


IS 


85 


14 


60 


13 


35 


12 


85 


12 


35 


II 


75 


11 


30 


II 


35 


los 


IS 


82 


14 


55 


13 


25 


12 


80 


12 


25 


II 


65 


II 


20 


II 


20 


no 


15 


79 


14 


50 


13 


15 


12 


70 


12 


15 


II 


55 


II 


10 


II 


10 


115 


15 


76 


14 


45 


13 


05 


12 


60 


12 


05 


11 


45 


II 


03 


II 


00 


120 


IS 


73 


14 


40 


13 


00 


12 


SO 


II 


95 


II 


35 


10 


94 


10 


90 


I2S 


IS 


70 


14 


35 


13 


00 


12 


45 


II 


85 


II 


25 


10 


83 


10 


80 


130 


IS 


67 


14 


30 


12 


97 


12 


42 


II 


75 


n 


IS 


10 


75 


10 


70 


135 


15 


64 


14 


25 


12 


95 


12 


39 


11 


6S 


II 


OS 


10 


67 


10 


60 


140 


15 


61 


14 


20 


12 


93 


12 


36 


II 


55 


10 


95 


10 


60 


10 


SO 


14s 


15 


58 


14 


17 


12 


91 


12 


35 


II 


4S 


10 


85 


10 


53 


10 


40 


150 


IS 


55 


14 


15 


12 


90 


12 


32 


II 


43 


10 


80 


10 


47 


10 


30 



Table 7. — Values of K in Formula (a) 



Engine type 



Throttling, single-cylinder, non-condensing . . . . 
Automatic cut-off, single-cylinder non-con 

densing. 
Automatic cut-off, single-cylinder, condensing . 

Compound, condensing 

Triple-expansion, condensing 



K 



27.938 to 25.952 
23.955 to 19-963 

21.959 to 19 963 
19.9s to 17.955 
16.758 to 15.96 



Bottom End 




Top End 




Fig. I. — Indicator cards for calculated example. 



Table 8. — Corrections for Values of K 



If -, is approximately 


Then K in Table's is to be 
multiplied by 


I 


.82 


1-25 


.87 


I -SO 


.91 


2.00 


1. 00 


2.50 


1.08 


3.00 


i-iS 


4.00 


1.29 


5.00 


1. 41 



426 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 









Table 9 


. — Mean Forward Pressure of Steam 


PER Lb. 


OF Initial Pressure 








Cut-off 


n fractions 
; stroke 












Percentage of clearance 














o£ th< 





I 


i^S 


2 


2.S 


3 


3-5 


4 


4^5 


S 


55 


6 


6.5 


7 


iV 


. I 


■ 3303 


• 3439 


• 3505 


• 3568 


• 3630 


.3690 


• 3750 


• 3808 


.3864 


■ 3919 


■3974 


.4027 


• 4076 


.4126 


i 


.125 


■ 3849 


■ 3966 


.4023 


.4078 


■ 4132 


.4187 


.4237 


• 4287 


■4338 


.4386 


■4433 


■ 4480 


• 4527 


• 4571 


J 


.167 


.4662 


• 4757 


.4802 


.4844 


.4890 


• 4933 


• 4973 


.5014 


.S056 


• S096 


.5134 


■ 5173 


• 5210 


■ 524s 


J-r 


.188 


• 5013 


■ 5097 


.5138 


.5181 


• 5217 


.5259 


.5295 


• 5332 


• S367 


• 5405 


■5440 


■5474 


• 5511 


■ 5546 


H 


.20 


■ 5219 


■ 5298 


• 5336 


• 5376 


•5414 


■5449 


■ 5482 


■ 5517 


■ SSS6 


• 5588 


■ 5623 


■ 5656 


• 5687 


■ 5716 


i 


• 25 


.5966 


.6025 


.6059 


.6090 


.6120 


• 6148 


• 6174 


.6207 


.6229 


.6258 


■ 6286 


■ 6312 


• 6336 


■ 6359 


IS 


■ 30 


.6609 


.6663 


• 6684 


.6712 


.6729 


.6755 


.6779 


.6803 


• 682s 


.6845 


• 6864 


.6882 


.6911 


• 6927 


i 


■ 333 


.6988 


.7029 


• 7047 


• 7076 


• 7092 


• 7106 


• 7132 


.7144 


■ 7168 


.7190 


.7212 


■ 7219 


• 7239 


• 7257 


i 


• 375 


.7433 


• 7458 


.7476 


• 7494 


• 7510 


• 7525 


• 7539 


• 7569 


■ 7582 


■ 7593 


• 7603 


.7630 


• 7639 


.7646 


5 


.40 


■ 7665 


.7691 


• 7719 


• 7729 


• 7738 


• 7765 


.7772 


.7778 


.7802 


.7806 


■ 7829 


.7831 


.7853 


• 7874 


i 


■ 50 


.8466 


• 8484 


.8492 


• 8503 


• 8513 


.8522 


■ 8530 


.8539 


.8548 


.8556 


.8565 


.8573 


• 8582 


.8590 


S 


.60 


.9064 


.9076 


.9081 


.9087 


.9092 


• 9097 


.9102 


• 9107 


.9112 


.9117 


.9122 


.9127 


.9132 


■ 9136 


f 


.62s 


.9188 


• 9194 


.9201 


.9206 


.9210 


.9215 


• 9220 


.9224 


■ 9228 


■ 9233 


■ 9237 


• 9241 


• 924s 


■ 9249 


f 


.667 


.9371 


.9378 


• 9382 


■ 938s 


■ 9389 


• 9392 


• 9396 


• 9399 


.9402 


• 9405 


.9408 


• 9411 


■ 9415 


■ 9418 


xa 


.70 


• 9497 


• 9502 


• 9S0S 


■ 9508 


■ 9511 


• 9513 


• 9Sl6 


.9518 


■ 9S2I 


• 9524 


.9526 


• 9528 


■ 9531 


.9533 


i 


■ 75 


■ 9657 


• 9661 


.9663 


.9665 


.9667 


.9668 


.9670 


.9672 


.9674 


■ 9675 


■ 9677 


• 9679 


• 9680 


.9682 



Cut-off =70.5 per cent. 

Condensation losses Sc same as for top end or 11 lbs. 
35-14 



Leakage losses Si = 



\/3oi.sXS-36 S-37 



^■62 
+TT- = 1-55 lbs. 



Cut-off 
1.00 ■SO .SO .70 .GO .50 


LOO ::-^-.JJ-j^.--L-_J 1 L 


:. :: :::: : !::±;:; : ':"-:": : ":: - _:: 






■* ^ 












\ 




' " " -_ _- _ ""i" . 




















g _ ._ -_ . -- - ^ - 


•B > _ _ 


3 — - - -- - _ ._ _ - -_ / -- 


< '- ' - / 


-" - - - ^^-._ 


" " --/ - -- 




S5 -^0 :":::::":'::::"::"-::";f ::":::':::":'": 








2 K 


f' lA 


p< --.::: :: : : _.:_- __ --. -. _ _- 


01 --- — - / ^. 


■S :: :::::. ,^_ __ _ _ __ 


3 <!n /I 


■3.°" 7 :. :...:_:_. _ _.. 


g . ^ ^ _ - - _. - _- 


m - - .-_ ^ . 


il 2 


"^ _ ._ :::::: ::. ..:::: :_ 


<H _ .. - _ : — : :':: : :: : : :::::: : 


: :. ;. -.:..::-Z: : : :::: 




S " :: :' "": ": . "": 




CJ.50 '' 






13 -. V : :::: : ::::.; ;: 


p — / - 


g - . — ^ — . ..::;:__ 


S _.._::: I" I" z : : ; _ .:::::::._ _ 


o> " ^ 


D Z 


•^ 40 .: ;:.-j.: .: ; ;..:: : 


jj .4U ^ --- 




-f ;_.: z_.: 


2 — : — _: 


p — ^. — _ .- _ — -. 




S 'l~ — " " — . - - 


g _.t. — . - _ - 


- - — - -- -_ _ 


&< Qd : - ... . .-:.: . . :.. : 


".30-- :: :::::::: 


^- - 


3 '-- - ' - - - - — — - 


•s :: : 2:::. :.:.::.: : : _ _. ._ _ 


a :: : t ::: :::::.:. __ _ -_ 


g 


5 : 7 --- --- --- - _:: ;: : ::__ 


•< t — — - ::::::: _:_:_ 












IB ^ — — --:-; 




t " " " .. -.. 


X- - - - ._ __ 




, „ _ _ 


\u -A....... z... ~ ~ — — ;::.: -. 






-.- ._ __ - --- 


F --- --- - -- - 






i — : " ----- — 


f - - — - - 


t _ .. .-- :; :: 





1.00 



.90 



.80 



.70 



.60 



.50 



.40 



.30 



.20 



,10 



Therefore, 

5 = 33-S + ii + i-55 =46.05 lbs. 
The steam-line losses are taken at 4 per cent.; hence the total 
steam consumption is 

46.05X1.04=47.9 lbs. per h.p.-hr. 
Therefore, the average steam consumption for the engine is 48.3 
bs. per i. h.p.-hr. 

Power Calculations 
The theoretical mean effective pressure of steam used expansively 
is given by the formula: 



M.e.p- 



^ i + hyp log r 



Line .10 .20 .30 .40 .50 

Cut-off 

Fig. 2. — Mean forward pressure of steam used expansively. 



In which P = absolute initial pressure, 
p = absolute back pressure, 
^ = ratio of expansion. 

The same results may be more quickly obtained from Table 9, 
by F. R. Low {Power, Sept. 26, 1911). The table gives directly the 
absolute mean forward pressure per lb. of absolute initial pres- 
sure. The quantities of the table are to be multiplied by the abso- 
lute initial pressure and from the result the absolute back pressure 
is to be subtracted, the result being the mean effective pressure. 

Essentially the same results, neglecting the effect of clearance, 
may be obtained from Fig. 2 by Professor Rankine {The Steam 
Engine and Other Prime Movers) which is self-explanatory. 

After obtaining the theoretical m.e.p. it is to be corrected for 
clearance and compression which may be done, with sufficient accuracy 
for most purposes, by multiplying the theoretical m.e.p. by .96. 

The actual or expected mean effective pressure may then be obtained 
by multiplying the result by the proper factor from Table 10 from 
Seaton's Manual of Marine Engineering. 

Table 10. — Factors for Obtaining Expected from Theoretical 
Mean Effective Pressure in Steam Engines 
Type of Engine 

Expansive engine, special valve gear or with separate \ 
cut-off valves, cylinders jacketed. / 

Expansive engine having large ports, etc., and good \ 
ordinary valves, cylinders jacketed. J 

Expansive engine with ordinary valves and gear as in 
general practice, unjacketed. 

Compound engines with expansion valve to h.p. cylin- \ 
der, cylinders unjacketed and with large ports, etc. / 

Compound engines with ordinary slide valves, cylinders \ 
jacketed and good ports, etc. J 

Compound engines as in general practice in merchant "1 
marine service with early cut-off in both cylinders, without r . 7 to .8 
jackets and expansion valves. ' 



Factor 



• 94 



.9 to .92 



. 8 to .85 



.9 to .92 



.8 to .85 



THE STEAM ENGINE 



427 



Table ii. — Actual Expansion Ratios 



Per cent. 
of clear- 
ance 



Points of cut-off 



.10 



.I2S 



.20 



.25 



• 30 



-333 



• 37S 



.40 



■ SO 



.60 



.625 



.70 



• 75 



.80 



.875 



.90 



.01 

.0125 
.0150 
.0175 

.02 

.0225 
.0250 

.0275 

.03 

.0325 

.0350 

.0375 

.04 

.0425 

.0450 

.0475 
■ OS 

.0525 
.0550 

.0575 

.06 

.0625 

.0650 

.0675 

.07 



9. 181 
9 ■ 

8.826 
8.659 
8.5 

8.346 

8.2 

8.088 

7-933 

7-792 

7.666 

7-545 
7.428 
7.315 
7.206 

7. 102 

7 

6.901 
6.806 
6.714 

6.625 
6.538 
6.454 
6.373 
6.294 



7.481 

7.363 

7.2s 

7.133 

7.034 

6.932 
6.833 
6.738 
6.64s 
6.555 

6.468 
6.390 
6.303 
6.229 
6.147 

6.082 
6 

S.98S 
5. 861 
5-794 

5. 729 
5.666 
5.605 
5.545 
5 -483 



4. 809 
4.764 
4.720 
4.677 
4-635 

4-595 
4-555 
4.S16 
4.417 
4.440 

4-404 
4.484 
4.333 
4. 298 
4.256 

4.232 

4-2 

4.168 

4.130 

4.106 

4.076 
4-047 
4-045 
3.990 
3 . 963 



3.884 
3.875 
3.830 
3.803 
3.777 

3.752 
3-727 
3.702 
3.678 
3.654 

3.631 
3.608 
3 -58 

3-564 
3-542 

3.521 

3-5 

3-478 

3.459 

3-439 

3-418 
3-407 
3-380 
3-362 
3.342 



3-258 

3.24 

3.222 

3.204 

3-187 

3-170 
3-153 
3-137 
3-I2I 
3.105 

3-089 
3-074 
3-058 
3-043 
3-028 

3.014 

3 

2.986 

2-971 

2-957 

2-944 
2-931 
2.917 
2.904 
2.892 



2.944 
2.930 
2.916 
2.902 
2-889 

2-876 
2.863 
2.850 
2.837 
2. 824 

2.812 
2.800 
2.788 
2.776 
2.764 

2.752 
2-741 
2.730 
2.719 
2.708 

2.697 
2.686 
2.675 
2.66s 
2.655 



2.623 
2.612 
2.602 
2.592 
2.582 

2.574 
2.562 
2.552 
2.543 
2.533 

2.524 
2.515 
2.506 
2.497 
2.488 

2.479 
2.470 
2.461 
2.453 
2-44S 

2-436 
2.428 
2.420 
2.412 
2.404 



2.463 

2.454 
2.445 
2.436 
2.428 

2.420 
2. 411 
2.403 
2.395 
2.387 

2.379 
2.371 
2-363 
2-355 
2.348 

2.340 
2.333 
2.325 
2.318 
2. 311 

2.304 
2.297 
2.290 
2.283 
2.276 



1.983 
1.975 
1.970 
1.966 
1.961 

1.956 
1.952 
1.947 
1.943 
1.938 

1.934 
1.930 

1.925 
1. 921 
1. 917 

1.913 
1.907 
1.904 
1.900 
1.896 

1.892 



1.877 



1.655 
1.653 
1.650 
1.647 
1.64s 

1.642 
1.640 
1.637 
1.634 
1.632 

1.629 
1.627 
1.62s 
1.622 
1.620 

1.617 
1.615 
1. 613 
1.610 
1.608 

1.606 
1.603 
1.601 
1.599 
1-597 



1.590 
1.588 
1.585 
1.583 
1.581 

1.579 
1.576 
1-574 
1-572 
1.570 

1.568 
1-566 
1.563 
1.561 
1.569 

1.557 
l.SSS 
1.SS3 
I. 551 
1.549 

1-547 
I -545 
1.543 
I. 541 
1.539 



1 .422 
I. 421 
1.419 
1. 418 
1.416 

I-41S 
I-413 
1.412 
1.410 
1.409 

1.408 
1.406 
1-405 
1.404 
1.402 

1.401 
1.400 
1.398 
1-397 
1.396 

1-394 
1.393 
1.392 
1-390 
1-389 



1.328 
1.327 
1.326 
1.325 
1.325 

1.324 
1.322 
1-321 
1.320 
1. 319 

1.318 
I-317 
I-316 
I.31S 
I -314 

I-313 
I-312 
I. 311 
1.310 
1-309 

1.308 
1.307 
1.306 
1.305 

1.304 



1.246 
1.246 
1.245 
1.244 
1-243 

1.243 
1 . 242 
I. 241 
1.240 
1.240 

1.239 
1.238 
1.238 

1.237 
1.236 

1.235 
1.23s 
1-234 
1-233 
1.233 

1.232 
1-231 
1-231 
I -230 

1.229 



1. 141 
1. 140 
1. 140 
1.140 
1. 138 

1.138 
1.138 
1. 138 
1.138 
1.138 

1. 137 
1.136 
1.136 
1.136 
1. 135 

1.135 
1.135 
1.134 
1.134 
1-134 

1-133 
1-133 
1-132 
I-132 
1. 132 



1. 109 
1. 109 
1. 109 
1.108 
1. 108 

1.108 
1. 108 
1. 107 
1. 107 
1. 107 

1.106 
1. 106 
1. 106 
1. 106 

1. 105 

1. 105 
1.105 
1. 104 
1. 104 
1.104 

1. 104 
1.103 
1.103 
1.103 
1. 103 



Table 12. — Horse-power of Single-cylinder Steam Engines 
PER Lb. of Mean Effective Pressure 



Diameter of 


Diameter 
of piston- 
rod, ins. 


Speed of piston in ft. per min. 


cyl., ins. 


ift. 


400 ft. 


500 ft. 


600 ft. 


700 ft. 


10 


Ii 


.00234 


.936 


1.17 


1.404 


1.638 


II 


li 


.00284 


1. 136 


1.42 


1.704 


1.988 


12 


. 2 


.00338 


1-352 


1.69 


2.028 


2.366 


13 


2 


-00397 


1-588 


1.98s 


2.382 


2.779 


14 


2i 


. 00460 


1.84 


2.30 


2.76 


3-22 


15 


2| 


.00529 


2.116 


2.64s 


3-174 


3-703 


16 


2i 


.00602 


2.408 


3-01 


3-612 


4-214 


17 


2f 


.0068 


2.72 


3-40 


4-08 


4.76 


18 


2i 


-00762 


3.048 


3-81 


4-572 


S.334 


19 


2i 


. 00849 


3.396 


4 -24s 


5-094 


S-943 


20 


3 


.00941 


3.764 


4-70S 


5.646 


6-587 


21 


3i 


.01038 


4.152 


5-19 


6.228 


7-266 


22 


3i 


.01139 


4.556 


5.69s 


6.834 


7-973 


23 


3l 


.01245 


4.98 


6.225 


7.47 


8.715 


24 


3h 


.01356 


5-424 


6.78 


8.136 


9.492 


25 


3S 


.01472 


5-888 


7.36 


8.832 


10.304 


26 


3i 


.01592 


6.368 


7.96 


9.552 


11.144 


27 


3s 


.01717 


6.868 


8.58s 


10.302 


12.019 


28 


4 


.01847 


7.388 


9.23s 


11.082 


12.929 


29 


4J 


.01981 


7.924 


9-905 


11.886 


13.867 


30 


4i 


.02121 


8.484 


10.605 


12.726 


14-847 


31 


4l 


.02264 


9.056 


11.32 


13.584 


15-848 


32 


4i 


.02413 


9.652 


12.065 


14.478 


16.891 


33 


4f 


.02566 


10.264 


12.83 


15.396 


17.962 


34 


4l 


.02724 


10.896 


13.62 


16.344 


19.068 


35 


4l 


.02887 


11.548 


14-435 


17.322 


20.209 


36 


5 


•0305s 


12. 22 


1S.27S 


18.33 


21-385 


37 


Si 


.03227 


12.908 


16.135 


19.362 


22.589 


38 


Si 


.03404 


13.616 


17.02 


20-424 


23.828 


39 


51 


.03585 


14.34 


17.92s 


21-51 


25.09s 


40 


Si 


.03772 


15.088 


18.86 


22.632 


26.404 


41 


Sf 


•03963 


15.852 


19-815 


23.778 


27.741 


42 


Si 


-04159 


16.636 


20.795 


24-954 


29.113 


43 


51 


.04360 


17.44 


21.80 


26.16 


30.52 


44 


6 


.04565 


18.26 


22.825 


27.39 


3I.9S5 



Actual expansion ratios at various points of cut-o£f when the clear- 
ance is taken into account are given in Table 11 by Robert Grim- 
SHAW {Amer. Mach., Jan. 20, 1883). 

The horse power of engines per lb. m.e.p. may be taken from 
Table 12. 

To lay out the hyperbolic or isothermal expansion curve proceed as 
in Fig. 3. Locate the clearance line AO and the line BO of absolute 
vacuum. Through any point, as C, draw CE parallel and CD 
perpendicular to the atmospheric line. Draw radiating lines OD, 
OL, OM, etc., and from £>, L, M, etc., and F, H, J, etc. draw horizon- 
tals and perpendiculars intersecting at G, /, iiT, etc., which are points 
of the required curve passing through C. 




B O 

Fig. 3. — Laying out the hyperbolic or isothermal expansion curve. 

Construction and Dimensions of Parts 

Current practice in the dimensions of steam-engine parts formed the 
subject of an investigation by O. N. Trooien {Bulletin No. 252 of 
the University of Wisconsin) from which the following is taken. 

Particulars were obtained of a large number of engines ranging 
between 20 and 400 rated h.p. The data secured were first tabu- 
lated and separated into classes and subclasses, the two main 
classes being high-speed or quick-revolution engines and low-speed 
or slow-revolution engines (the latter class being principally the 



d28 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Corliss). Divisions into subclasses were made in the treatment of 
such parts as the crank pin for center-crank and side-crank engines, 
while in dealing with such parts as the piston rod or crosshead pin, 
no such division was thought necessary. 

The following symbols of notation are used in the formulas given : 
D = diameter of piston, ins. 
A =area of piston, sq. ins. 
i= length of stroke, ins. 

^ = unit steam pressure, taken as 125 lbs. per sq. in. above ex- 
haust as a standard pressure. 
H.P. = rated horse-power. 
iV = revolutions per minute. 
C = a constant. 
K = a. constant. 

d = diameter of unit under consideration, ins. 
Z = length of unit under consideration, ins. 
The commercial point of cut-off was taken at one-fourth of the 
stroke. 

Other notation than the above is explained as used. 
Diameter of pislon rod : 

d = C\/DL 
L being the free length. 

Values of C for high-speed engines: Mean .15; maximum .187; 
minimum .125. For Corliss engines: Mean .114, maximum .156; 
minimum .1. 

Thickness of cylinder wall : 

t = CD + . 2& in. 
in which i = thickness, ins., 

£) = diameter of piston, ins., 
C = a constant. 
Values of C: Mean .054; maximum .072; minimum .035. No 
characteristic difference was found between high- and low-speed 
engines. 

Diameter of cylinder-cover stud holts : 

d = CD+i in. 
in which (i = diameter of bolts, ins., 

D == diameter of cylinder, ins. 
C = a constant. 
Mean value of C = .o4. 
With only one exception the smallest diameter of bolts used in the 
high-speed engines was f in., and in the Corliss engines the smallest 
value was i in. 

The mean thickness of cylinder flanges for holding cylinder covers, 
where these were bolted to cylinder flanges, was found to be 1.12 
times the thickness of cylinder wall, for both high-speed and Corliss 
engines. 

The thickness of cylinder cover at the center seems to vary a great 
deal, but for the engines examined it may be taken as 2.75 times the 
thickness of the cylinder wall for high-speed engines and 1.12 times 
the thickness of cylinder wall for Corliss engines. 
Number of stud holts for cylinder covers: 

N = CD 

in which N = the number of bolts 

Mean value of C = .72 for high-speed engines, and .65 for Corliss 
engines. 

The least number of bolts used for any engine was found to be six. 
For additional information on cylinder-cover joints and bolts, in- 
cluding a chart for the diameter and number of bolts, see below. 

The clearance volume was found to vary from 5 to 11 per cent. 
in high-speed engines and from 2 to 5 per cent, in Corliss engines. 

Ratio of length of stroke to diameter of cylinder in engines having 
a speed greater than 200 r.p.m.: 

L = CD 

Values of C: Mean 1.07; maximum 1.55; minimum .82. 



Ratio of length of stroke to diameter of cylinder in engines having a 
speed between no and 200 r.p.m.: 

L = CD 
Values of C: Mean 1.36; maximum 1.88; minimum 1.03. 
Ratio of length of stroke to diameter of cylinder in engines having a 
speed less than no r.p.m. (Corliss engines) : 
L = CD+?, ins. 
Values of C: Mean 1.63; maximum 2.40; minimum 1.15. 
Face of piston in terms of diameter: 

■w = CD 
or w = CD-\-i in. 

in which w = width of piston, 

D = diameter of piston, 
C = a constant. 

Using w = CD: 

Values of C for high-speed engines: Mean .40; maximum .47; 
minimum .30. For Corlis engines: Mean .32. 

Using the equation •w = CD+i in: 

Values of C for high-speed engines: Mean .32; maximum .40; 
minimum .24. For Corliss engines: Mean .26. 

The box type seems to be the prevailing form of piston. The 
thickness of shell of piston in high-speed engines is about .6 of the 
thickness of cyhnder wall, and for Corliss engines this ratio is 
about .7. 

The prevailing number of rings used for the piston is two, and 
the rings are usually turned to a diameter J in. larger than the bore 
of the cylinder. For additional details of pistons see below. 

Piston speed — high-speed engines: 

Mean 605, maximum goo, minimum 320 ft. per min. 

Piston speed — Corliss engines: 

Mean 592, maximum 800, minimum 400 ft. per min. 

Area of cross-head shoes: 

a = CA 
in which a = area of cross-head shoes : 

Values of C: Mean .53; maximum .72; minimum .37. 

Pressure on cross-head shoes, steam being assumed to follow as 
far as half stroke: 

n C 
in which 5 = pressure on shoes, lbs. per sq. in., 
length of connecting rod 
length of crank 
For high-speed engines n may be taken as 6 and for Corliss engines 
as 5.5. Values of 5 for high-speed engines: Mean 39.5; maximum , 
57; minimum 28. For Corliss engines: Mean 43; maximum 61; 
minimum 32. 

Under normal conditions of shorter cut-off these values are 
materially reduced. 

Length of hearing part of cross-head pin in terms of its diameter: 

l = Cd 

Values of C for high-speed engines: Mean 1.25; maximum 1.5; 
minimum i. For Corliss engines: Mean 1.43; maximum i.g; mini- 
mum I. 

Dimensions of cross-head pin: 

dl = KA 

Values of K for high-speed engines: Mean .10; maximum .15; 
minimum .037. For Corliss engines: Mean .115; maximum .19; 
minimum .037. 

Cross-section of connecting rod of high-speed engines at the middle 
of its length: 

h=-Ch 
in which /? = height, 
J = breadth. 



THE STEAM ENGINE 



429 



Values of C: Mean 2.28; maximum 3; minimum 1.85. 
Dimensions of cross-section of connecting rod of high-speed engines 
at the middle of its length: 

b = C\/DLc 
in which Z,c = length of rod between centers. 

Values of C: Mean .073; maximum .og4; minimum .05. 
Dimensions of cross-section of connecting rod of Corliss engines 
(circular section only) : 

d = CVDLc 
Values of C: Mean .092; maximum .104; minimum .081. 
Length of crank pin in terms of its diameter: 
l = CD 

Values of C for high-speed engines: Mean .87; maximum 1.25; 
minimum .66;. For Corliss engines: Mean 1.14; maximum 1.30; 
minimum i. 

Diameter of crank pin: 

d = CD 

Values of C for high-speed center-crank engines: Mean .40; maxi- 
mum .526; minimum .28. For side-crank Corliss engines: Mean 
.27; maximum .32; minimum .21. 

Diameter of main journal of high-speed center-crank engines: 

= \H P 

Values of C: Mean 6.6; maximum 8.2; minimum 5.4. 
For Corliss engines this dimension seems best expressed by the 
form: 



d = C 



3 JH.P. \ 



Values of C: Mean 7.2; maximum 8; minimum 6.4. 

Length of main journal in terms of its diameter: 
l^Kd 

Values of K for high-speed center-crank engines: Mean 2.1; maxi- 
mum 2.9; minimum 1.6. For Corliss side-crank engines: Mean 1.9; 
maximum 2.2; minimum 1.62. 

Projected area of main journal in terms of piston area: 

dl=FA 

Values of F for high-speed center-crank engines: Mean .48; maxi- 
mum .78; minimum .32. For Corliss side-crank engines: Mean .6; 
maximum .66; minimum .5. 

For additional data on bearings of steam engines see Index and 
below. 

Weight of fly-wheel : 



w=cx 



H.P. 



-DhN^ 

in which IF = total weight of wheel, lbs. 

This relation gives fairly satisfactory results for high-speed engines 
up to about 175 horse-power, and for this range the values of C are: 
Mean 1,300,000,000,000; maximum 2,800,000,000,000; minimum 
660,000,000,000. 

When high-speed engines of larger size are considered, the relation 
seems better expressed by: 

H.P. 



W-- 



^CX;p 2^^3+1000 



Values of C:Mean 720,000,000,000; maximum 1,140,000,000,000; 
minimum 330,000,000,000. 

A somewhat greater uniformity seems to exist among the builders 
of standard Corliss engines. In these engines the relation seems best 
expressed by: 

DhN' 
Values of C: Mean 890,000,000,000; maximum 1,330,000,000,000; 
minimum 625,000,000,000. Corresponding values of K: Mean 4000; 
maximum 6000; minimum 2800. 



For additional information on the weight of steam-engine fly- 
wheels see Fly-wheels. 

Diameter of fly-wheel in terms of length of stroke: 
Di = CL 
in which Z'i = outside diameter of wheel, ins. 

Values of C for high-speed engines: Mean 4.4; maximum 5; mini- 
mum 3.4. For Corliss engines: Mean 4.4; maximum 5.25; minimum 

3-25- 

Belt surface per indicated horse-power: 

S = CXH.P 
in which 5 = velocity of wheel rim, ft. per min., multiplied by the 
width of belt, ft. 

Values for C for high-speed engines; Mean 26.5; maximum 55; 
minimum 10. 

For Corliss engines, this relation seems better expressed by: 
S = CXH. P-\-iooo 

Values of C: Mean 21; maximum 35; minimum 18.2. 

For additional data on main belts for steam engines see Belts. 

Velocity of fly-wheel rim in ft. per sec; for high-speed engines : 
Mean 70, maximum 82, minimum 48 ft. per sec. 

For Corliss engines: Mean 68, maximum 82, minimum 40 ft. per 
sec. 



Weight of reciprocating parts: 



W = CX 



PL 

LN 



in which IF = weight of reciprocating parts (piston, piston rod 
cross-head and one-half the connecting rod), lbs. 

Values of C for high-speed engines (data not obtained for Corliss 
engines): Mean2,ooo,ooo;maximum3,4oo,ooo; minimum 1,370,000 

For the cases where the information was obtainable, the balance 
weight opposite the crank pin was found to be about 75 per cent, 
of the weight of the reciprocating parts. 

Total weight of engine in terms of horse-power: 

IF = CXH.P. 

in which IF = total weight of engine, lbs. 

Values of C: Mean 82; maximum 120; minimum 52. 

For direct-connected engines, the weight of the engine without 
the generator was found to be from 10 to 25 per cent, greater than 
the weight of belt-connected engines of the same capacity. 

Values of C for Corliss engines: Mean 132; maximum 164; mini- 
mum 102. 

The dimensions of main bearings of large engines, to avoid undue 
heating, according to the practice of the late Edwin Reynolds, should 
be such that the product of the square root of the speed of rubbing 
surface in ft. per sec. multiplied by the pressure in lbs. per sq. in. 
of projected area should never exceed the constant number 375 
for an horizontal engine, or 500 for a vertical engine when the shaft 
was lifted at every revolution. 

Locomotive main driving boxes in some cases give a constant as 
high as 585, but this is accounted for by the cooling action of the 
air. 

Using this principle, Figs. 4 and 5 have been constructed by F. 
W. Salmon {Amer. Mach., Sept. 17, 1903). Fig. 4 gives the velocity 
of rubbing in ft. per min. and per sec. for shafts from 4 to 1 7 ins. 
diameter and for speeds from 60 to 140 r.p.m., and Fig. 5 gives the 
loads per sq. in. for various velocities and for various constants. 

The dimensions of the main frames of steam engijies have received 
less attention in discussion than any other feature. When the Cor- 
liss (girder) frame was more popular than now, the author made an 
examination of a good many such frames {Amer. Mach., Feb. 14, 
1895). While the resulting data have small application to other 
types of frames they are given here in the absence of others. The 
method of comparison was to compute from measurements of the 
frames the number of sq. ins. in the smallest cross-section, that 



430 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 




90 100 110 120 130 140 250 300 400 500 600 

Revolutions per Minute of Journal Velocity of Rubbing. Ft. per Min, 

Fig. 4. Fig. 5. 

From the r.p.m. in the base line of Fig. 4 trace upward to the diagonal for the diameter of the journal, thence horizontally and 
read velocity of rubbing. Find this velocity in the base line of Fig. 5, trace upward to the curve for the selected constant, thence 
horizontally and read the appropriate pressure per sq. in. The journal is then to be of a length which will bring the pressure per sq. 
in. down to this figure. 

Figs. 4 and 5. — Rubbing velocity and safe bearing pressures on main journals of steam engines. 



is, immediately behind the pillow block, also the compute the total 
maximum pressure upon the piston, and to divide the latter quantity 
by the former. The result gives the number of lbs. pressure upon 
the piston allowed for each sq. in. of metal in the frame. This, 
while not the actual strain upon the metal, is strictly compara- 
tive and that is all that is required for the purpose. 

Representative figures resulting from the examination are given 
in Table 13. 

Table 13. — Dimensions of Smallest Section of Corliss Engine 

Frames 



Size of engine 


Lbs. 


persq. 


in. of smallest section of frame 


10X30 






217 


12X36 






248 


18X42 






278 


24X48 






360 


24X48 






395 


28X48 






S7S 


30X60 






350 



It will be observed that, speaking generally, the strains increase 
with the size of the engine and that more cross-section of metal is 



allowed with relatively long strokes than with short ones, both of 
which are as we should expect. Other data of a more miscellaneous, 
character show loads of about 300 lbs. on short stroke engines of 
about 10 ins. diameter of cylinder, while one memorandum of a 32- 
in. engine which had been running for many years without any indi- 
cation of weakness gives a strain of 667 lbs. 

From the above the author formulated the general rule that in 
engines of moderate speed, and having strokes up to one and one- 
half times the diameter of the cylinder, the load per sq. in. of smallest 
section should be for a lo-in. engine 300 lbs., which figure should be 
increased for larger bores up to 500 lbs. for a 30-in. cylinder of the 
same relative stroke. For high speeds or for longer strokes the load 
per sq. in. should be reduced in accordance with good judgment. 

The following additional particulars of steam-engine parts are from 
Seaton's Manual of Marine Engineering: 

Fratne bolts: Stress not to exceed 4000 lbs. per sq. in. at bottom 
of thread or, for a large number of small bolts, 3000 lbs. When 
possible add 20 per cent, to the cross-section as given by this rule. 

Cylinder covers: When above 24 ins. diameter for high- and 40 
ins. for low-pressure cylinders, cylinder covers should be made hollow 
with a depth at the center of about j the diameter of the piston. 



THE STEAM ENGINE 



431 



Pitch of cylinder-cover bolts should not exceed 



ViXioo . 
1 
P 



in which 



i = thickness of cover flange in i6ths in. and ^ = pressure on cover, 
lbs. per sq. in. 
Flat surfaces of cast-iron sustaining steam pressure should be stifi- 

1^2 y CO 

ened by ribs of a pitch not greater than -vl =_ in which i = 

thickness in i6ths in. and p= pressure, lbs. per sq. in. Ribs to be 
of the same thickness as the flat surface and of a depth = 25 times 
the thickness. 
Piston of the follower type: Compute 

x = ~ V*+i 
SO 



the ring should be released from the face plate and be reclamped 

before making the finishing cut. 

A reverse procedure is used in making the pattern. If the pattern 
is turned up round, it will be found when the gap is cut out and closed 
up, that the casting will have so departed from its circular shape that, 
unless excessive finish has been allowed on the pattern, the ring will 
not clean up at all points, and where the cut is heavy, the ring will 
be thin after turning. To obviate this, make the pattern of pattern 
size with usual finish, as though the ring were to be solid and of the 
same size as the cylinder bore. Next saw the pattern apart where 

the gap of the ring is to be, and insert a piece the size of the gap. 



n 



in which D = diameter of piston, ins, 

^ = effective pressure, lbs. per sq. in.. 

Number of ribs in piston 



D+20 
12 

Thickness of ribs in piston =.i8a; 

Thickness of front of piston near hub = .2X 
Thickness of front of piston near rim = .170; 



Thickness of back of piston 


= .i8:*; 


Thickness of hub around rod 


= .3X 


Depth of piston near center 


= l.4X 


Diameter of follower bolts 


= .ixXi in. 


Pitch of follower bolts 


= 10 diameters 


Slide valve rod: 




Diameter of valve rod = - 


JLXBXp 





- ^mmmm 

Fig. 8. Action of a Snap 
Piston Ring under 

Pressure 




Fig. 6. A Snap Piston Eing 



JFiG. 9. Steam Blowing Past 
a Lapped Joint 



;n which Z, = length of valve, ins., 
-B = breadth of valve, ins., 
^ = maximum pressure, lbs. per sq. in., 
/^ = 12,000 for long steel rods, 
jF = 14,500 for short steel rods. 

Slot links for link motion: 

Let i) = diameter of valve rod as above, taking 
F= 12,000 

Diameter of block pin if secured at one end 
only =D 

Diameter of block pin if secured 
at both ends =.7S-D 

Diameter of eccentric-rod pins = .7D 

Diameter of suspension-rod pins 
if secured at both ends =•55^' 

Diameter of suspension-rod pins 
if secured at one end only =-75D 

Breadth of link = .8D to .gD 

Length of block = 1.6Z) to i.8Z> 

Thickness of bars of link =-7-D 

Diameter of suspension rod if but 
one =-7D 

Diameter of suspension rods if 
two' =•55-0 

The dimensions of snap piston rings may be de- 
termined from Figs. 6 and 7 by the author 
{Amer. Mach., June 3, 1909). The ring is cast 
large with two lugs on the inside as shown in 
Fig. 6. After being cut off and faced to thickness, the gap is cut 
out. The ring is then sprung together by a clamp on these lugs, 
when it is strapped to a face plate, or, better, clamped between two 
flanges of a special fixture, and is put in the lathe and turned to the 
true size of the cylinder bore. When the clamp is relieved, the ring 
expands again, but not to a circular shape; but when put in its 
place in the cylinder, it resumes its circular form, or very near it, 
and is a fit all around, as it should be. To secure the best results 































































^ 
























































^ 


^ 






2 
















































,..-' 


^ 
























































^ 




























































^ 




















































c 


^ 


^ 


^ 
























LH 
















































































































































^ 


■^ 
























































^ 


^ 












































1 








































































" 








































. 






— 






































h 







— 




' 










































■— 










n 



















— 






— 






h 










— ' 






















d 




— 





































_ 


















































































































n 




i- 


















_ 




__, 
















_ 


L 














_ 





6 8 10 12 14 16 18 20 

Diameter of Cylinder^ Ins, 

a= .0268 <i+.o893 
6= .0312 d-\- . 25 
c= . 1071 d-Y . 1071 
d being the diameter of the cylinder and the remaining notation, as in Fig. 6. All dimen- 
sions in inches 
Fig. 7. — Dimensions of snap piston rings. 
Figs. 6 to 9. — Snap piston rings and their action. 

This will spring the pattern outward to a non-circular form such that 
when the casting is cut and sprung inward for turning, it will be nearly 
a true circle in the rough, with a fairly uniform allowance for finish 
all around and with a gradually tapering thickness, as intended. 

Should the rings not come out exactly as intended, the case can be 
met to a certain extent by changing the width of the gap as there is 
no nicety about this dimension. 

Large numbers of rings have been made to the dimensions of the 



432 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



chart and with entire success — the extreme size made being of 28 ins. 
diameter. 

Regarding the large increase in the width of gap over the more 
customary dimensions there is nothing to be said against it, as the 
strength of such rings is not sufficient to bring about any undue 
pressure against the bore of the cylinder, while it has the advantage 
that it reduces the danger of breakage when putting the rings in place 
— especially with the smaller sizes. The fact is that the rubbing of 




Table 


14 


. — Dimensions for Piston Ends 


OF 


Piston Rods 




A 


B 


C D 


E 


F 


G 


7/ 


7 


K 


L 


M 


N 


P ! R 1 5 


ins. 


ins. 


ins. 


ins. 


ins. 


ins. 


ins. 


ins. 


ins. 


ins. 


ins. 


ins. 


ins. 


ins. 


ins. 




2i 


3i 


3 


2 J 


4\ 


3f 


3j 


2f 


sl 


3l 


3i 


3 


li 


2 


4l 


8 


3 


3i 


3i- 


2\ 


A\ 


3i 


3i 


2| 


5 J 


4i 


4l 


1 


li 


2 


Si 


8 


3l 


3l 


3J 


3 


5^ 


4i 


4i- 


2f 


61 


4l 


4i 


1 


li 


2 


Sl 


8 


3 5- 


4 


3i 


3i 


5^ 


A\ 


3l 


2f 


7l 


5 


4i 


r 


li 


2I 


6 


6 


3i 


4 


3! 


3i 


b\ 


S\ 


4J 


2i 


81 


5 


4 5 


r 


12 


2i 


6 


6 


3i 


4i 


4 


Z\ 


(>\ 


s\ 


4i 


2| 


8i 


5i 


Sl 


I 


1) 


2i 


6i, 


6 


4 


Ah 


Ai 


31 


61 


%\ 


4l 


2\ 


8i 


Sl 


Sl 


I 


Is 


2| 


6i 


6 


4 


4l 


4i 


Z\ 


7 


6 


Sf 


25 


8f 


Si 


Sl 


I 


li 


2i 


6i 


6 


4i 


4l 


4i 


4 


7 


6 


5f 


3\ 


9s 


61 


6 


I 


2 


3 


7i 


6 


4i 


5 


A\ 


4i 


7 


6 


Si 


3l 


9\ 


61 


61 


I 


2 


3 


7i 


6 




Table 15. — Dimensions for Crosshead Ends of Piston Rods 



A 


B 


c 


D 


E 


f 


G 


H 


7 


K 


L 


ins. 


ins. 


ins. 


ins. 


ins. 


ins. 


ins. 


ins. 


ins. 


ins. 


ins. 


2| 


2| 


iH 


Sl 


Sl 


1 


6 


1 5 


2| 


1 


9 


3 


2i 


iM 


Sl 


Si 


1 


6 


li 


2f 


4 


9 


3J 


3^ 


2A 


6i 


6 


1 


61 


If 


2| 


1 


9 


3i 


3f 


2A 


61 


6 


1 


61 


il 


2| 


1 


9 


31 


3l 


2 11 


7 


6h 


3 


7i 


il 


3 


1 


loj 


4 


3l 


2II 


7 


(>i 


1 


7i 


il 


3 


i 


loj 


4i 


4J 


3^ 


7-1 


7 


1 


7l 


2 


3i 


I 


Hi 


4i 


4i 


3-h 


li 


7 




7l 


2 


3i 


I 


Hi 



the rings against the cylinder wall introduces a sort of peening action 
and no matter what their original strength they soon lose most of it. 

The force which presses the ring against the cylinder wall is chiefly 
the steam pressure, compared with which the force exerted by the 
strength of the ring is insignificant. 

Referring to Fig. 8, in which the clearances are exaggerated for 
clearness, the steam comes down the joint between the piston and the 
cylinder bore from right to left, and flowing down the joint ab, 
establishes full pressure in that joint and below the ring. As shown 



experimentally by Prof. S. W. Robinson, the steam also estab- 
lishes a creeping film in the joint ad, beginning at full pressure at a 
and ending at such pressure at d as may exist at the left of d, the 
average pressure of the film being about the mean of the initial and 
the terminal pressures. Under these circumstances the outward 
pressure prevails and the ring is forced against the cylinder bore. 
It is doubtful if the eccentric construction has much value beyond 
satisfying the feeling that it is appropriate for the purpose. 

A consideration of the action described in connection with Fig. 8 
will show that the practice which some follow, of placing two rings 
in a groove is wrong. The average pressure per sq. in. of the surfaces 
in contact with the cylinder bore is substantially the same with two 
rings as with one, while the two rings, having twice the surface, exert 
twice the total pressure and double the tendency to wear the cylinder. 
On the other hand if the rings are placed in separate grooves the 




Table 16. — Dimensions of Crossheads 



Diameter 




















of rod 




















A 


B 


C 


D 


E 


F 


G 


H 


J 


K 


ins. 


ins. 


ins. 


ins. 


ins. 


ins. 


ins. 


ins. 


ins. 


ins. 


2I 
3 


6i 


Sl 


6 


Il 


3 


S 


loi 


li 


2M 


3i 
3i 


7i 


6i 


7 


2 


3i 


Sl 


12 


il 


3i 


3i 
4 


8 


7 


7i 


2 


3I 


Ci 


13 


II 


3i 


4i 

4i 


9 


8 


8i 


2i 


61 


61 


14 


^ 


3l 



second one has only to deal with the pressure due to the leakage past 
the first, there being a progressive reduction of pressure from ring to 
ring, the tendency to leak and to wear being measured by the difier- 
ence of pressure on the two sides of a ring. The fact that two rings 
in the same groove may be so placed as to break joints has no value, 
as the action is precisely the same as that of the lapped joint referred 
to in the next paragraph. 

A feature of these rings, as sometimes made, which serves no useful 
purpose is the lap joint, shown in Fig. g. With such a Joint the 
steam foUows the course indicated by the arrow and escapes as 
freely as though the lap were absent. The only good effect of this 
form of joint is to prevent the tendency to streak the cylinder, due 
to a plain square joint, but this can be obviated just as effectively 
by cutting the joint at an angle. 

Eccentric rings being heaviest opposite the joint, they have a 
tendency, in horizontal cylinders, to work around to a position with 
the joint at the top of the piston, where the steam may blow through 
freely. To prevent this these joints should be placed at or near the 
bottom of the piston and pins be inserted in the grooves to keep them 
there. 

The practice of scraping the rings into their grooves when followers 
and junk rings are used, represents wasted eSort as a consideration 
of Fig. 8 will show. Moreover, the accumulation-of oil residue tends 
to stick such rings fast and prevent their expansion. Up to the point 
where noise results, a slight degree of looseness sideways is advan- 
tageous, as it tends to prevent this action. 



THE STEAM ENGINE 



433 



Locomotive practice in dimensions of pistons, piston rods and 
steel crossheads, as drawn up by a committee of the Amer. Ry. M.M. 
Asso. {Amer. Mach., June 29, 1911) is given in Tables 14, 15 and 16. 

The taper fit of piston rods in pistons and crossheads is almost 
universal, but the author can see no reason for it. His own practice 




Fig. 10. — Piston valve without packing rings. 




A construction -of piston valve which avoids the use of packing rings 
is. shown in Fig. 10 {W . H. Booth, Amer. Mach., May 21, 1896). 
After rough turning, the valve is slotted longitudinally, compressed 
by an encircling clip, turned at the ends and a V-piece fitted in a 
V-recess on the inside of the valve, the slot through the body being 

along the apex of the V. The V-piece 
is fitted with a spring and wedge com- 
bination or other means of setting up. 
The ends are then bolted on, the en- 
circling clip removed and the valve 
is finished to its correct diameter. 
Thus made, it has an initial elas- 
ticity and does not require any ex- 
panding pressure from the V, the 
duty of which is merely to close the 
longitudinal slot. 

Large piston valves of this construc- 
tion as made by the Union Iron 
Works {Amer. Mach., Oct. 12, 1905) 
are shown with a few leading dimen- 
sions in Fig. II. The object of the 
deep circumferential recesses aa is 
to facilitate heating of the seats and 
thus diminish distortion. The ad- 
mission of steam by the high-pres- 
sure valve is by its inside and by 
the intermediate and low-pressure 
valves by their outside edges. The 
rings forming the valves proper are 
split longitudinally and have tongue 
pieces — not shown — in the joints as 
shown in Fig. 10. 



-6'9'4'' 



Cylinder Cover Joints 




Fig. II. — Large piston valves for steam engines. 



was to make them a straight sliding fit, bottoming at the end for the 
crosshead and against a shoulder for the piston. The straight fit is 
much cheaper, not only as regards the actual fits but because the 
rod can be made to measure. The taper fit is chiefly a matter of 
habit and tradition. 
28 



The diameters of cylinder and tank 
head bolts may be obtained from 
Fig. 12, by F. K. Caswell {Amer. 
Mach., July 7, 1898), the use of 
which is shown by the example 
below it. 

The unit stress for cylinder head 
bolts in good practice is about 4000 
lbs. per sq. in. on the net section, or 
3500 lbs. if the bolts are less than f 
in. diameter, though stresses up to 
8000 lbs. are used. 

The adjustment of the diameter to 
the number of bolts to give the re- 
quired total cross-section is so made 
that the distance between bolts 
shall not be so great as to endanger 
tightness of the joint. For informa- 
tion on this point see above. Pro- 
vision for tightness with small cylin- 
ders gives an excess of strength 
when customary sizes of bolts are 
used. 

The common gasket joint of cylin- 
der covers is a common nuisance. 
Fig. 13 shows the joint used on the 
Straight Line Engine and on the engines of the Ball Engine Co. 
The joint is not ground but simply faced in a lathe without special 
care or workmanship. The only essential for its success is that it be 
narrow — not over | in. wide. The distance between stud centers 
should not exceed about four times the thickness of the cover flange. 



434 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Initial Steam Pressure or Static Pressure. 

Lbs. per Sq. In, 
50 100 150 200 250 




40 



5. 



35 30 25 20 15 10 

Total Bolt Area: Sq. Ins.of Net Section 
To find the number and diameter of bolts for a steam cylinder head of i8 ins. diameter, subjected to a pressure of 175 lbs. per 
sq. in. : Find 175 on the pressure scale at the top, trace downward to the cylinder diagonal, 18, thence horizontally to the stress 
diagonal, say 4500 lbs. per sq. in., thence downward to the J-in. diagonal and thence horizontally and read, at the right, 25, the 
number of bolts, or, from the intersection with the stress diagonal trace downward to the bottom and read 9.75 sq. ins. the total 
net bolt section. By tracing vertically from the pressure per sq. in., 175 lbs., to the cylinder diagonal, 18, and thence horizontally 
to the left, the total pressure on the head, 44, may be read in thousands of lbs. 

Fig. 12. — Diameter and number of cylinder and tank head bolts. 



THE STEAM ENGINE 



435 



Stuffing box glands should be made as shown in Fig. 15 — not as 
usual, as shown in Fig. 14. With the form shown in Fig. 14, leakage 
is apt to take place around the outside of the packing, though the 
gland be more than tight enough to stop leakage around the rod. 
The bottom of the box should be flat — not beveled, as in Fig. 14. 

Areas of Ports and Pipes 

The areas of steam ports and pipes, as determined by an investigation 
of 165 single-cylinder engines ranging from 20 to 740 h.p. by Prof. 
John H. Bare {Trans. A. S. M. E., Vol. 18), may be expressed by the 
formula : 

AV 

in which' a = area of port or pipe, sq. ins., 
A =area of piston, sq. ins., 
F = velocity of piston, ft. per min., 
C = mean velocity of steam in port or pipe, ft. per min. 




Fig. 13. — Cylinder cover joint of the Straight-Line engine. 




Fig. 14. Fig. 15. 

Figs. 14 and 15. — Correct and incorrect construction of stuffing 
box glands. 

For high-speed engines using the same port for both admission and 
exhaust, the values of C are: Mean 5500; maximum 6500; minimum 
4500. For Corliss engine steam ports: Mean 6800; maximum 9000; 
minimum 5000. For Corliss engine exhaust ports: Mean 5500; 
maximum 7000; minimum 4000. 

For high-speed engine steam pipes the values of C are: Mean 6500; 
maximum 7000; minimum 4800; For Corliss engine steam pipes: 
Mean 6000; maximum 8000; minimum 5000. 

For high-speed engine exhaust pipes the values of C are: Mean 
4400; maximum 5500; minimum 2500. For Corliss engine exhaust 
pipes: Mean 3800; maximum 4700; minimum 2800. 

In the case of plain slide-valve engines it has long been taught that 
the port should be opened to steam about f of its width, but prevail- 
ing practice with high-speed single-valve shaft-governor engines has 
shown that a simple constriction in a steam passage does not obstruct 
the flow of steam as much as has been supposed and that so large 
an opening is unnecessary. 

Fig. 16 gives indicator cards from such an engine, with loXio-in. 
cylinders {Amer. Mach., Dec. 6, 1900), taken under the following 
conditions: The engine was loaded with a friction brake, so that the 



load could be varied, and cards were taken at various loads. The 
valve rod had a sharp point attached, which was made to scribe a line 
on a strip of tin pressed against it at the instant of taking the card. 
In this way a record of the exact valve travel at the instant was 
obtained, and by working backward through the known dimensions 
of the valve and ports, the exact openings which gave the various 
cards were determined. As it was impracticable to insert the steam 




Card 1. Speed 301 R.P.M. Valve Travel 
1.745 Ins. 




Card 2. Speed 301 R.P.M. Valve Travel 
1.755 Ins. 




Cards. Speed 300 E.P.M. Valve Travel 
1.86 Ins. 




Card 4. Speed 301 R.P.M, Valve Travel 
1.98 Ins. 




Card 5. Speed 303 R.P.M. Valve Travel 
2.37 Ins. 

Fig. 16. — Effect of small port openings on indicator cards. 

gage in the steam chest, it was placed in the steam pipe 20 ft. from 
the engine. The gage had been recently tested. No doubt the pres- 
sure in the chest was somewhat below that shown by the gage, and 
this loss due to the 20 ft. of pipe is, in the diagrams, added to the loss 
due to the ports. The cards are, however, fairly comparative, and 
they show clearly how little efifect is produced by the reduced open- 
ings at the earlier cut-offs. 

The standard rule for steam ports which calls for an area such that 
the velocity of the steam in them shall not exceed 6000 ft. per min., 
would, at the speed of this engine, call for a port area equal to 8.35 



436 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



per cent, of the piston area, while the actual area of the ports was 1 2 
per cent, of the piston area. Similarly the rule for the port opening 
would call for an area of opening of 6.25 per cent, of the piston area. 
The cards are numbered in order, beginning with the shortest cut-oS, 
and Table 1 7 gives the opening figured as a percentage of the piston 
area and a comparison of this area with the area called for by the rule. 

Table 17. — Poet Areas in Shaft Governor Engines 



Number 
of card 


Area of port opening as 


Area of opening divided 


a percentage of piston 
area 


by area called for by the 
old rule 


I 


2.27 


.363 


2 


2.36 


• 377 


3 


3.38 


• 524 


4 


4-32 


.69 


S 


7.76 


1.24 




Fig. 17,— Indicator card from a high-speed passenger locomotive. 
Average Steam Pressure, Lbs, perSq. In. Abs. 



go o 
u:> o 

03 <N W 



o o o o 
jg 540 CO 



CQ <M ca'-» »-• iH »-• 



21/2 



V-2 



— 




■--^ 










— 




---.. 


^ 


^^^ 




^ 




3 


■-S 


^ 




~^ 


■^ 








■^^ 














.^ 




'~- 












~^ 






[--^ 




''"^ 




^ 










■^ 






"~ 


■^ 


















^ 




■^ 




■^ 










■^ 




"" 


~^ 


^ 




















■-~. 


-- 




"^ 






^ 


^ 


^ 




"^ 


^ 


■^ 






-^ 












^ 


~~^ 


^ 




"" 










^ 




^ 


r- 


-^ 




















~^ 






^--^ 














^■-^ 
















■~-^ 






"^ 


~-«; 


■" 




^ 










■^ 






^ 


■^ 


















^ 




^ 














■-- 






"- 


~~^ 


















- 




--, 




~~~- 










^ 




^ 


-- 


^ 
















^ 




"" 










-. 


^. 


^ 


^ 


^ 






^ 














-^ 


^ 




.^ 










^ 




"^^ 


■- 


^ 
















H 


^ 


-<r 


^ 


^ 


-- 










,^ 




-^ 


-^ 












--^ 








-- 


^^ 


~^ 




- 




> 






"- 


- 


^ 


"^ 


- 
















■^ 


-< 


"^ 




^ 




1 


"^- 


■- 


- 


^ 


^ 




_. 












i- 






^ 


- 


-. 




^ 




i- 
-1. 






: 
















■--- 


-^ 


^ 




- 




1 
1 






^^ 




- 


- 


^ 








^ 
^ 










^ 




- 




^ 




1 
-1 






- 




- 




^ 


















^ 








i:i 


k-1 


s 


^ 


^ 


_i 




^ 


- 


^ 




3& 




^ 




ii 


^ 




I^L 





7000 
6000 
5000 
4000 

3000 

2000 
1500 
1200 
1000 
800 
600 
500 
400 
300 

200 

150 
120 
100 



60 ^ 

50 « 

40 ;§ 

<u 

30 O 

20 i 

15^5 

12 

10 

8 

6 

5 

4 

3 

2 
1.5 
1 



.1 



.15 .2 .3 .4 .5 .6 .8 1 1.5 2 3 4 5 6 
liOss of Pressure per 100 Ft. of Pipe, Lbs.per Sq.In. 



8 1.0 



That is, in cards i and 2 the actual area is but little more than one- 
third of that called for by the rule, while in 3, which represents about 
an average point of cut-off with economical load, it is but a little 
over one-half. Comparing the performances, the drop in the steam 
line of cards i and 2 is greater than in cards 3 and 4, measured at the 
most favorable point of the latter, but less when measured at less 
favorable points, while it is slightly less than in card 5, although the 
opening for the latter card has 3.41 times the area of that for the 
former. 

The influence of the steam pipe between the pressure gage and 
the steam chest vitiates the comparison to a certain extent, as, whUe 
its size remains fixed, more steam must be drawn through it with 
late cut-offs than with early ones, but the inference is unmistakable 
that a very decided constriction in the steam passage has a very 
slight effect on the flow of steam. 

It is common to explain this action of the small ports by refer- 
ence to the fact that they go with early cut-offs. The velocity of 
the piston being less at the early cut-offs, the velocity of the steam 
through the ports is correspondingly less, and hence it is argued that 
it should be expected that smaller ports would answer. The author 
is convinced that the importance of this action is much exaggerated. 
Were it true to an appreciable extent, the effect of the increased 
velocity at mid-stroke would appear in the exhaust line. The 
steam is forced through the exhaust port at various velocities at 
different positions of the piston, and if the action described had an 
appreciable influence, the exhaust line would arch upward; but, in 
point of fact, it is almost invariably straight. 

Average Steam Pressure, Ins. of Mercury 
15 10 8 6 5 4 3 2 1.5 1 




.1.12 



,15 .2 .3 .4 .5 .6 .8 1.5 2 8 4.-.-- 

Loss of Pressure per 100 Ft. of Pipe, Lbs.per Sq.In. 



Fig. 18. 
From the desired presstire loss say .3 lb. per sq. in. per 100 ft. of pipe 
3 ins., thence diagonally to the vertical from the steam pressure, say 80 
quantity of steam delivered 20 lbs. per min. 

Figs. 18 and 19. — Drop of pressure in steam pipe lines. 



Fig. 19. 
length trace vertically to the diameter of the pipe, say 
lbs., thence horizontally to the right where read the 



THE STEAM ENGINE 



437 



The ports and port openings of locomotives are invariably smaller 
than those of stationary engines and yet well-designed locomotive 
valve gears give surprisingly good results, as shown in Fig. 17, espe- 
cially after making allowance for the back pressure due to the blast 
nozzle which is still smaller in area than the ports. The results of an 
examination of this subject by the author may be found in his Slide 
Valve Gears. The calculations were based on time-card speeds, which 
are necessarily much less than running speeds, and, on this basis, 
velocities of steam through the ports (that is, values of C in Professor 
Barr's formula) were found as high as 11,000 ft. per min., and even 
this high velocity is still farther increased at the blast nozzle. The 
use of a single nozzle for both cylinders makes the comparison of 
steam velocities through ports and nozzle unsatisfactory, a better 
comparison being that between port and nozzle areas. The area of 
the nozzles was found to range between 36 and 44 per cent, of the 
area of two ports. 

Tke drop of pressure in steam-pipe lines is given by the following 
formula, due to Professor Unwin: 



W 



= 87.sJ — 



.1^) 



in which W = weight of steam delivered, lbs. per min., 
P = drop in pressure, lbs. per sq. in., 
y = density of steam, lbs. per cu. ft., 
(i = diameter of pipe, ins., 
Z, = length of pipe, ft. 

This formula has been accepted with slight and unimportant 
changes in the coefi&cient and after extended tests by Prof. R. C. 
Carpenter and G. H. Babcock. It has been reduced to chart form 
by Prof. H. V. Carpenter {Power, Dec. 17, 1912 and Jtine 10, 1913), 
these charts being given here as Figs. 18 and 19, instructions for use 
appearing below them. The charts are subject to the caution due 
to the fact that they are extended far beyond the range of any experi- 
ments that have been made. They, however, represent the best 
existing knowledge of the subject. Fig. 18 is for high and Fig. 19 for 
low pressures, including those below the atmosphere. The great 
velocities permissible at low pressures increase the relative importance 
of elbows. 

I The charts relate to actual, not nominal, pipe diameters. They 
apply to saturated steam. For superheated steam instead of the 
actual pressure use the pressure at which saturated steam has the 
same weight per cu. ft. 

Experiments on the resistance of pipe fittings are few in number and 
give very discordant results. The formulas by Robert Briggs for 
these losses may be used in the absence of anything better. They 
are, for one standard 90-deg. elbow : 

76d 



1-- 



and for one globe valve: 



■+¥ 



li4(f 



1 + 



3-6 



in both of which / = length of pipe, ins., equivalent to one fitting, 
(^ = diameter of pipe, ins. 

The resistance of gate valves is negligible. 

For the resistance of screwed pipe fittings to the flow of water, 
see Index. 

Steam Pipe Coverings 

A remarkably complete investigation of the insulating properties of 
commercial steam pipe coverings was made at the University of 
Wisconsin by L. B. McMillan {Trans. A. S. M. E., 1915) of which 
the following is an abstract. The coverings tested are indicated in 
Fig. 21. 



Fig. 20 summarizes the tests on bare pipe and gives the loss of heat 
per unit of surface and of temperature difference and, similarly. 
Fig. 21 summarizes the tests on pipe fitted with single thickness 
coverings as indicated. One result of the investigation is to show 
that the best covering is usually the most economical. The first 
cost is usually recovered many times each year and almost ceases to 
be a factor. At prevailing prices of coverings and taking the cost of 



5.2 

5.0 
4.8 
4.6 
4.4 
4.2 
4.0 
3,8 
3,6 
3,4 
3,2 
3.0 
2,8 
2,6 
2,4 
2,2 
2,0 
1,8 
1.6 
1.4 
1.2 
1.0 
.8 
.6 
. .4 
.2 
ft 
















































































/ 








































/ 






































/ 






































/ 






































..^ 


v 




































JV 








































































,■? 






































^^Y 






























" 




..-:i 






































/ 


































A<^ 


^y 


' 
































i^ 




r 


































d 


y 
































<>< 


?'h 






























t> 


.,.v°!l 


lV 




r 













































































































































































































































































































































































































































""0 50 100" 150 200 250 300 350 400 450 500 
Temperature Difference-Degrees Fahrenheit 
(Pipe Temp.-Eoom Temp.) 

Fig. 20.^ — Heat losses from bare pipe. 

steam at 30 cents per 1000 lbs. representative figures for the interest 
return on the cost of single thickness magnesia and air-cell coverings 
are shown in Table 18. 



Table 18. — Annual Interest Return on Cost of Single 
Thickness Coverings 



Kind of 
covering 


Temperature 

difference 

(Fahr.) 


Actual tempera- 
ture (Room = 
80 deg. Fahr.) 


Interest on 
investment 




50 


130 


87.0 




100 


180 


214.0 


85 per cent. 


200 


280 


S/6.0 


magnesia 


300 


380 


1099.0 




400 


480 


1869.0 




SCO 


580 


3078.0 




50 


130 


89.0 


Air cell 


100 


180 


218.0 




200 


280 


581.0 




300 


380 


IIOI.O 




400 


480 


1865.0 




500 


580 


3060 . 



The thickness of magnesia covering to give the maximum saving 
at prevailing prices may be obtained from Fig. 22 which applies to 
any temperature difference, any price of steam and any number of 
hours service per year. The chart does not show values for length 



438 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



of service, but to use it for other periods than 365 days at 24 hrs. 
a day, multiply the price of steam by the number of hours per year 
the steam line considered is in service and divide by 8760 and, using 
the result as the price of steam on the chart, find the proper thickness. 



gage pressure will be about 365 deg. Fahr., and, assuming a room 
temperature of 80 deg., the temperature diSerence between pipe and 
room will be 285 deg. Now on the chart, using the curve for steam 
at $0.10 per 1000 lbs., the proper thickness corresponding to 285 deg. 




60 100 150 200 250 300 350 400 450 
Tempeiature Difference-Degrees Fahrenheit 
(Pipe Temp.-Boojn Temp.) 

Fig. 21.' — Heat losses from single thickness coverings. 




100 200 300 400 500 

Temperature Difference, Degrees Fahrenheit 
(Pipe Temp. Room Temp.) 

Fig. 22. — Proper thickness of 85 per cent, magnesia covering for 
maximum net saving 

For example, suppose that the steam pressure is 150 lbs. per sq. 
in. gage, that it costs $0.30 per 1000 lbs. generated, and that the line 
is in use 12 hr. a day and 9 months out of the year. The number of 
hours per year that the steam is on is therefore 2920, or one-third of 

(202o\ 
8~6~) 
=$0.10. The temperature of the pipe containing steam at 150 lbs. 



3.2 
3.1 

3.0 
2.9 
2.8 
2.7 
2.6 


















































































































































































































































1 














..-per-yeaLatiTempJDi^Jf 


500°Fa 


u. 




— 


1 






kf\ 


OoUa 




— 


2.5 

2.4 

2.3 

2.2 

2.1 

2.0 

1.9 

1.8 

1.7 

1.6 

1.5 

1.4 

1.3 

1.2 

1.1 

1.0 

.9 

.8 

.7 

.6 

.5 


1 




9,'' 










r 






















1 


^9 




































1 1 








































1 1 


i 












i.O 


























1 








S 
















CIV"* 


wf 


iv<. 






111 










1 3.0 

as 

■a si 
S §2.0 

r 
■§ 1.0 










L'6sl»^ 


•""^r.^ft 


t^ 








ill 












^^ifi^^^' 


IfllV 






















\ 


m 


■^ 


^ft^ 


















11' 












' 






















11 










1 

1 


































1 
























1 1 
1| 


\ 






























'1' 


1 








1) 100 200 300 400 500 1 


I 


1 


-Temp.Di 


ff=5MpFahx.-^''"'P«^''^'^^«h'ienheir''' ■"'^'^^' | 


^,. .. =100° " 




1 




ff;; 


=30 


. 






A*, 


1 .UU\nD°>i 


irs 


per 


Year at i'erap.D 


)Vi 


hr. 






w 


\^ 


ev.'- 


■ — 


































,11' 


'\ 




































/ 

-r 
f 




« 


N. 




































* 


^ 


1 
































1 








p 


























• 














^' 


^^.Te™ 


>,n 


iff 
















.'2 


j 
















■"" 


=^^^^^^i£ipipelH|. 


















.„.^_ 


1 1 1 r T 1 T~- 


t 


~~ 


,1 




et-S 


*v 




n"T>ol>ars 


pel 


Year at Temti. Di[f.= inn"Ifnh 


y 















.00 



7.00 



6.00 



5.00 «• 



4.00 -a 



3.00 fl 



2.00 s 



1.00 



4.0 



6.0 



Fig. 



1.0 2.0 3.0 

Thickness of Covering, Ins 

23. — Gross and net savings due to 85 per cent, magnesia 
coverings. 



150 
il40 
|l30 
-S120 

»iio 
sioo 

I90 

S 80 
M 70 

60 

1 SO 
i ^° 

^30 
120 
g 10 




































^ 


'^ 
































^ 


y' 
































^ 


y 
































y 


^ 


































A 




































y 


































y 


/ 


































/ 


/ 


































A 


/ 


































/ 


/ 


































/ 


/ 




































/ 




































/ 




































/ 




































/ 


Y 




































/ 







































Fig. 



24.- 



40 80 120 160 200 240 280 320 360 
Heat Loss per Sa.F-t.of Outer Surface of Covering, B.I.U. 

-Relation of heat loss to temperature difference between 
covering surface and surrounding air. 



temperature difference is found to be 2.1 ins. This then is the proper 
thickness for maximum net saving under the given conditions. 

Fig. 23 gives the law of the gross and net saving for steam cost- 
ing 30 cents for 1000 lbs. and for prevailing prices of coverings. 
The central chart shows, for example, that for a temperature differ- 



THE STEAM ENGINE 



439 



ence of 300 deg. the thickness for maximum net saving is 3 ins. but, 
consulting the chart line immediately below, we see that the net 
saving with 2 ins. thickness is but a trifle less. The heat loss curves 
show a material increase in the heat loss but this is largely offset 
by the increased cost of covering. 

Fig. 24 supplies a means of determining the losses due to pipe 
coverings already in place. In order to find the loss from any pipe 
covering having its surface finished with white canvas, place a ther- 
mometer under the canvas and another in the air 4 or 5 ft. from the 
pipe; take the diilerence between the two temperature readings and 
on the curve find the corresponding loss. Such a test might give 
results as much as 5 per cent, in error due to the chances of not getting 
the average temperature difference closer than that, but at that it 
would be accurate enough for some purposes. 

A cheap and efeclive steam-pipe covering may be made of sawdust 



Find by trial point k, such that a circle struck from it as a center 
will be tangent to Oa' , Jg and cd. The radius, kl, of this circle is 
the lap and the diameter, mn, is the travel of the valve. The advance 
angle is equal to i'OB', the center of the eccentric being at p, such that 
pq = kn. If the valve has no inside lap, Oi' is the crank position 
and i the piston position for release and compression. If the valve 
has inside lap equal to the radius ko, compression takes place at ft 
and release at b. With negative inside lap, release and compression 
change places. 

The above construction assumes the slotted cross-head construction 
or its equivalent, a connecting rod of infinite length, and, when applied 
to the actual connecting-rod construction, it gives the mean positions 
of the events of the stroke. To find the actual positions, project, 
as in Fig. 26, the extremities of the crank positions to the diameter 
by circular arcs of which the radius equals the length of the connect- 




FlGS. 



a" a hlibti\ 



Fig. 26. 

25 to 27. — The Bilgram diagram applied to a plain slide valve 



^,-^-- 


— -,^ 


/' 


"^^ 


/ _. 








X ,''""' 


^'^^y^'^ \^"\ 








/ ^^ \a'\ 


/ / 


/ ^jj'^^'n \ 




i /^^ V'^ 1 


' / 


\yf'' \ \ \ / 


/ / 


yy\. \ \ X 


1 / 








1 *'' 


y^' ^ fl ! 


jL -^ ^^ 


\a 1 




t , 




\ 1 






A 11 \ ^>\ 


1 1 




1 1 


\ ji ><^'^ \ 


1 1 


\i|,^^S? 


/ 1 


\\ \J 


/ / 


\ \ >c 


/ 


^On -^"-^^ 




"^ 


• 


■^•^^ 


^^ 


• — 





Fig. 27. 



and lime. The mixture is made up like sand mortar, using one 
barrel of lime to five of sawdust and allowing several days for it to 
dry before turning on the steam. A steam line of 197 ft. of 8-in., 
219 ft. of 7-in., and 258 ft. of 6-in. pipe was covered with this mix- 
ture encased in a wood box 1 2 ins. square inside and tamped down. 
A test of 20 days with bare pipe showed a condensation of 1440 
lbs. of water per hour or a fraction under ij lbs. per sq. ft. of 
external surface per hour. The covered pipe showed a condensa- 
tion of 19s lbs. of water per hour or 2j oz. per sq. ft. of external 
surface of pipe per hour, the loss covered being 14 per cent, of that 
uncovered. The working steam pressure was 90 lbs. and the air 
temperature averaged about 64 deg. Fahr. If the wood box is not 
desired "a little fire clay or flour mixed in makes it possible to 
wrap it on under a covering of muslin." The mixture is regarded 
as fire-proof (F. A. Nystrom, Amer. Mach., Mar. 7 and Apr. 4, 
1901). 

Steam-pipe lines should incline about i in. in 10 ft. in the direc- 
tion of the flow of steam. 

Laying out the Slide Valve 

Laying out a slide valve may be most conveniently done by the 
Bilgram diagram, for the demonstration and many additional appli- 
cations of which see the author's Slide Valve Gears. To lay out a 
plain slide valve prpceed as in Fig. 25. Let AB be the length of 
stroke to any convenient scale, a being the desired point of cut-off. 
Draw the crank circle A'a'B' and project point a to it, giving Oa', 
the cut-ofi position of the crank. Made de equal to the desired lead 
opening and draw/g with radius Og equal to the desired port opening. 



ing rod, giving points a', a", b', b", h', h", of which the single primed 
letters refer to the outward and the double primed letters to the in- 
ward stroke — the cylinder being assumed to lie at the left of the 
diagram. 

To equalize compression and release, lay down the desired compres- 
sion and release points as in Fig. 27 at a and b. Project these points 
to the crank circle by circular arcs with radius equal to the length of 
the connecting rod, giving points a' and h'. Draw the corresponding 
crank positions and give the a end of the valve an inside positive 
lap and the b end an inside negative lap equal to the radii of the 
small circles. 

The action of a shifting eccentric upon a slide valve may be 
determined as in Figs. 28 and 29. In Fig. 28 the eccentric swings . 
from a center located on the center line of the crank and on the same 
side as the crank pin, that is, the center of the arc di°. With the 
radius of dd° and with a center in the vertical center line, strike the 
arc QQ°. With the eccentric center at the full throw position, d, 
the action of the accentric on the valve is given by the lap circles 
struck from Q as a center and, similarly, with the eccentric at any 
other point, d', the action on the valve is given by the lap circles 
struck from Q' as a center, the three cut-off positions of the crank 
being shown by tangents to the lap circles. The increasing distance 
of the lap circles above the horizontal center line as the cut-off is 
shortened shows the increase of the lead vnth. shortened cut-off, and 
the actual lead for any point of cut-off may be measured from the 
diagram. 

In Fig. 29 the eccentric is swung from a center on the center line 
of the crank but opposite the crank pin. The centers of the lap cir- 
cles are now located on an arc which is convex downward instead of 



440 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



upward, the general eSect being the same but with the important 
exception that the lead decreases as the cut-off is shortened, as shown 
by the lap circles approaching the horizontal center line as the cut- 
off is shortened. 

The action of a Stephenson link motion on a slide valve is essen- 
tially the same as that of a shifting eccentric. If the eccentric rods 
are "open," that is, if they are not crossed when the eccentrics are 
placed as in Fig. 30, the lead increases as the cut-off is shortened and 
the Bilgram diagram is similar to Fig. 28. If the rods are "crossed," 
that is, crossed when the eccentrics are placed as in Fig. 30, the lead 
decreases as the cut-off is shorteded and the Bilgram diagram is 
similar to Fig. 29. Crossed rods, however, are used but little if at all. 

The amount of variation in the lead depends upon the length of the 



horizontal distance from the vertical center line equal to the lap, 
and locate c, d, c', d', at additional horizontal distances equal to 
the full gear lead, these last points being those which the eccentric 
centers occupy when the crank is on the centers. From these points 
lay down the full and dotted midgear positions of the link when the 
crank is on the centers, and measure the midgear travel, ef. Divide 
this in half, subtract the lap oi or on and obtain the midgear lead if 
or en. Note that if the rock shaft has unequal arms, leading to 
inequality between the eccentric throw and the valve travel, it is 
most convenient to use the valve travel as the diameter of the dotted 
circle, the eccentric rod lengths and the link dimensions being changed 
from the actual in the proportion of the valve travel to the eccentric 
throw. 





Fig. 29. 
28 and 29. — The Bilgram diagram appHed to a shifting eccentric valve gear. 





Fig. 30. — Open eccentric rods. 

eccentric rods — the variation increasing as the rods are shortened. 
The proper radius of the link is the length of the eccentric rods plus 
such distance as there may be between the geometrical link arc (the 
curved center line of the link) and the eccentric-rod pins. With any 
other radius the variation in the lead with varying cut-off differs 
for the two ends of the cylinder. 

The layout of the Bilgram diagram for a link motion does not 
differ essentially from that for a shifting eccentric. While, however, 
the full gear lead is commonly given in advance, the midgear lead 
being dependent on the length of the rods mus be found by the method 
shown in Fig. 31. Make the diameter of the dotted circle equal to 
the full gear travel of the valve and lay down points a, b, a', h' at a 



Fig. 31. — Finding the mid-gear lead of link motion. 

To construct the Bilgram diagram proceed as in Fig. 32. Draw 
the inner dotted circle equal in diameter to the full stroke valve travel, 
and lay down ab equal to the lap, make be equal to the full gear lead, 
and de equal to the midgear lead, as found in Fig. 31. Now, a 
circle struck through ejg will give the path on which a single shifting 
eccentric must travel to produce a valve movement equivalent to 
that given by the link of Fig. 31. Laying off h'f equal to hf, and 
Oe' equal to Oe, the circular arc e'f is easily drawn, on which the 
center of the lap circle for all points of cut-off must lie. Drawing 
the outer dotted circle to represent the path of the crank pin to scale, 
and selecting, say, the cut-off at one-third stroke for study, the point 
i is laid down such that ij equals one-third of the stroke, and by the 
perpendicular ik the crank line Ok for one-third stroke is located. 
Drawing a lap circle tangent to Ok and with its center on the line 
e'f, we have, for the one-third cut-off: lead=/M, port opening = Oo, 



THE STEAM ENGINE 



441 



valve travel = 20^, exhaust opening and closure (assuming no inside 
lap) at crank position Og. Similarly we have for the full gear a lead 
rs equal to be, a port opening Ot, a travel to twice Of', and an exhaust 
opening and closure at crank position Ot. To investigate the reverse 
motion extend the arc e'f to g'. 

For particulars regarding practice with negative lead in the full 
gear and unequal leads in the forward and reverse gears see the 
author's Slide Valve Gears. 

Friction of Slide Valves 

The friction of slide valves formed the subject of experiments by 
J. A. F. AspiNALL (Proc. I. C. E., 1898). The experiments were 
upon two horizontal locomotive valves, one an ordinary unbalanced 
valve of phosphor bronze and the other a Richardson relieved valve 
of cast-iron. 

As a sight-feed lubricator was used in the experiments, it was easy 
to watch the result of increasing the number of drops of lubricant per 




Fig. 32. — The Bilgram diagram applied to a Stephenson link motion. 

min., and it was found that there was a perceptible improvement in 
the ease of movement of the valve when the lubricant was increased. 

The experiments show that the friction of slide valves is somewhat 
greater against a horizontal than against a vertical face; the coelScient 
of friction found in previous experiments for valves on a vertical 
face was .068, while in the experiments dealt with here, the average 
coefficient was found to be, for the unbalanced valve .0878, and for 
the partially balanced valve .0919. 

The coefficient of friction, as given in Table 19, together with the 
other results of the experiments, is calculated from the whole area 
of the back of the plain valve, supplementary experiments by Mr. 
Aspinall having convinced him of the correctness of that procedure. 
In the author's opinion this conclusion was not warranted by the 
experiments, but, if the friction of other valves is calculated in the 
same way and from Mr. Aspinall's determinations of the coefficient 
of friction, the results should be sufficiently correct for all practical 
purposes and doubtless within the variations due to varying condi- 
tions. For the Richardson valve, the balanced area was taken as 
that portion which is enclosed between the strips, excluding the area 
of the strips themselves. 



Table 19. — The 


Friction or Locomotive Slide Valves 


1- 





"3 
0. 

u 



•S 
3 


c 
_o 

'•4-> 

CS 



■c 

3 


4) 
> 

> 


0) 

ft 


4) 

u u 


<u ft 


h 

v 

2 


tJ p 

0^ 


V 

> 


eg 

sfi-c 
° 








Drops 






















per min. 




Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 




F 


I 


Pull 


4 






IS6 


ISS.O 


24,149-0 


1. 549-5 


2,099.0 


.086 


F 


I 


Pull 


4 






IS2 


ISI-S 


23.562.5 


1,589-0 


2,126.0 


.090 


F 


2 


Pull 


4 






1S8 


IS4-0 


23.405-0 


1,510-5 


2,056.5 


.087 


F 


2 


Pull 


4 






157 


153-0 


23.037-0 


1,497-0 


2,039.5 


.088 


F 


3 


Pull 


4 






144 


137.0 


19. 513-0 


1,090.0 


1.575-5 


-080 


F 


3 


Pull 


4 






IS4 


150.0 


21,619.0 


1,562.5 


2,094-5 


.092 


F 


4 


Pull 


4 






146 


140.0 


19.455-5 


1,234.0 


1.730-5 


.088 


F 


4 


Pull 


4 






158 


ISO.O 


21,077.5 


1,392-0 


1,924.0 


.091 


B 


I 


Pull 


4 






158 


155. 5 


23,894-5 


1,497.0 


2,048.5 


.085 


B 


I 


Pull 


4 






149 


154-5 


23,929.0 


1.497-0 


2,045-0 


.08s 


B 


2 


Pull 


4 




> 


152 


ISO.O 


21,924.0 


1.444-5 


1,976.5 


.090 


B 


2 


Pull 


4 






ISS 


147 -5 


22,052-0 


1.313.0 


1,836.0 


-083 


B 


3 


Pull 


4 






ISO 


145.5 


20,824.0 


1,300.0 


1,816.0 


-087 


B 


3 


Pull 


4 




Q 


156 


151. S 


21, 801. S 


1,300.0 


1,837.0 


-084 


B 


4 


Pull 


4 




(U 


ISS 


152.0 


20,819.5 


1,103.0 


1,642.0 


.078 


B 


4 


Pull 


4 




(4 

a 


147 


140.0 


19,248-5 


1,129.0 


1,625.5 


-084 


F 


I 


Push 


3 







160 


IS7-S 


24,461.5 


3.326.0 


1,767-0 


.072 


F 


I 


Push 


6 






& 



IS8 


156-S 


24,291.5 


2,866.0 


2,311.0 


-095 


F 


2 


Push 


6 




IS9 


ISO.O 


22,907.0 


1,709.0 


1.177.0 


-051 


F 


3 


Push 


5 




160 


136.0 


20,s68.o 


2,347-5 


1,865.5 


.090 


F 


3 


Push 


3 




Cm 


IS9 


152.0 


23,032.5 


1,503-0 


964.0 


.041 


F 


4 


Push 


3 






160 


148.0 


20,941-5 


2,620.5 


2,095-5 


. 100 


F 


4 


Push 


3 






161 


148-S 


20,736.0 


2,532.0 


2,005-0 


.096 


B 


I 


Push 


3 






160 


147-0 


22,502.5 


2,973-5 


2,452.0 


.108 


B 


I 


Push 


3 






1S8 


IS2-0 


23.732.0 


I,9IS-0 


1,376.0 


-057 


B 


2 


Push 


3 






IS8 


142.0 


21,184.5 


2,826-S 


2,322.5 


. 109 


B 


2 


Push 


3 






164 


160.0 


23,605 -S 


3,326-0 


2,758-5 


. 112 


B 


3 


Push 


3 






1S8 


147.0 


21,292.5 


2,914-5 


2,393-0 


, 112 


B 


3 


Push 


3 






160 


150.0 


21,677.0 


2,944-0 


2,412.0 


.III 


B 


4 


Push 


3 






1S8 


140.0 


19.995 -0 


2,468.5 


1,972.0 


.098 


B 


4 


Push 


3 




, 


160 


145-0 


21,165.5 


2,503-0 


1,988.5 


-093 


B 


I 


Pull 


4 






144 


140 -0 


9,467.0 


420.0 


916.5 


.097 


B 


I 


Pull 


4 




0) 


142 


140.5 


9.514-5 


393.5 


892 .0 


-094 


B 


2 


Pull 


4 




s. 


140 


133-0 


8,633.0 


288. 5 


760. 5 


-088 


B 


2 


Pull 


4 






IS8 


154-5 


9.976-0 


393-5 


941 -5 


-094 


B 


3 


Pull 


4 




m 


143 


140.0 


8,704.0 


3S4-0 


850.5 


-097 


B 


3 


Pull 


4 






IS2 


149 -0 


9,238.0 


236.0 


765.0 


.082 



Note. — -The throttle valve was full open in all the experiments. 
Poppet Valves 

Double beat poppet valves as usually made are, as is well known, 
difficult to keep tight. Slight differences in the coefficient of expan- 
sion of the metals composing the valve and its case, or slight differ- 
ences of temperature due to the accumulation of water will cause one 
or other seat to lift slightly and thus leak. 

Fig. 33 is a sketch showing the usual construction, from which it 
will be apparent that any difference of expansion between valve and 
case will open one or other seat. Should the valve expand the more, 
the seat a will open; while should the case expand the more, seat b 
will open. Fig. 34 shows the construction used by the Nordberg 
Mfg. Co. {Amer. Mach., Aug. 14, 1902) whereby this difficulty is 
overcome. Its essential feature is that the cone surfaces of the two 
seats have a common apex at a. Should the valve expand the 
more, its vertical expansion will tend to open the seat b; but its 
horizontal expansion, having the same increment of excess, will tend 
to close the seat, and the two actions will offset one another. The 
reverse action will take place should the case expand the more. 
Looked at in another way, the expansion of both valve and case is 
from the common center a and any difference of expansion is accom- 
panied by a slight sliding of valve and seat upon one another on the 
line of the joints between them, but without any tendency to open 
either joint. This action will take place wherever the common apex 
a may be and regardless of the angle of the two seats. An actual 
valve by the Nordberg Mfg. Co. (a lo-in. regulating valve) is shown 
in Fig. 35. The lower seat is here flat but the two seats intersect 
at b and the action described in connection with Fig. 34 still holds. 



442 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Dimensions and Lift of Poppet Valves for a Given Size of 
Opening 
D = Smaller diameter of valve seat. 

d = diameter of piping to whiich valve opening must correspond. 
D 



r = - 



Flat-seated Valves 

Forr = i Lift = dX.2So = Z>X.2So 

For r = 1. 25 Lift = <iX .2oo = Z?X . 160 

Fory = i.5 Lift = dX.i66 = Z)X.iii 

Fory = 2 Lift = <^X.i25 = DX.i62 

Forr = 2.5 Lift = iX . ioo = Z)X .040 



It will be noted that cone-seated valves require a lift from one-fifth 
to one-quarter greater than the corresponding flat-seated valves. 

The proportions of lift given in connection with flat-seated valves 
are geometrically correct. It should, however, be borne in mind 
that flat-seated valves generally introduce a certain amount of wire 
drawing of the incoming charge. A slight increase over the theoret- 
ically correct Kfts should consequently be provided. A definite 
coefficient cannot be given, as this wiU depend considerably upon the 
valve seat and valve chest design, as well as upon the proportions of 
the fillet between valve stem and valve head. The matter is one of 
personal intuition by the designer; in the best French designs the 
extra allowance seldom exceeds 25 per cent, of the theoretical lift. 
It is well to so arrange the contour of the valve and valve chamber 




Fig. 33. — Incorrect construction of double beat poppet valves. 



$?$$^^$$§$$$$$$$$$$^S^^^. 



I 1 




Fig. 34. — Correct construction of double beat poppet valves. 




Fig. 35. — Nordberg Mfg. Co's double beat poppet valve. 



Cone-seated Valves; 45 Deg. Angle of Cone 

For r = i 
For r = i. 25 
For r = 1 . 5 
For r = 2 



Lift = <iX.307 = -DX.307 
Lift = (fX.2s6 = Z)X.205 
Lift = rfX.2i9 = Z)X.i46 
Lift = (fX.i7o = DX.o84 



Forr = 2.s Lift = (ZX .I38 = I'X .055 



profiles that a minimum of lift be required, as this is favorable to 
silence in running and to flexibility. 

Condensing Water 

The approximate quantity of water required to condense i It), of 
steam with a jet condenser may be obtained from the formula: 



THE STEAM ENGINE 



443 



C= 



{H-\-Z2)-T 



(T-t) 

in which Q = lhs. of water required to condense i lb. of steam, 
r = temperature of discharge water, Fahr., 
< = temperature of injection water, Fahr., 
fl^ = total heat above 32 deg. Fahr. in i lb. of steam to be 
condensed. 
Table 20 by W. F. Fischer {Power, Sept. 26, 1911) is based on the 
steam tables of Marks & Davis and is given as a guide in figuring the 
condensing water required per lb. of steam in condenser installations. 
Example: With a vacuum of 28 ins. of mercury referred to a 30- in. 
barometer (H+32), is found from Table 2 to be 1137 B.t.u. Sub- 
stituting in the formula, 

1137-r 



densers In the ordinary surface condenser of the single- or double- 
flow type, Tc may be taken from 10 to 20 deg, lower than the 
temperature due to the vacuum. 

Table 20. — Condensing Water per Pound of Steam 



Q= 



T-t 



for a 28-in, vaccuum. In column 5 the volume in cu. ft. per lb. of 
steam is given, and in column 6 the weight of i cu. ft. of steam at the 
given pressure and temperature corresponding to the given vacuum. 

In determining the proper value to substitute for T, care should 
be taken to allow a suitable drop between the steam in the condenser 
and the temperature of the discharge water. In practice the tem- 
perature of the discharge water is assumed to be 15 deg. lower 
than the steam temperature and it is customary to allow for 10 per 
cent, more water than the estimated quantity where actual condi- 
tions are unknown. 

When estimating the quantity of water required per lb. of steam 
in surface condensers it is customary to take into account the tem- 
perature of the condensed steam; that is, the hotwell temperature. 

Hence for surface condensers 

{H+32)-Tc 
^~ T-t 
in which Tc equals the temperature of the condensed steam and Q,H, T 
and / represent the same quantities as in the formula for jet con- 



Vacuum in 

ins. of mercury 

referred to a 

30-in. 

barometer 


Absolute 

pressure 

lbs. per 

sq. in. 


Tempera- 
ture of 

steam and 
water at 

condenser 
pressure 


B.t.u. 's in 

I lb. of 
steam -f32 


Volume 

in cu. ft. 

per lb. of 

steam 


Lbs. 

of steam 

per cu. 

ft. 


29.82 


.09 


32 


1105 


3294 


.0003 


29. so 


• 25 


59 


1117 


1249 


.0008 


29.00 


• SO 


80 


1127 


636.8 


.0016 


28.50 


■ 74 


92 


1132 


442.2 


.0023 


28.00 


1. 00 


102 


1 137 


331 S 


.0030 


27.50 


1.24 


109 


1 140 


272.9 


• 0037 


27 .00 


1. 51 


116 


1 143 


225.8 


.0044 


26.50 


1.72 


121 


114s 


197.9 


.0050 


26 .00 


1.99 


126 


1 147 


1739 


.0057 


25.50 


2. 22 


130 


1 149 


157-1 


.0064 


25 .00 


2.47 


134 


USD 


142.2 


.0070 


24.50 


2.73 


138 


1152 


128.9 


.0077 


24.00 


2.96 


i4t 


IIS3 


119.9 


.0083 


23.50 


3-19 


144 


IISS 


III. 6 


.0089 


23.00 


3.4s 


147 


1156 


104.0 


.0096 


22.50 


3-70 


ISO 


1157 


97.0 


.0103 


22.00 


3-96 


IS2 


IIS8 


93 


.0108 


21.50 


4.18 


ISS 


1159 


86.4 


.0116 


21.00 


4.40 


IS7 


1160 


82.6 


.0121 


20.50 


4.70 


IS9 


I161 


78.0 


.0125 


20.00 


4.90 


162 


1162 


73.8 


■ 0135 


18.00 


5.80 


169 


1 165 


63.3 


.0158 


16.00 


6.85 


176 


Ii68 


54 -5 


.0183 


14.00 


7.85 


182 


1171 


48. 12 


.0207 


.00 


14.70 


212 




26.79 


• 0373 



THE GAS ENGINE 



Current practice in the dimensions oj gas-engine parts formed the 
subject of an investigation by the Department of Machine Design of 
Cornell University, the results being reported by Sanfoed A. Moss 
{Amer. Mach., Apr. 14, 1904) and given below. The investigation 
included an analysis of the dimensions of engines of 76 different sizes 
by 20 builders. 

The computed stresses are perhaps open to criticism, since the 
formulas may not take everything exactly into account. The 
numerical coefficients given are absolute, however, being taken from 
the actual data, and may safely be used, even though the exact 
stresses, bearing pressures, etc., to which they correspond may not be 
known. 

There are also given in the last column rough formulas for average 
cases. For instance, in the case of the cylinder wall, the rational 
average formula is t = .ooo204pD+l. This gives a thickness vary- 
ing with the maximum pressure p. In an average case p is 300, and 
if this value is substituted for p we have t={.ooo204X3oo)pD+i, 

or very nearly / = — + j. This formula, of course, should not be used 

where the pressure is much different from 300. Formulas like those 
in the last column are given in works on gas-engine design, without 
qualification, which is not correct, as these formulas have a limited 
range. The rational formulas given, with the mean values of the 
numerical coefficients substituted, are the proper formulas for general 
use. 

The maximum explosion pressure in the engines examined varied 
from about 250 to 350 lbs. per sq. in., the average being 300 lbs. per 
sq. in. The compression pressure varied from about 50 to 100, the 
average being 70 lbs. per sq. in. The lower values of compression 
pressure and maximum pressure are for engines using gasoline, and 
the higher values for natural gas. This is, of course, due to the fact 
that pre-ignition must be avoided. 

The maximum horse-power which an engine can develop is found 
to average very closely i| times the rated horse-power for which the 
engine is sold. 

The mechanical efficiency averages about 80 per cent. The engines 
examined were single-cylinder horizontal or single or multicylinder 
vertical engines, all single acting, varying from 5 to 100 h.p., and the 
formulas given apply only to such engines. 

The maximum probable brake horse-power of gas engines may be 



Fig. i.i 



■■mmj^ 



W 



^//////////Ma 



¥ 



Fig. 4. 



Fig. 2 




"»il fcj 



^ 





^ 



? 



^ 



^ 








¥ 



Figs, i to 5. — Types of gas engines in relation to weight of fly-wheels 



Table 2. — Horse-power Constants 


FOR Gas Engines 


• Single-acting engines j 


Double-acting engines 


Cylin. 


Natural 


Producer 


lUum'g 


Cylin. 


Natural 


Producer 


Illum'g 


diam. 


gas 


gas 


gas 


diam. 


gas 


gas 


gas 


S 


.00162 


.00140 


.00175 


10 


.0122 


.0105 


.0132 


5i 


.00179 


.00154 


.00193 


lOj 


.0135 


.0116 


■0145 


Si 


.00197 


.00169 


.00212 


11 


.0148 


.0128 


• 0159 


5i 


.00215 


.00185 


.00232 


Hi 


.0161 


.0139 


• 0173 


6 


.00234 


.00202 


.00252 


12 


.0175 


.0151 


.0189 


6i 


.00254 


.00219 


.00274 


I2i 


.0191 


.0164 


.0206 


6i 


.00275 


.00237 


.00296 


13 


.0207 


.0178 


.0222 


6f 


.00297 


.00255 


.00319 


I3i 


.0222 


.0191 


.0239 


7 


.00319 


.00274 


.00343 


14 


.0239 


.0206 


• 0257 


7i 


.00342 


.00294 


.00368 


I4i 


.0257 


.0222 


.0277 


n\ 


. 00366 


.003 IS 


.00394 


IS 


.0274 


.0236 


.0295 


n 


.00390 


.00336 


.00421 


16 


.0313 


.0270 


■ 0337 


8 


.00416 


.00358 


. 00448 


17 


.0354 


.0305 


.0381 


8i 


. 00443 


.00381 


. 00476 


18 


.0395 


.0340 


.0425 


81 


.00470 


. 00405 


.00506 


19 


.0441 


.0380 


.0474 


8i 


.00498 


. 00429 


.00536 


20 


.0489 


.0421 


.0526 


9 


.00526 


. 00454 


.00567 


21 


.0538 


.0464 


.0579 


9l 


.00587 


.00505 


.00632 


22 


.0597 


.0510 


■0637 


10 


.00650 


.00560 


.00700 


23 


.0646 


.0557 


. 0696 


lOi 


.00717 


.00617 


.00772 


24 


.0703 


.0606 


■ 0759 


11 


.00786 


.00678 


.00847 


25 


.0763 


• 0657 


.0827 


Hi 


.00860 


.00741 


.00926 


26 


.0825 


.0711 


.0889 


12 


.00936 


.00806 


.0101 


27 


.0890 


.0767 


• 09S9 


I2i 


.0101 


.00875 


.0109 


28 


.0958 


.0825 


.103 


13 


.0110 


. 00946 


.0118 


29 


.103 


.0885 


.III 


13 1 


.0118 


.0102 


. 0127 


30 


. 110 


.0947 


.118 


14 


.0127 


.0110 


.0137 


31 


.117 


. lOI 


.126 


I4l 


.0137 


.0118 


.0147 


32 


.125 


.108 


•135 


15 


.0146 


.0126 


.0157 


33 


.133 


.115 


•143 


16 


.0166 


■ 0143 


■ 0179 


34 


.142 


. 122 


.152 


17 


.0188 


.0162 


.0202 


35 


.149 


.129 


.161 


18 


.0210 


.0181 


.0227 


36 


.158 


.137 


.171 


19 


.0234 


.0202 


.0252 


37 


.l68 


.144 


.180 


20 


.0260 


.0224 


.0280 


38 


■ 177 


.152 


, 190 


21 


.0287 


.0247 


.0309 


39 


.186 


. 160 


.20a 


22 


.0315 


.0271 


.0339 


40 


.195 


.168 


.210 


23 


.0344 


.0296 


.0370 


41 


.205 


■177 


.221 


24 


.0374 


.0323 


.0403 


42 


.216 


.186 


.232 


25 


.0406 


.0350 


■ 0437 


43 


.226 


.193 


.243 


26 


• 0439 


.0379 


.0473 


44 


.236 


.204 


.255 


27 


.0474 


.0408 


.0510 


45 


.247 


.213 


.266 


28 


.0510 


.0439 


• 0549 


46 


.258 


.223 


.278 


29 


.0547 


.0471 


.0589 


47 


.270 


.233 


.291 


30 


.0585 


.0504 


.0630 


48 


.282 


■ 243 


.304 



obtained from Table 2 by Cecil P. Poole {Power, Mch. 23, 1909) in 

connection with the formula : 

Probable brake h.p. = constant from table X stroke, ins. X r.p.m. 

The constants for double-acting engines include an allowance of 6 
per cent, for the effect of the piston rod. 

The weight oj flywheels j or gas engines may be determined from the 
formula, by R. E. Mathot, {Engineering Magazine, June, 1907), 

D^an^ 
in which P = the weight of the rim (without arms or hub), tons, 
£> = diameter of the center of gravity of the rim, ft., 
(Continued on page 386, first column) 
444 



THE GAS ENGINE 



445 





Table r. — Current Practice in the 


Dimensions of Gas Engines 
















Assumptions made in 




Engine dimension and 
name of design con- 
stant upon which it 


Notation: All dimensions in ins., 
all pressures and stresses in lbs. 


Rational formula 
for engine dimen- 
sion in terms of the 


Maximum, 
mean and min- 
imum values of 


Corresponding 
numerical values 
of coefficient of 


deducing formula for 

average cases from 

rational formula; mean 


Reduced for- 
mula for average 


depends 


per sq. in. 


design constant 


design constant 


formula 


value of design con- 
stant always used 


cases 


Thickness of cyl. wall. . . . 


/ = thickness of cyL walls. 


'=(^)*^ + i 


S = 


I 


P = i00 


'=^> 


Stress in cyl. wall 


5 = stress in cyl. walls. 


1,62s 


2S~ 






p = max. pressure. 




2,450 


.000308 








D = dia. of cylinder. 




3-750 


.000204 
.000133 






Thickness of jacket walls 


r = thickness of jacket wall. 
t = thickness of cyl. wall. 


T = cl 


c = 
.86 
.60 






r=.6< 


Thickness of water jacket 


J = thickness of jacket space. 


j = ct 


■43 
c = 






j^iit 


space. 


< = thickness of cyl. wall. 




1.8s 
I. 25 








Number of cyl. head studs 


g = number of cyl. head studs. 
D = diameter of cyl. 


q = cD+2 


I . 00 

c — 

1. 14 

• 67 






g=fD-|-2 


Outside dia. of cyl. head 


= outside diameter of cyl. head 


o^-^JId 


.40 

s = 


I 


P = 300 


D 

"=72 


studs. 


studs. 


V.TS \ g 






Stress in cyl. head studs . 


q = number of cyl. head studs 
^ = maximum pressure. 




10,900 
7,800 


.oiis 
• 0135 


g = 8. This is correct 
for 9-in. cylinders 






D = diameter of cyl. 




4,500 


.0179 


and nearly correct 






i = stress at root of thread 








for quite a range on 
either side. 




Length of stroke in terms 


L = length of stroke. 












of cyl. diam. 


r' = cyl. diameter. 


L = cD 


c = 
1.8 
l-S 






Z- = iJD 


Length of con. rod 

Ratio of con. rod to crank 


C = distance from center to center 

of connecting rod. 
u = ratio of con. rod to crank. 
L = length of stroke. 


c-.\ 


I.O 

« = 
4. 10 
5-15 
6.00 






-4 


Weight of piston 


W = weight of piston. 

H = area of cyl. = 7-02 
4 










W=i.3H 


Weight of con. rod 


V = weight of con. rod. 

H = area of cyl. = — D2 
4 










F= .8ff 


Total weight of recipro- 


W = total wt. of piston. 












cating parts. 


V = total wt. of con. rod. 


W=iV+wH 


w = 






W-\-W=-i-lH 


Wt. of recip. parts per sq. 


H = area of cylinder. 




1.02 








in. of cyl. 


If = weight of truly reciprocating 
parts per sq. in. of cyl. 




1.70 

2.42 








Length of piston 


B = length trunk piston. 


\4 b J u 


fc = 


.018 
.025 
.036 


^ = 300 


B=iW 


Bearing pressure on pis- 
ton due to con. rod 


& = bearing pressure on projected 
area of piston (mean value dur- 


9.6 
6.9 
4.8 


« = S 




thrust. 


ing working stroke. 










u = ratio of con. rod to crank. 












p = maximum pressure 














Z) = cyl. diameter. 












Bearng pressure on pis- 


i' = bearing press, on proj. area of 










6'= .89 


ton due to weight. 


piston due to wt. of itself and 


-wD 






1v=l.^ 




portion of con. rod supported 


^'=V 






B^iiD 






by it. 














TO = wt. of recip. parts per sq. in. of 














cyl. 














D = diameter of cyl. 














B = length of piston. 


/.4l\ ^ 




/•4l\ 




D 


Thickness of rear wall of 


2 = thickness of rear wall of piston 


.= (;_) v^z> 


s = 


u)= 


P = 300 


'=ro 


piston. 


s = stress in rear wall of piston 




2.860 


.00766 






Stress in rear wall of pis- 


p = max. pressure. 




5,320 


.00562 






ton. 


£' = cyl. diameter. 




TO, 200 


.00405 







446 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table i. — Current Practice in the Dimensions of Gas Engines — (Continued) 













Assumptions made in 




Engine dimension and 
name of design con- 
stant upon which it 


Notation: All dimensions in ins., 
all pressures and stresses in lbs. 


Rational formula 
for engine dimen- 
sion in terms of the 


Maximum, 
mean and min- 
imum values of 


Corresponding 
numerical values 
of coefficient of 


deducing formula for 

average cases from 

rational formula; mean 


Reduced for- 
mula for aver- 


depends 


per SQ. m. 


design constant 


design constant 


formula 


value of design con- 
stant always used 


age, cases 


Length and diam. of wrist 


d" = diam of wrist pin. 


4/V 


i = 




^ = 300 


d"= .22D 


pin or piston pin. 


/" = length of wrist pin. 


<i"=\-—VpD 


13,300 








Stress and bearing pres- 


p — max. pressure' 


\4sb 


10.500 








sure on wrist pin. 


£) = diameter of cyL 
s = stress in wrist pin. 


IT, 


10,000 










b = bearing pressure on projected 


l"=\-d" 


fc = 






l" = iid" 




area of wrist pin due to maxi: 


\4& 


3,900 










mum load. 




2,800 








Area of mid-section of 
con. rod. 


a = area of mid-section of con. rod. 
k = factor of safety of rod or ratio 




2,260 
k = 
5-44 


k 




R= .23D 


44.560 


44,560" 


^ = 300 

Round rod assumed. 


Factor of safety in con. 


of breaking load by Ritter's 


^, / , .OOOI2C2\ 


3 90 


.0001220 


C2 




rod considered as a long 


formula to actual load. 


2 23 


0000857 


I-f .000I2X-^ 




column. 


C = distance, center to center of rod 

R = diam. of mid-secton if round. 

= height of mid-secton if rec- 
tangular. 
r = radius of gyration of mid-sec- 
tion. 

j-2=i?2/i6 or QV12 

D = diameter of cyl. 






0000500 


is given the average 
value 1 . 6 




Length of arm of bending 


I = length of crank-pin journal. 


M = cD 


c = 






M=.6£> 


moment on crank pin in 


r = length of main bearing journal. 




■450 








terms of cyl. diam. 


2m = distance from center to center 
of main bearings. 




.609 
.850 








- 


M = m-(ll-\-\n)= arm of effective 
bending moment on crank pin, 
for reaction on main bearing due 














to explosion. 
I = length of crank-pin journal. 




s = 


(-:) = 


M= .6D as found 




Diameter of crank pin.. . 


d=\(-^MpD'' 


d=.4iD 


Stress in crank pin 


I' = length of main bearing journal. 


y \ i/ 


18,800 


.000213 


above 






2m = distance from center to center 




10,600 


. 000379 


f> = 300 






of main bearings. 




7.500 


.000533 








M==m-(.il + in. 












- 


d = diam. of crank pin. 














i = stress in crank pin. 














D = diameter of cyL 














p = max. pressure. 












Length of crank pin 

Bearing pressure on crank 
pin. 


I = length of crank pin journal. 
d = diam of crank pin. 
6 = bearing pressure on projected 
area of crank, due to the aver- 


^.145'!\PD^ 
'-V 4b ) d 


b = 
158 
213 
348 


.I4ST 
46 

.000720 
.000535 


d= .41D 
from above 

p=300 


l=.95d 




age value of load for a complete 




.000327 








cycle. 












Thickness of crank 


X = thickness of crank throws (in 


x = cd 


c = 






x=U 


throws. 


direction of shaft axis). 
d = diam. of crank pin. 




.46 
.63 
.80 






1 


Breadth of crank throws. 


ji = breadth of crank throws (per- 
pendicular to shaft axis). 

« = thickness of crank throws (in 
direction of shaft axis). 


y = cx 


c = 
I. so 
2.12 
3 00 






y = 2ix 


Length of arm of equiva- 


V = length of main bearing journal. 


M' = cD 


c = 






M'=.4D 


lent bending moment 


L — length of stroke. 




■324 








on crank shaft, in terms 


I> = diameter of cyl. 




.400 








of cyl. diam. 


il^' = (-325'' + -09oZ.) = arm of equiv- 
alent bending moment on 
crank shaft (at inner edge of 
main bearing) for reaction on 




.468 










main bearing due to explosion, 
s = stress in crank shaft at inner 




s = 


a) = 


M'= .4D 




Diameter of crank shaft. 


d'='\J(^)pDm' 


d = iD 


Stress in crank shaft. , . . 


edge of main bearing journal. 


^ \ 0/ 


14,400 


.000278 


from above. 






d' = diam. of crank shaft at main 




9,500 


.000422 


^ = 300 






bearing. 




6,200 


. 000644 








I' = length of main bearing journal. 














M' = (.325r-f.090L). 














D = diameter of cyl. 














p = max. pressure. 













THE GAS ENGINE 



447 



Table i. — Current Practice in the Dimensions of Gas Engines — (Continued) 













Assumptions made in 




Engine dimension and 


Notation: All dimensions in ins., 


Rational formula 


Maximum, 


Corresponding 


deducing formula for 


Reduced for- 


name of design con- 
stant upon which 
it depends 


all pressures and stresses in 
lbs. per sq. in. 


for engine dimen- 
sion in terms of the 
design constant 


mean and min- 
imum values of 
design constant 


numerical values 

of coefficient of 

formula 


average cases from 
rational formula; mean 
value of design con- 
stant always used 


mula for average 
cases 


Length of main bearing 


r = length of main bearing journal. 


■■HiY-? 


b = 


\7^b)'^ 


d' = iD 


l' = 2id' 


journal. 


d' = diam. of main bearing journal. 


174 






Bear'g pressure on main 


6 = bearing pressure on projected 




123 


.000752 


p = 300 




bearings. 


area of main bearing, due to av- 
erage value of load for a com- 
plete cycle. 




98 


.001068 
.001334 






Outside diameter of fly- 


F = outside diam. of fly-wheel in ins. 


'-(^)i 


K = 


(^> 




p_ 12,300 


wheel. 




4,490 




n 


Velocity of fly-wheel rim . 


K = velocity of fly-wheel rim in ft. 




3,220 


17,140 








per min. 




2,290 


12,300 








« = revolutions per min. 






8,750 






Weight of fly-wheel 


U = total weight of all fly-wheels in 
lbs. 


f7 = 
272,300,000,000 


/= 
■ 034 


272,300,000,000 


Above value of rim 
velocity 


U = 


Speed fluctuation coeffi- 


/ 


33.000 H.P. 


cient. 


H.P. = rated horse-power. 


/- 


■ 054 


= 


J. 12,300 
n 


« 




F = outside diam. of fly-wheel in 


HP. 


.091 


8,000,000,000,000 






ins. 


That is 


5,000,000,000,000 








n = revolutions per min. 




3-4% 


3,000,000,000,000 








/= speed fluctuation coefBcient, or 




5-4% 










ratio of total variation in r.p.m. 




9-1% 










to the mean value. 












Rotation speed 


n = rotation speed, r.p.m. 


^70,382/ 

K = T^ 

VwL 


/ = 


\/70,382/ = 


a)= 1 .7 


800 


Inertia force at end of 


/ = inertia force at end of stroke 


8.14 


757 


as found above 


» = Vl 


stroke, per sq. in. of pis- 


per sq. in. of piston. 




15.40 


1,041 




This is equiva- 


ton. 


w — weight of truly recip. parts (pis- 
ton-)- i con. rod) per sq. in. 
of piston. 

L - length of stroke, ins. 




30.80 


1,472 




lent, to tak 
ing the piston 
speed in ft. 
per min. as 
133 VI 


Exhaust pipe diameter. 
Nominal speed of gases 


E = exhaust pipe diam. 
» = nominal speed of gases thro' 


E=(~\Dvni 


8,850 


(V6".^ = 


800 


E=.2iD 


through exhaust pipe. 


exhaust pipe, ft. per min. 




S.730 


• 00434 


as found above. Then 






n = revolutions per minute. 
L = length of stroke. 




3,120 


•00539 
.00732 


E depends on .y/Z, and 






D = diameter of cyl. 








hence varies little for 
different values of L. 
L taken as 12. 




Exhaust valve diameter. 


e = exhaust valve diam. 




f = 


(V6i) = 
■ 00497 


Same as above. 


e= .3D 


Nom'l speed through ex- 
haust valve. 


» = nominal speed through ex- 
haust valve. 


'-(V6i>^''" 


6,750 
5,200 








n, L and D as above. 




3.630 


.00566 
.00678 






Inlet valve diameter, . . . 


i = inler valve dia., when there is a 




V=: 


. 00447 


Same as above 


8= .27D 


Nom'l speed through 
inlet valve. 


valve admitting whole charge. 
V — nominal speed thro' inlet valve. 


'■=(vV>^^" 


8,330 
6,400 








M, L and D as above. 




4,680 


.00510 
.00598 






Gas pipe diameter 


G = gas pipe diam., natural gas. 




v = 


.00158 


Same as above 


G= .llD 


Nom'l speed through gas 
pipe. 


V = nominal speed thro' gas pipe, 
n, L and D as above. 


^=(vW>^^" 


6,670 

3.700 












2,380 


.00212 
.00264 






Gas valve diameter 


g = gas valve dia., natural gas. 




v = 


( ' )- 
.00224 


Same as above 


S=-15D 


Nom'l speed through gas 
valve. 


V = nominal speed thro' gas valve, 
n, L and D as above. 


^==Ut>^^" 


3,330 
2,080 












1,110 


.00283 
.00387 






Air pipe diameter 


A =air pipe diameter, natural gas. 


A = 


v = 


( ' ) 
^\/6.67zi '' 

.00374 


Same as above 


A = .25D 


Nom'l speedt hrough air 
pipe. 


V = nominal speed through air pipe. 
K, L and D as above. 


(-y^)DVrn 


10,700 
6,900 






■ 






4,500 


.00466 
.00577 






Maximum brake H.P. 


M.P. = maximum brake H.P. 


D^LnP 

M P = 

■'"■^ • 1,008,500 


P= 

50 






D'Ln 

M.P. = 

14,400 


Nominal mean eflective 


p' = mean effect, press, from area of 


(for four stroke cycle 


70 








pressure. 


indicator card. 
h = mechanical efiiciency, or ratio 

of brake to indicated power. 
P = hp' = nom'l M.E.P. 
D = cylinder diameter. 
L = length of stroke. 
M = revolutions per minute. 


engine.) 


85 









448 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



a = the amount of allowable variation, 

n = the revolutions per minute, 

N = the brake horse-power, 

K = coefScient varying with the type of engine. 
The coefficient K, is determined as follows: 

iir = 44,000 for Otto-cycle engines, single-cyUnder, single- 
acting. (Fig. I.) 

Jr = 28,000 for Otto-cycle engines, two opposite cylinders, 
single-acting, or one cylinder double-acting. (Fig. 2.) 

if = 25,000 for two cylinders single-acting, with cranks set 
at 90 deg. (Fig. 3.) 



K = 2i,ooo for two cylinders, single-acting. (Fig. 4.) 
K = 7000 for four twin opposite cyhnders, or for two tandem 
cyUnders, double-acting. (Fig. 5.) 
The factor a, the allowable amount of variation in a single revolu- 
tion of the fly-wheel is as follows: 

For ordinary industrial purposes -h to 3V 

For electric lighting by continuous current ^ to sV 

For spinning mills and similar machinery — '. ' xk's to xs 0^ 

For alternating-current generators in parallel xsir 

The total weight of the fly-wheel may be considered as equal to 
PXi.4- 



1 



COMPRESSED AIR 



Table i. — Pneumatic Constants 
Weight and Volume of Air 
I cu. ft. = .076097 lb. = 1.217 oz. 
I lb. = 13.141 cu. ft. 
Value of one Atmosphere of Pressure 
Lbs. per Column of Column of 

sq. in. water, ft. mercury, ins. 

14-7 33-947 3° 

Pressure Equivalents 
I lb. per sq. in. =2.04 ins. of mercury = 2.309 ft. of water. 
I in. of mercury =.49 lb. per sq. in. =1.132 ft. of water. 
I ft of water = .433 lb. per sq. in. = .883 in. of mercury. 
Temperature 62 deg. Fahr.; pressure 14.7 lbs. per sq. in. 

Table 2. — Barometric Pressure at Various Altitudes 



Altitude, ft. 


Mercury 
column, ins. 


Lbs. per sq. in. 


Water 
column, ft. 





30 


14.7 


33-95 


1,000 


28.88 


14- IS 


32.68 


2,000 


27.80 


13 -62 


31.46 


3.000 


26.76 


13-II 


30.28 


4,000 


25.76 


12 .62 


29 15 


S.ooo 


24.79 


12.15 


28.05 


6,000 


23.86 


11 .69 


27.00 


7,000 


22.97 


II .26 


25. 99 


8,000 


22. II 


10.83 


25.02 


9,000 


21.28 


10.43 


24. 08 


10,000 


20.48 


10.04 


23.18 


11,000 


19.72 


9.66 


22.32 


12,000 


18.98 


930 


21.48 


13,000 


18.27 


8.95 


20.67 


14,000 


17-59 


8.62 


19.90 


13,000 


16.93 


8.30 


19- 16 



Table 3. — Equivalents of Ounces per Sq. In., in Ins. of Height 
OF Columns of Water and Mercury 



Ozs. per sq. in. 


Ins. of water 


Ins. of mercury 


.146 


•25 


.018 


.292 


•SI 


•037 


.438 


.76 


■05s 


.584 


1. 01 


.074 


I 


1-73 


.127 


2 


3-46 


•255 


3 


S.20 


.382 


4 


6.93 


.510 


5 


8.66 


■637 


6 


10.39 


.765 


7 


12.12 


.892 


8 


' 13.85 


1.019 


9 


15-59 


1. 148 


10 


17.32 


1-275 


II 


19.05 


1.402 


12 


20.78 


1.529 


13 


22.52 


1.658 


14 


24^25 


1-785 


IS 


25.98 


1913 


16 


27,71 


2.036 



The word efficiency has two special meanings as applied to air com- 
pression, these being called volumetric and compression efficiency. 
The former refers to the volume of air taken in compared with the 
piston displacement. Loss of volumetric efficiency is chiefly due to 
re-expansion of air from the clearance spaces as the suction stroke 
begins. It, hence, increases with the volume of the clearance spaces 
and with the receiver pressure. The clearance spaces being the same, 
the loss is less with compound than with simple compressors because 
of the reduced pressure produced by the first cylinder. It is com- 
monly measured by dividing the actual length of the suction line of 
the indicator card by the total length of the card, although this 
ignores a known but unmeasured source of loss due to the warming 
of the air as it enters the hot cylinder. 

The compression efficiency compares the developed with the theoret- 
ical air horse-power, in which comparison two practices prevail. 
The first compares the actual power with that due to isothermal 
compression, while the second compares it with single-stage adiabatic 
compression. Isothermal compression being an impossible condition, 
the first practice has little real significance, while, adiabatic compres- 
sion being the normal condition, the second practice furnishes a 
ready means of expressing the actual gain (often actual loss) of com- 
pound over simple compression. 

Air compressors should not draw air from warm engine rooms. A 
suction flue connecting with the cooler out-door air gives rise to con- 
tinuous economy. The gain is approximately i per cent, for each 
5 deg. Fahr. difference of temperature, the gain appearing in increased 
delivery of air which costs nothing. 

Compressed-air Power Calculations 

The fundamental formulas for the adiabatic compression of gases are: 



General 


For air 


For natural gas 




Pi W 


Pi_ /vi\ 1-41 
Pi~ W 


p2 M\ 1-26S 

Pi~ W 


{a) 


ti /lli\ n-l 

<1~ w 


t2_ /"iV" 

tl~ \V2/ 


h /»l\-266 

h~ w 


(b) 


h (P2\ "-' 
tl \pl/ » 


ti /p2\ -29 
h \pj 


ti /Pi\ -21 
h \PJ 


ic) 



in which ^i = initial pressure, abs., 
^2 = final pressure, abs., 
i>i = initial volume, 
t)2 = final volume, 
n= initial temperature, abs., 
<2 = final temperature, abs. 
_ specific heat at constant pressure 
specific heat at constant volume 
= 1.408 or, for practical purposes, 1.41 in the case of air 

The work of air compression, including that due to explusion of the 
air from the cylinder, adiabatic compression being assumed, may be 
most conveniently expressed by the formula: 

OT.e.^.=3.4S/'i('"^'-i) (o) 

p2 

in which r= —. 
Pi 

Calculations of the mean effective pressure may, in most cases, be 

abbreviated by the use of Table 4, of which the constants of the 

third column multiplied by the initial pressure, lbs. per sq. in. abs., 

give directly the m.e.p., lbs. per sq. in. For compression at sea level 

the multiplication has been carried out to give both the m.e.p. and 



29 



449 



450 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



the air h.p. per loo cu. ft. free air compressed per min. It should, 
however, be noted that raost compressors do not realize full 
atmospheric pressure in the cylinder and, in such cases, the actual 
pressure at the beginning of compression should be used when cal- 
culating the theoretical m.e.p. The assumption of too high an 
initial pressure results in a credit for cooling to which the compressor 
is not entitled. 

Affecting the actual m.e.p. may be mentioned such cooling as there 
may be from the cylinder jacket, the re-expansion of the air in the 
clearance spaces, and the effect of the clearance spaces in reducing 
the actual compression ratio of volumes from the apparent ratio, all 
of which tend to reduce the mean pressure, while the "camel backs" 
due to the opening of the discharge valves tend to increase it. 

The effect of jacket cooling is always small and the actual m.e.p. 
should not differ much from that given by formula (d). 





Table 4 


. — Constants 


FOR Single 


-STAGE Compression 




a 


£ II 

t-i 

a 

is 

'S 


V 

a 
> 

u 


w 
!i 

3". 
« >- 

a II 

0, 

is 

'S 


J 


Compre 


ssion from 14. 


7 lbs., initial 


i 

u< 
O. 

u 
flj 

> 

o 

V 

Pi 


Gage 

pressure, 

lbs. 


M.e.p., 
lbs. 


H.p. per 100 

cu. ft. free air 
per min. 


I -23 


1 .067 


.231 


3.7 


3-4 


1.48 


I-S 


1. 127 


.431 


7-3 


6.3 


2.76 


I-7S 


1. 176 


.607 


II. 


8.9 


3.89 


2 


I . 222 


.766 


14.7 


II. 3 


4.91 


2.2S 


1.26s 


■ 914 


18.4 


13.4 


5.86 


2.5 


1.304 


1.049 


22.0 


15.4 


6.73 


2.7S 


I. 341 


I. 176 


25.7 


17-3 


7.54 


3 


1-375 


1.294 


29.4 


19-0 


8.30 


3-25 


1.408 


1.408 


33-1 


20.7 


9.03 


3S 


1.438 


I. 511 


36.7 


22.2 


9.69 


3-75 


1.467 


1 .611 


40.4 


23.7 


10.33 


4- 


1.495 


1.708 


44.1 


25 I 


10.96 


4-2S 


1. 521 


1.797 


47.8 


26.4 


11-53 


4.S 


I. 547 


1.887 


51.5 


27". 7 


12. 14 


4-75 


I. 571 


1.970 


55. 1 


29.0 


12.64 


5. 


1.595 


2.053 


58.8 


30.2 


13.17 


S-2S 


1. 617 


2 . 129 


62.5 


31.3 


13-66 


SS 


1.640 


2.208 


66.2 


32.5 


14. 16 


5-75 


1. 661 


2. 280 


69.8 


33.5 


14.63 


6. 


1. 681 


2.349 


735 


34-5 


15-07 


6.2S 


1 .701 


2.418 


76.2 


35.5 


IS. 51 


6.5 


I. 721 


2.487 


80.9 


36.5 


15-95 


6.75 


1.740 


2.553 


84.6 


37-5 


16.38 


7- 


1.758 


2.615 


88.3 


38.4 


16.77 


7-25 


1.776 


2.677 


91.9 


39.3 


17.17 


7-5 


1.794 


2.739 


95-6 


40.3 


17.57 


7.75 


i.8n 


2.798 


99-3 


41. I 


17-95 


8 


1.828 


2.8S7 


103 .0 


42.0 


18.33 



Compound or stage compression is resorted to in order to save power 
and to reduce the final temperature and thereby lessen the danger of 
explosion of decomposed oil in the air receiver and pipes. Such 
explosions — whatever their explanation — have happened too many 
times to permit the danger to be ignored. In high-pressure work 
compounding becomes a mechanical necessity. 

For the most economical results in compound (two stage) compression 
the work should be equally divided between the cylinders, and that 
is accomplished by making the number of compressions in each 
cylinder equal the square root of the total number of compressions — 
the number of compressions being understood as the higher divided 
by the lower pressure, abs. 



The work of compound compression, including that due to expulsion 
of the air from the cylinders may be most conveniently e.xpressed by 
the formula: 

m.e.p. = 6.qopi{r..^^^ — i) (e) 

in which pi = initial pressure, lbs. per sq. in., abs. 
final pressure, abs. " 
~ initial pressure, abs. 

This equation gives the m.e.p. reduced to the low-pressure cylinder, 
under the assumption that the cylinders are proportioned as called 
for above, that the intercooler reduces the temperature of the air to 
that at which compression began and than the valves and passages 
offer no resistance to the flow of air. 

As with the formula for simple compression, calculations of the 
mean effective pressure may, in most cases, be abbreviated by the 
use of Table 5 of which again the constants of the third column 

Table s- — Constants for Two-stage Compression 







3 


1 








to 


^_^^ 


XI 


J3 


. 


rt 


a 


.0 ' 


J3 






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5 

5.5 
6 
6.5 

7 

7-5 

8 

8-5 

9 

9.5 

10 



13 

14 



1.263 
1.280 
1.297 
1. 312 
1.326 

1.339 
1.352 
1.364 
1.375 
1.386 

1.396 
1 .416 
1-434 
1-450 
1 .466 



I. 815 
1-932 
2.049 
2.153 
2.249 

2.339 
2.429 
2.512 
2.587 
2.663 

2.732 

2. 870 
2.993 
3-105 
3.215 



Compression from 14.7 lbs. initial 



Gage 

pressure, 

lbs. 



58.8 
66.2 
73. 5 
80.9 
88.3 

95.6 
103.0 
no. 2 

117 .6 
125-O 

132-3 

147-0 
161 . 7 
176.4 
igi . I 

205-8 



M.e.p., lbs. 
reduced 

to low 
pressure 

piston 



26. 7 
28.4 
30.1 
31.7 
33-1 

34.4 
35-7 
36-9 
38.0 
39-2 

40.2 
42 . 2 
44-0 
45.6 
47-3 



H-p. per 

100 cu. ft. 

free air 

per min. 



Possible 
saving by 
compound- 
ing, per 
cent. 



II .64 
12.39 

13.14 
I3-8I 
14-43 

15-00 
15.58 
16. II 
16.59 
17-08 

17-52 
18.41 

19. 21 
19.92 
20.62 



II. 6 

12 . 2 
12.8 
13-4 
14.0 



14.6 
15.2 



multiplied by the initial pressure, lbs. per sq. in., abs., give directly 
the m.e.p., lbs. per sq. in., reduced to the low-pressure piston. For 
compression at sea level, the multiplication has been carried out to 
give both the m.e.p. and the air horse-power per 100 cu. ft. free air 
compressed per min., while the last column gives the theoretical sav- 
ing due to compounding under the assumptions of the last paragraph. 

The air horse-power for each 100 cu. ft. of free air compressed per 
min. for simple or compound compression. 

= .436Xm'.e.p. (/) 

As applied to pressures in common use for industrial purposes — 
say 80 to 100 lbs. per sq, in. — the margin of saving by compounding 
which, as Table 5 will show, is not large, may be more than offset by 
defective design. 

This is shown in Fig. i from actual indicator cards. Because of 
deficiant capacity of the intercooler, the volume of air entering the 
high-pressure cylinder is not reduced to the isothermal line as it 
should be, while the overlapping of the high- and low-pressure cards 
due to inadequate valves more than offsets such gain as the inter- 
cooler of itself brings about, the final result being an actual loss of 
power due to compounding. That good results can be obtained by 
compounding, is shown in Fig. 2 (from a Nordberg compressor) in 
which the volume of air entering the high-pressure cylinder is carried 
back to the isothermal line, while the overlapping of the high- and 
low-pressure cards is almost negligible. If suflSciently cold water is 



COMPRESSED AIR 



451 



available, there is no reason why an efficient intercooler should not 
reduce the temperature of the air below that at which compression 
began and thus carry the volume of air entering the high-pressure 
cylinder within the isothermal line. As a matter of fact, this has 
often been done. 

Air. Compression at High Altitudes 

The effect of altitude on air compression is to decrease both the deliv- 
ery of air and the consumption of power but not in the same ratio, 
the net result being to increase the power consumed in producing a 
given volume of compressed air. 

The relation of the volume of air delivered by a given compressor at 
sea level and at an altitude, the gage pressure of delivery being the 
same and ignoring clearance losses, is given by the equation: 

(«) 



Table 6. — Relative Amounts of Air Deliveked by a Given Com- 
pressor AT Various Altitudes 



»2 



Pi 



P 

p2 

in which vi = volume of delivery at sea level measured at the delivery 
pressure and after the heat has dissipated, 

1)2 = volume of delivery at an altitude measured at the 
delivery pressure and after the heat has dissipated, 

/>! = barometric pressure at sea level, 

^2 = barometric pressure at an altitude, 

P = gage pressure. 
Table 6 has been calculated from this formula. The actual reduc- 
tion of delivery due to altitude is, however, greater than the table 
shows. The loss due to clearance in- 
creases with the ratio of compression and 
since, for a given gage pressure, this 
ratio increases with the altitude, the 
clearance losses increase likewise. The 
heat due to compression is also a func- 
tion of the ratio of compression and 
hence, for a given gage pressure, the 
temperature of the compressed air in- 
creases with the altitude. For both 
reasons compounding is of increasing 
importance with increase of altitude. 

Graphic Compressed-air Power 
Calculations 



Altitude, ft. 


Relative output at gage pressures 


70 lbs. 


100 lbs. 


o 


1. 000 


1. 000 


IjOOO 


.969 




968 


2, GOO 


• 938 




936 


3,ooo 


.908 




90s 


4, GOO 


.880 




875 


S,ooo 


•853 




846 


6,ooo 


.826 




818 


7,OOG 


.798 




790 


8,GGG 


.772 




763 


9,OGG 


.746 




737 


IO,GGG 


.720 




712 


1 1, GOO 


.697 




688 


12,000 


.67s 




665 


13,000 


• 654 




642 


I4,GGG 


.632 




620 


15,000 


.611 


•S99 



To find the final temperature find the value of — on the scale at 

pi 

the top, trace vertically to the line for the suitable initial tempera- 



The foregoing calculations may be 
made graphically by the aid of Figs. 3 
and 4, by J. A. Brown (Amer. Mach., 
June 12, 1913), Fig. 3 being for single 
and Fig. 4 for two-stage compression, 
the assumption in the latter being that 
the intercooler reduces the air to its 
initial temperature. 

To use the chart. Fig. 3, for single- 
stage compression: Find the absolute 
final pressure on the scale at the left by 
adding the barometric pressure for the 
required altitude, Table 2, to the gage 
pressure. Trace horizontally to the 
line for the altitude, vertically to the 

(pi\ -29 

line marked (— I and horizontally to 



Lbs, 




Lbs 



Lbs. 
100 



80 



CrankEnd 



Fig. 



E.P.M. 52 



Head End 




Crank End 



Fig. 2. 



Head End 



the right where read the value of I — 



-(!:)■ 



Figs, i and 2. — Good and poor compound air-compressor practice. 



find the resulting 



Subtract, mentally, i from the value of 

value on the lower scale and from it trace vertically to the altitude 
line and then horizontally to the right where read the m.e.p. from 
the middle scale and the horse-power per 100 cu. ft. free air per min. 



ture and then horizontally to the scale at the center of the chart where 
road the final temperature, Fahr. absolute. 

Fig. 4 for two-stage compression is used in precisely the same 
way. 



452 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 
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Fig. 3. — Mean effective pressure, horse power and temperature of compressed air. Single stage compression. 



The Index of the Compression Curve 

To find, the index of an actual compression curve corresponding to 
the theorectical value 1.41 in equation (a), measure the pressure and 
volume at two points on the indicator card, as widely separated as 
possible, for which any scale may be used, inches divided into tenths 
being most convenient. Call the lower pressure pi the corre- 
sponding volume vi, the higher pressure p2 and the corresponding 
volume Vi. The index being unknown we have then to solve the 
equation : 

p2_ fvA X 

pi~ W/ 



The solution of this equation will give the required index. The 
work may be abbreviated by the use of Table 7 as follows: Select 
such values of p2 and pi that the former shall be an exact multiple — 
2, 3, 4, 5, 6, 7 or 8 times the latter. 

Measure V2 and Vi to correspond with p2 and pi; divide Vi by V2 

and find the resulting ratio in the column of — standing under the 



P2 



Vi 



Opposite the ratio for — will be found the 



Let 



^2 r, , ^1 

~ = R and — = r 

pi V2 



giving R = r^ 

That is, log i2=log, r^ 

or log R = x log r 

log 2? 

or x = , 

logr 



W 



ratio for — selected. 

Pi 
value of the index. 

The index of the compression curve may also be found graphically 
by the aid of Fig. 6, by J. A. Brown {Amer. Mach., June 28, 1900). 
Take for example the card shown in Fig. 5. From the card measure 
V and p at two points and by any scale — inches and tenths being 
most convenient — one at the beginning of compression where p 
measures .44 in. and v 6.1 ins. Take any other point, say ^ = 1 in., 
and » = 3.i2 ins. On the chart Fig. 6 mark the intersection of .44 
on the scale of pressures with 6.1 on the scale of volumes; mark the 
intersection ol p=i and ?; = 3.i2 in the same way. Using two 



COMPRESSED AIR 



453 



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Fig. 4. — Mean effective pressure, horse power and temperature of compressed air. Two stage compression. 



triangles as a parallel ruler find the index diagonal to which a line 
through these points is most nearly parallel and read the figures 
for the approximate index — in this case 1.25. If through the inter- 
section of p and V at beginning of compression, diagonals parallel 
to the isothermal and adiabatic lines be drawn, intersections with 
these lines give corresponding values of 11 and p, and for every 
value of V, the three values of ^ corresponding to isothermal, adia- 
batic and the actual compression curve plotted on Fig. 5. 

The Friction of Compressed Air in Pipe 

The formula for the friction of compressed air in pipe, taking into 
account the increase in volume and velocity that accompanies drop 
in pressure, was first established by Professor Unwin {Transmis- 
sion of Power) . SHghtly transformed to make it read volume instead 
of weight it is as follows: 



7 = 3 



.oA^sf7P 



-P2') 



in which V = volume, cu. ft. free air per min. at sea level and 60 
deg. Fahr., 
i = diameter of pipe, ins., 
/»i = initial pressure, lbs. per sq. in., abs. 
^2 = terminal pressure, lbs. per sq. in., abs., 
/ = coefficient of friction, 
/ = length of pipe, ft. 
The coefficient of friction is not constant but varies with the diam- 
eter of the pipe. According to Professor Unwin it is expressed 
by the equation: 

/=. 0027(1+-^-^) 

in which d = diameter of pipe in ft. 

Problems involving the friction of compressed air in pipe may be 
solved by the use of Fig. 7 by Jas. A. Brown {Amer. Mach., July 10, 
1913) which incorporates both the above formulas. The use of the 
chart is explained below it. It will be recognized that the method of 



454 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 

Table 7. — Values of the Index of Compression Curves 

ire Higher nressiirfi Hialier nrpssnre TTiVhpr nrPRsnrft 



Higher pressure 


Higher 


pressure 


Higher pressure 


Higher pressure 


Higher pressure 


Higher pressure 


Higher pressure 


Lower pressure 
Pi 


Lower 

p2_ 
Pi 


pressure 
R = 3 


Lower pressure 

f-^ = R = 4 
Pi 


Lower pressure 


Lower pressure 

7^ = R = (> 

Pi 


Lower pressure 
Pi 


Lower pressure 
!-' = K = 8 

Pi 


a 

1 


0) 

a 
.3 II 

^ J'., 

i> B BJ 
•3 11 

8 

CO 


0) 

> 

3 

^ II 
« 
K II 
OJ 

•0 

C 


a 

3 

1 

u 


a 

il 

■3 II 
a 

CO 


S,i 




a 
3 

> 

u 

E? 


a 

It 

■3 II 
a 

CO 


^11 

« 

K II 


a 

_3 

"o 

> 


Smaller volume 
_vi _ _ 

V2~ ~ 


0) 

> 

3 
U II 

V 

•a 
a 


a 

3 

1 

1 


a 

> •- 
t li 

.HSIS 

•3 11 

CO 


ill 

•^ H 
«ll 

c 




a 

3 

"o 
> 

u 
OJ 

ij 


Smaller volume 

V2 


3 II 

•a 
c 


a 
3 

> 

u 


a 

•s II 
■3 II 

1 


II 
to 


1. 95 


1.04 


2.9s 


1.02 


3-90 


1.02 


4.90 


1.02 


5.80 


1.02 


6.80 


1.02 


7.80 


1. 01 


1.90 


I.c8 


2.90 


1.03 


3.80 


I. 04 


4.80 


1.03 


S.60 


1.04 


6.60 


1.03 


7.60 


1.03 


1.88 


1. 10 


2.85 


I. OS 


3.70 


1.06 


4.70 


1.04 


5.40 


1.06 


6.40 


1.05 


7.40 


1.04 


r.86 


1. 12 


2.80 


1.07 


3.60 


1.08 


4 60 


1.06 


5.20 


I .09 


6. 20 


1.07 


7.20 


1.06 


1.84 


1.14 


2.75 


1.09 


3.50 


l.II 


4-50 


1.07 


5.00 


I. 12 


6.00 


1.09 


7.00 


1.07 


1.82 


1. 16 


2.70 


r.ii 


3.40 


I. 13 


4.40 


1.09 


4.90 


I 13 


5.80 


I. II 


6.80 


1.09 


1.80 


r.i8 


2.6s 


1. 13 


3-30 


1.16 


4-30 


I. 10 


4.80 


I. 14 


5.60 


I -13 


6.60 


1. 10 


1.78 


1.20 


2.60 


I. IS 


3.20 


I. 19 


4.20 


I. 12 


4.70 


I. 16 


5 40 


I. IS 


6.40 


1. 12 


1.76 


1.23 


2. 55 


1. 18 


3.10 


1.23 


4.10 


I. 14 


4.60 


I. 18 


5.20 


1. 18 


6.20 


1. 14 


1.74 


I. 25 


2.50 


1.20 


3.00 


1.26 


4.00 


I. 16 


4-50 


I. 19 


5-00 


I. 21 


6.00 


I.l6 


1.72 


1.28 


2.4s 


1.23 


2.95 


1.28 


3.90 


I. 18 


4.40 


I. 21 


4.90 


1.23 


5.80 


1. 18 


1.70 


1. 31 


2.40 


1.26 


2.90 


1.30 


3.80 


I. 21 


4.30 


1.23 


4.80 


1.24 


5.60 


1. 21 


1.69 


1-33 


2.3s 


1.29 


2.85 


1-33 


3.70 


1.23 


4.20 


I. 25 


4.70 


1.26 


5.40 


1.23 


1.68 


r-34 


2.30 


1.32 


2.80 


1.35 


3.60 


1.26 


4. 10 


1.27 


4.60 


1.28 


5.20 


1.26 


1.67 


1.36 


2.28 


1.34 


2.78 


1.36 


3.50 


1.29 


4.00 


1.29 


4-50 


1.30 


5.00 


1.29 


1.66 


1-37 


2.26 


1.35 


2.76 


1.37 


3.45 


1.30 


3-90 


1.32 


4.40 


1.32 


4.90 


1. 31 


1.6s 


1.39 


2.24 


1.36 


2.74 


2.38 


3.40 


1.32 


3.80 


1-34 


4-30 


1-34 


4.80 


1-33 


1.64 


1. 41 


2.22 


1.38 


2.72 


1.39 


3.3s 


1-33 


3-70 


1.37 


4.20 


1.36 


4.70 


1. 35 






2.20 


1.40 


2.70 


1.40 


3.30 
3.2s 
3- 20 


1. 35 
1.36 
1.38 


3.60 


1 .40 


4. 10 
4.00 


1.38 
1 .40 


4.60 

4. SO 

4.40 


1.36 
1.38 
1. 41 



from which factors for other lengths may be calculated. 



introducing the length factor is due to the fact that, unlike the case 
of liquids, the friction loss is not in simple proportion to the length 
of the pipe. The factors of the table below the chart are values of 

/lOO 

I 

Plotting the Compression Curve 

The plotting of the adiahatic curve involves the solution of equation 
(i) which, for this purpose and for air, may be more conveniently 
written: 

p2Vi ^■'^^ = plVi 1-" 



Values of pi and vi are to be measured from the indicator card 
near the beginning of the compression when various smaller values 
of V2 being taken, the corresponding values of p2 may be calculated 
The process may be greatly abbreviated by the use of the constants 
of Fig. 8, which gives the relative increase of pressure for the re- 
duction of volume as the compression goes on. To use the diagram 
the indicator card should be divided into tenths and half-tenths, as 
this diagram is divided. Then, taking the absolute pressure at the 
beginning of compression, the product of this pressure by the multi- 
plier at the left of each ordinate of the diagram will give the pressure 




.4 .5 .6 .7 .8 .9 

Relative Volumes 
Fig. 8. — Ordinates of the adiabatic curve for air. 




Fig. 9. — Construction of the adiabatic curve. 



COMPRESSED AIR 



455 



due to the adiabatic curve at the corresponding ordinate of the indi- 
cator card. The diagram is applicable to high and low pressure cards 
alike, the proper initial absolute pressure being taken, of course. 

For compression from 14.7 lbs. initial the multiplications have 
been made and entered on the right side of the vertical lines of the 



c» >A a» 

00 «3 T-1 




Fig, 5. — Compressed air indicator card analyzed by the chart. Fig. 6. 



diagram, from which the pressures at the various ordinates may be 
read directly for that initial pressure. 

The use of this initial pressure is, however, seldom justified be- 
cause of the suction loss which exists in most cases. In such cases 
the assumption of full atmospheric pressure results in a curve which 
is too high for the truth, and gives the compressor credit for a degree 
of cooling to which it is not entitled. 

The plotting of the adiabatic curve for any value of the index 
may be done graphically as in Fig. 9 {Amer Mach., June 21, 1900). 
Draw OJ at any convenient angle « with OX. Determine the 
angle /? from the relation 

i+tan /3 = (i+tan a)" 

in which w = the required index. 

Draw OC at the angle ^ with OY. Through A draw AB parallel 
to OX and A J parallel to OY. Lay off BC at an angle of 45 deg. 
with OY, and from the intersection of BC and OC draw a horizontal 
line CE. From the intersection of AJ and OJ draw a line at 45 
deg. with OX and cutting OX at H. At H erect a perpendicular 
cutting CE at E and OJ produced at Ji. Then £ is a point on the 
curve, and so proceed. The smaller angle a is taken the more closely 
the points of the curve will be located, but the greater the opportunity 
for instrumental error. Obviously the construction may be begun 



S 





1 




















2 










Relative Pressures 

8 4 


5 6 


7 8 


■:■■: ::: 

IeI! hi 
nil 111 

111! Ill 

iEJiK:!!! 


1 


^ 







1 


^ 


1 


1 


^ 




^ 




=u« 




.-^ 














.9 

.8 
.7 


-^ 


== 




— 


= 
= 
= 


s 
= 

= 




wfi 

— Jvt— pJ- J 

-^S^s — EEEEEEE 


- --: :::::::::: 


■■■■■■£S>SiSS! 

issssEzzHsEs:: 

aESSSESiSSSSSS 

iiiinii 

nuisHiliill 

hSskk::;:;; 




^ - - — 


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— 






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— 




— 


- 




" ^- ^"' T"-f-- 


























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1 


1 




*^^'- - ^ 








1 














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SS 










p 


- 


1 n"=F-T-r^--ZD 


rr 1 1 rvYtrniTii T^T'TiTni i ^Tnn ni imTTI 


5 

















— 














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r- 




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s 




> > s 


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S 




































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\ 








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s 


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:;:.::... .+.5. ...:,,. h 


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_.:: .:: :.:...:..:.,...!s,.:|s,..--- .-. 


































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":.: :::::::._ i :!,--'k _... 


































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S V^^^V^S^^^^^ 


Sri :::-'-'--- > :■> :'i-... 






































is^^^^^ N-i "-^^ 


s, :: :-.,...:!, 




S 


































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S'^^v, :>iL.., : ,_.: 




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X^v^^^.5,^^ 


s.^s^^, , Z^... 






s 
































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, s>,^ V :;,.. fA^iu 




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V 
































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.3 






s 






























"\\^'^^^ 


s 'v V S ^ V, ::, 






\ 


-N 


^ 




























^ ^.\ ^ 


. ., 






^ 


••^ 


•\ 




























^"'s •^^ 


%,s '-S h.N,: ;; i 




\ 




\ 




N 


























L \ "^ 


s !s \ \,s. :>.-;;. 




\ 






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fs, 
























-^^"■^ 


.i_!..^,_^..:^.:^::;:=,..., 






N 




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s 


s 






















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\js.:^.\..^,.:s ■■=.:$^i 






\ 






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V 






















\ 


. \ ^ \ V i/--. ■■,::# 


m 








S, 






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s 


s 




















t^:.s.s;:.,. :.,.:::.:::,: 


syortf 
















\^ 






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[ jjijSW 


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V 








\, 






S 
















....'^.::..::^.::^.::... .:::.':■ 


^^ &■, 








^ 


\. 








\ 




1 


X 














^\ 'n ~-^ "^^ ■■•■I^ 


!^mH '■' 












\ 










\ 






S, 


s 










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^ 










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s 


. 






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"•■^l''. ■• • 
















\ 








1 


\ 










s 


s^ 


' ""m 


li'''':'ri 


















\ 






1 
1 




v 


\ 










i- 

- 


^Iwt 




















^ 




1 
1 










\ 


s 


"^N 


' y''r\r''-ri 






















\ 


1 
\ 1 
















""" - \;^-:'^^';'^^ 
























1 


s 














; rx"'x ■■'. 
























1 




\ 










S --- -- 

s 


. , . ^. .1 Si. .. .. 


1 






















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\ 




N. 


^» 11 



Fig. 6. — Finding the index to an actual compression curve. 



456 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



at point 5 as readily as at point A and conversely, if we have a curve 
for which we wish to derive an exponent, we can, by working back- 
ward, locate the lines OC and OJ, measure the angles a and 13, and 
solve for n. 

For the index value 1.41 the method of Fig. 8 is to be preferred as, 
from its nature, the method of Fig. 9 involves an accumulation of 
error which impairs the accuracy of the result. 

The isothermal curve has comparatively little application to com- 



pressed air as actual compression is always nearly adiabatic. 
equation is: 

Pi 1)1 
in which /'i = initial pressure, abs., 
^2 = find pressure, abs., 
i'i = initial volume, 
112 = find volume. 



Its 



10,000 100,0001,000.000 




Directions for use : Double the initial absolute pressure and from the product subtract the desired pressure loss. Find the re- 
sult on the left hand vertical scale ; trace horizontally to the right to the line for the desired pressure loss ; thence vertically to the 
line for the diameter of the pipe ; thence horizontally to the right hand scale where read the volume of free air in cu. ft. per min., 
supposing the pipe to be 100 ft. long. For other lengths multiply this volume by the proper factor from the following table : 



Length, 
ft. 


Factor 


Length, 
ft. 


Factor 


Length, 
ft. 


Factor 


Length, 
ft. 


Factor 


100 


1. 00 


7SO 


•36s 


2,500 


. 20 


7,000 


.119 


200 


■71 


1,000 


.316 


3,000 


.183 


8,000 


. 112 


300 


■578 


1,250 


.283 


3,500 


. 169 


9,000 


.105 


400 


•SO 


1,500 


.258 


4,000 


.158 


10,000 


.10 


500 


• 448 


i,7So 


• 24 


5,000 


.141 


15,000 


.082 


600 


• 41 


2,000 


.224 


6,000 


.129 


20,000 


.079 



Fig. 7. — Friction of compressed air in pipes. 



COMPRESSED AIR 



467 



The chief use of this equation in compressed-air work is in laying 
down the isothermal curve on combined indicator cards in order to 
determine the efficiency of the intercooler in reducing the tempera- 
ture and volume of the air before entering the high-pressure cylinder. 
For a graphical method of constructing the curve see Isothermal 
Curve. 




5 10 15 20 25 30 35° Fah. 

Difference in Temperature between Air Leaving and Water Entering 

Fig. io. — Relation of surface and capacity of intercoolers. 




The Intercooler 

The design of intercoolers should be such that the air and water 
pass through them in opposite directions in order that the incoming 
or coolest water may act on the air at the last stage of the cooling. 
The outer surface of the tubes should be the air surface as its greater 
area compensates, in part, for its lesser efficiency. There is no 
advantage in copper or brass over iron tubes, in fact, in such com- 
parative experiments as have been made iron tubes have been found 
the more efficient — due probably to their greater roughness. 

The cooling area required for any given final effect may be deter- 
mined from Fig. lo by H. V. Haight, Chief. Engr., Canadian Inger- 
soU-Rand Co., {Amer. Mack., Aug. 30, 1906) which represents the 
ormula (determined by experiment) : 

in which y = free air capacity of intercooler, cu. ft. per min. per 
sq. ft. of cooling surface measured on the air side, 
a; = difference in temperature, deg. Fahr., between the air 
leaving and the water entering. 
The Nordberg construction of intercooler, Figs. 11 and 12, has an 
unusual provision to compel the water to flow equally through all 
the tubes in addition to the counter-current direction of air and water. 
The tube plates are shown at aa while complete tubes are shown at 
bh and others — cut off to avoid confussion — are shown at cc. The 
air enters the intercooler at d and is discharged at e, while the water 
enters at / and is discharged at g. Baffle plates hhh guide the air 
in the manner indicated by the arrows Hi, while baffles in the water 
heads insure the flow of the water in the opposite direction as indi- 
cated by the arrows jjj. 

Reheating Compressed Air 

The gain due to reheating compressed air formed the subject of 
experiments by W. G. Edmondson and E. L. Walker {Amer. Mach., 
July 31, 1902) a 2 h.p. shaft governor engine being used. The 
consumption of air per h.p. was, no doubt, greater than with 



FiG. i2.'-Sectional Elevation 

Figs. 11 and 12. — The Nordberg 
intercooler. 



50 100 150 200 250 300 850 400 450 
Temperature of Reheated Air 

Fig. 13. — Reduced consumption of com- 
pressed air due to reheating. 































, --- 


40 




,^' 






ti 


^ 














--/--- 








...i _ 




35 










ik 


'^y 






w/ 


^ 


J'' 




1^ 


S 


:;;: :::::: 




a'/ts^ 


y j,.^ 


/' 




?i^^tX' 


M 






Mtm, 


^■^ 




c: 


?(.../ 


- kX-- 







.1 /. 


tsg' 




^ "jc; 


L../...I 


^J. 






1 / t 


1^ 






' 






s 


" ' .J -/ -../ 






2 


/ -/ . 






a 


. :.J.....i.... 






8 20 


/ 7 :; 








/..I..../ 






B 


.. / L. 






C 


. _.J 






rt 


,.J....i 






15 


I I ... , 








/ / 








/ / 








' .. ' 








,,.., 






10 


7 / 








/ /• 








jt 1 








I 








.... ]..1 








/ 








...., ;::::._... ::_ 








f! / 








1 xt 


n T 













50 100 150 200 250 300 350 400 450 
Temperature of Air entering Engrine 

Fig. 14. — Increased economy of com- 
pressed air due to reheating. 



L 



458 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



larger engines but there is no apparent reason why the gain due to 

reheating should not hold. The tests were made at three pressures 

and various degrees of reheating. The results on the air consumption 

per brake h.p. are given in Fig. 13, while Fig. 14 shows the 

gain in economy, including the cost of reheating. In this chart, 

. . B.t.u.'s per h.p. reheated 

gain in economy = i — — ; ; ; :■ The results found 

B.t.u. s per h.p. not reheated. 



in its groove. The piston is thus a floating piston, the weight of 
which is carried by the piston rod. This rod, as befits its duty, and 
as shown in Fig. 16, is extremely light. It was made of steel pipe, 
its diameter providing ample stiffness. Each end carries a shoe for 
supporting the weight of the piston, the shoes and the guides being 
shown in Fig. 16. 

When the slight deflection of a piston rod which suffices to transfer 



Oil Groove 




m 



Fig. 15. Fig. 16. 

Figs. 15 and 16. — Nordberg construction of piston and piston rod of blowing engines for very light pressures. 



exceed those to be expected from 
the calculated expansion of the air 
due to the reheating. This is ex- 
plained by the increased mechanical 
efficiency of the engine when heated 
air was used. Indicator and brake 
tests showed large friction losses 
due to the low temperatures when 
unheated air was used — losses that 
grew markedly less when the air 
was heated. The tests showed the 
fuel cost of the air obtained by 
compressing to be from 8 to 19 
times that obtained by heating. 
Were the first cost of the com- 
pressor plant to be compared with 
that of the heater, the comparison 
would be still more striking. 




Fig. 17. — Nordberg construction of piston rod for moderate air pressures. 



Details of Air Compressors 



Piston and piston-rod constructions for large blowing engines under 
light air pressure, as used by the Nordberg Mfg. Co., (Amer, Mach., 
Feb. 7, 1907) are shown in Figs. 15 and 16. The pistons. Fig. 15, 
which are of 70 ins. diam. and designed for an air pressure of 40 oz. per 
sq. in., were made of f-in., boiler plate, each side in one piece and 
dished to the form shown. Lateral stiffness was provided by riveting 
the plates at the outer diameter to a ring spider in halves, the piston 
ring groove being between the halves. The piston was turned 2 f ins. 
smaller than the bore of the cylinder and the ring did not bottom 



its weight to the cylinder is considered, the conclusion seems inevita- 
ble that the provision of slides at both ends of rods of the usual pro- 
portions is of more than doubtful value as a means of transferring 
the weight of the piston from the cylinder to the slide. 

Another piston-rod construction used by the Nordberg Co. for air 
pistons under somewhat heavier pressures (in this case 7 lbs. per sq. 
in.) is shown in Fig. 17 {Amer. Mack., Nov. 23, 1905). With such low 
pressures, the objection to large stuffing boxes disappears and the 
rod is of enormous size — 16 ins. diam. for an air cylinder of 62 ins. 
diam. It is of cast-iron, hollow and within it is a forged rod, the 
twoJaeing so connected that the forged rod carries the tensile and the 



COMPRESSED AIR 



459 



cast rod the compression strains. A key at a serves to put the forged 
rod under initial tension. 

Air vahes for a large blowing engine (62X42-in. air cylinder, pres- 
sure 7 lbs. per sq. in., speed 75 r.p.m.) as made by the Nordberg 
Mfg. Co. are shown in Fig. 18 (Amer. Mack., Nov. 23, 1905). The 
valves are of the Corliss type and are essentially similar to steam 
cylinder valves, except that their functions are reversed, the inlet 
air valves being similar to exhaust steam valves and the outlet air 
valves similar to admission steam valves. All valves are double- 
ported, a provision which gives to the air valves the unusually gen- 
erous effective port area of 13.4 per cent, of the piston area. Fig. 





r— , 












'T 




; 


r 1 


r 1 


■r 






1 
■• 1 


1 


i 1 






lE 


3 


1 i lalet 

U c'H 


//^ 


>■ 





Fig. 18. — Nordberg construction of double ported Corliss valves for blowing engine cylinders, 




plate made of a steel punching, which, by means of two arms fixed 
by plugs, is guided without friction, the valve guard and a helical 
spring between valve and guard. Fig. 3 shows all these parts in 
detail. The only purpose that the spring has to serve is to efiect 
the closing of the valve plate at the proper time. Suction- and 
discharge-valve parts are identical. 

The method of inserting the valves into the casings is also shown 
in the illustration. The valve is inserted between the casing and 
guard, which former is provided with stems connecting it with the 
cover plate. The tightening of the valve and the cover is effected 
by rings made of a suitable graphitic material. 

Poppet valves for air-compressor 
service {Riedler system) are shown in 
Fig. 20 {Amer. Mack., Oct. 16, 1902). 
The special feature of this system 
of valves is that while they are 
closed positively by mechanism they 
are opened by the air or water, as 
the case may be. A moving lever 
closes the valve at the end of the 
stroke, and then, before the time for 
the valve to open arrives again, the 
lever withdraws and leaves the valve 
free to open whenever the conditions 
require it. The action is the same 
with both suction and discharge 
valves, and much of the mechanism 
is common to both. Both sets of 
valves require closing at the same 
-at the extreme end of the stroke — and since that is all 
mechanism does, the same movement is obviously as 



Fig. 19. — Borsig construction of air-compressor valves. 




Fig. 20. — Poppet valves for air compressors — Riedler system. 

4 shows the valves in relation to their seats and the method by which 
the inlet valves are given a double port. 

Spring-closed relief or safety valves are also provided (not shown) 
to provide for any possible failure of the regular valves (which are 
positively actuated) to act. 

Poppet valves for air compressor service, as made by A. Borsig, 
Tegel, Germany, are shown in Fig. 19 {Amer. Mack., Nov. 5, 1908). 

The valve arrangement consists of a cast-iron seat, a thin valve 



moment— 

that the 

appropriate to one as to the other. 

Fig. 8 is a longitudinal section of the upper end of the low-pressure 
air cylinder and its valves, the surrounding casing being broken 
away. The valves will be seen to be double seated, the direction 
of the air currents being shown by the arrows. The manner in which 
the levers ab act to close the valves will be apparent on inspection. 
At cd are air gag or choke pots, the office of which is to prevent any 
rebound of the valves from the stops which limit their opening when 
the compressor is running at high speed. The valves of the high- 
pressure cylinder are similar to those of the low except that they are 
single seated. 

Packing for a high-pressure air plunger of a 4-stage air compressor 
for 1000 lbs. per sq. in. pressure, by H. V. Haight, Chf. Engr., 
Canadian IngersoU-Rand Co. {Amer. Mach., Apr. 23, 1908), is shown 
in Fig. 21. The construction gave excellent results, although it is 
not easy to see the office of so many packing rings outside the oiler 
ring or lantern. The plunger should be ground to reduce wear of 
the packing rings. 

The cuts in the inside rings do not lead to appreciable leakage. At 
most they form a long and tortuous passage with many enlargements 
to destroy the energy of the moving air. Moreover they fill with 
oil and soon close up. 

For pipe fittings for high-pressure compressed air see Pipe Fittings 
for High-pressure Air. 



Consximption of Compressed Air 

The consumption of compressed air by various pneumatic tools as 
made by the IngersoU-Rand Co. is given in Table 8. The figures are 
not mere estimates based on piston displacement but are the re- 
sults of careful tests by the makers, the air being accurately measured 
by a water-displacement meter. 

The consumption of compressed air by Curtis direct-lift air hoists 
is given in Table 9. 



460 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 8.- 
Crown Hammers 



-Consumption of Air by Pneumatic Tools 

Imperial Hammers 



Size 


Cyl 


inder 


Weight, 
lbs. 


Cu. ft. free 
air per min. 


Uses 


no. 


Bore, 


Stroke, 


60 Ibj. 


100 lbs. 




ms. 


ms. 




pres. 


pres. 




Sl-H 


i-l 


1 


9 


10 


17 


Chipping and calking bath 
tub and range boilers and 
other light work. 


S2-H 


i| 


2 


10 


II 


i8 


Light chipping and calking, 
beading flues and sealing 
castings. 


53-H 


i| 


3 


II 


12 


20 


General chippingand calking. 


S4-H 


li 


4 


IS 


13 


22 


Heavy chipping and calking. 
Riveting light tanks and 
heavy sheet iron. 



Little David Piston Drills 



Size 
no. 


Weight 
lbs. 


Max. 

diam. 

twist 

drill, ins. 


Max. 

diam. 

wood 

bit, ins. 


Max. 

diam. 

reamer, 

ins. 


Max. 

diam. 
tap, 
ins. 


Max. 
diam. 

flue 
roller, 

ins. 


Cu. ft. free 

air per 

min. at 

90 lbs. 

pres. 


I 

2 

, 3 


S3 
40 
23 
31 
20 


2 


4 
2 


2 

I 
1 


2 

I 

i 


3 

2i 


SS 
30 


13 








30 









Crown Sand Rammers 





2ngth 
er all, 
ins. 


CO n 


Cylinder 


Cu. ft. free air 


per min. 


at pres. 




Bore, 


Stroke, 


40 


SO 


60 


70 


80 


90 


100 




J s 


^ 


ins. 


ins. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


Bench ram- 


20 


10 


I 


4 


12 


145 


17 


19J 


22 


2S 


28 


mer lO-SR. 
























Floor ram- 


46 


22J 


li 


S 


12 


IS 


18 


21 


24i 


28 


32 


mer 20-SR. 

























Table 9. — Consumption or Air by Direct-lift Air Hoists 



Nominal inside 
diameter of 
hoist in ins. 



Capacity in lbs. at 80 lbs. 

air pressure. 10% 

allowed for friction loss 



Cu. ft. of free air re- 
quired to lift full load I ft. 
at 80 lbs. air pressure 



4 


861 


•54 


5 


1.356 


■ 85 


6 


2,0S0 


1 .22 


7 


2,791 


1.73 


8 


3-616 


2.24 


9 


4,592 


2.8s 


10 


S.636 


3-29 


12 


8,154 


S06 


14 


11,270 


7-13 


17 


i6,soo 


10. 10 


19 


20,900 


12.50 



The consumption of compressed air by direct-acting steam pumps 

may be determined from the formula: 

h , 

V = a (.0307 X;^ +.799) 





Cylinder 




Cu. ft. free 




Size 






Weight, 


air per 


Uses - 


no. 


Bore, 


Stroke, 


lbs. 


min. at 80 




ins. 


ins. 




lbs. pres. 




I 


lA 


I^ 


10 


20 


Chipping and calking bath 
tubs and range boilers and 
other light work. 


2 


lA 


2 


Hi 


20 


Light chipping and calking 
beading flues and sealing 
castings. 


3 


lA 


3 


I2i 


20 


General chipping and calking. 


4 


I A 


4 


i3i 


21 


Heavy chipping and calking. 


5 


lA 


S 


IS 


21 


Extra heavy chipping and 
calking. 


40 


lA 


4 


13 1 


21 


Driving riviets J-in. diameter 
and less. 


SO 


lA 


5 


IS 


21 


Driving rivets j-in. diameter 
and less. 



Imperial Motor Hoists 



Size 
no. 


Capacity, 
lbs. 


Ft. lift per min. 
at 80 lbs. pres. 


Max. lift, 
ft. 


Cu. ft. free air 

per ft. lift of 

max. load 


I 


1,000 


32 


20 


1. 41 


2 


2,000 


16 


20 


2.82 


4 


4,000 


8 


20 


5.63 


7 


7,000 


8 


20 


10 


10 


10,000 


7 


20 


II. 4 



in which F = volume of air consumed, measured at atmospheric 
pressure, cu. ft. per min., 
a = area of steam piston, sq. ins., 
/! = head of water, ft. 

diameter of steam cylinder 
diameter of water cylinder' 
The formula is the full rational formula reduced to its lowest terms 
with the added assumption that 20 per cent, of the power is consumed 
in friction of the pump and of the water in the pipes and that 15 per 
cent, of the piston displacement is lost in clearance and leakage. 
The compressor is assumed to operate under 14 lbs. atmospheric 
pressure. A useful modification of the formula is the following: 
F = .628/«+i6.9r2 



'nSV ^V^^VsSSVVVl=/-SVVvVVxV\<i 







Packing Rings 



Oiler Ring 



Packing Rings 



^^§^?:^^-:<SS<Sf:>S:;:fSSSS:if:^^ii::ii^>:^ 





Every Alternate Ring on Outside of Red Brass 



17 Inside Rings I 2 lbs. Genuine Babbitt 

8 Outside Rings \ 1 lb. Lead 

9 Outside Rings Tied Brass 



Fig. 21. — Haight's packing for high-pressure air plungers. 



COMPRESSED AIR 



461 



in which F = volume of air measured at atmospheric pressure re- 
quired to pump loo gals, of water, cu. ft. 

The formulas have been verified by tests on compressors driving 
pumps exclusively. 

The discharge of air through orifices is given in Table lo. 



Water 




Water -^ X 



<_Air 




The Air-lift Pump 

The air lift pump of Dr. E. S. Pohle, a remarkable invention in 
point of simplicity and eSectiveness, is shown in its original form 
in Fig. 22. Other forms are used, including that of Fig. 23, by the 
IngersoU-Rand Co. Whatever the arrangement, the principle is the 
same. The water pipe is well submerged and, air of adequate pres- 
sure being admitted, it rises in the pipe and forms a mixture of air 
and water which being lighter than the surrounding water rises and, 
if the depth of submergence be sufficient, overflows at the outlet. 
Numerous installations exist in which water is raised from 600 to 
1200 ft. 

The theory of the pump is obscure, but the following formula by 
E. A. Rix and the Ingersoll-Rand Co. gives the volume of air 
required : 

h 



Fig. 22. Fig. 23. Fig. 24. Fig. 25. 

Figs. 22 and 23. — The air Figs. 24 and 25. — Air lift pumps 
lift pump. tested by Mr. Kelly. 



F = - 



Clog 



g+34 

34 



in which V =air piston displacement cu. ft. (ordinary volumetric 
efficiency assumed) per gal. of water 
h = vertical lift, ft., 
H = submergence of water pipe, ft., 
C = a constant from Table 1 1 . 

It must be borne in mind that the working water level, when 
pumping from wells, is commonly below the standing level and by 
an amount that cannot usually be known in advance. It is hence 
customary to assume certain conditions of lift and submergence 
based on experience and pipe the wells accordingly. After the pip- 
ing is installed and working conditions arrived at, the submergence 



Table 10. — Discharge of Compressed Air through Orifices 
By The Ingersoll-Rand Company 



Receiver gage pres- 
sure, lbs. per 
sq. in. 



Diameter of orifice, ins. 



Discharge, cu. ft. of free air per min. 



2 

S 
10 

IS 



25 

30 
35 
40 

45 

50 
60 
70 
80 
90 

100 

125 



.038 

• 0597 

.0842 

.103 
.119 

.156 

• 173 
.19 
.208 

.225 
.26 

• 295 

•33 

• 364 

.40 
.486 



•153 
. 242 
■ 342 
.418 
.485 

•54 
.632 

•71 
• 77 
•843 

.914 

1-05 
1. 19 

1-33 

1-47 

1. 61 
1.97 



.647 

•965 
.136 
1.67 
^•93 

2. 16 
2.52 
2.80 
3^07 
3-36 

3-64 

4.2 

4.76 

5-32 

5.87 

6.45 
7.85 



2-435 
3-86 

5-45 
6.65 

7-7 

8.6 
10 

11. 2 

12. 27 
13-4 

14-50 
16.8 

19 
21. 2 

23-50 

25-8 
31-4 



9-74 
15-40 
21. 
26. 70 
30.8 

34-5 
40 

44-7 

49-09 

53-8 

58-2 

67 

76 

85 
94. 

103. 
125 



21-95 
34.60 

49 
60 
69 

77 

90 
100 

110.45 
121 

130 

151 

171 
191 
211 

231 
282 



39 
61.60 

87 
107 
123 

138 

161 

179 

196-35 

215 

232 
268 
304 
340 
376 

412 
502 



6r 

96.50 
136 
167 
193 

216 
252 
280 

306 . 80 
336 

364 
420 
476 
532 
587 

64s 
785 



87.60 

133 
196 
240 

277 

310 
362 
400 

441-79 
482 

522 
604 
68s 
765 
843 

925 



119-50 
189 
267 
326 

378 

422 
493 
55° 
601 .32 

658 

710 
622 
930 

1004 



156 
247 
350 
427 

494 

550 
645 
715 

785 - 40 
860 

930 



242 
384 
543 
665 
770 

860 
1000 



350 
550 
780 
960 



625 
98s 



462 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table i i. — Values of C in Formula for Air-lift Pumps. Proper 
Submergence Assumed 
h =Lift. C 

lo ft. to 60 ft. inclusive 245 

61 ft. to 200 ft. inclusive 233 

201 ft. to 500 ft 216 

501 ft. to 650 ft 185 

651 ft. to 750 ft. inclusive 156 

is altered to suit by raising or lowering the pipe until the best rates 
are established. 

The necessary percentage oj submergence varies with the lift. Mean- 
ing by percentage of submergence the percentage of the total length 
of pipe submerged when pumping, the range, according to the In- 
gersoll-Rand Co., is as follows: 

For a lift of 20 ft 66 per cent. 

For a lift of 500 ft 41 per cent. 

The average best percentage in the class of work usually encoun- 
tered will lie between 50 and 65 per cent. 

The air pressure required does not depend upon the lift h but upon 
the submergence H and is greater when the pump is being started 
than when at work, because the submergence is greater with the water 
at the standing level. The starting pressure must slightly exceed 
the pressure due to the submergence or, say: 

starting pressure = .44 H 
the pressure being in lbs. per sq. in. and the submergence in ft. 
The working pressure is equal to the working submergence multi- 
plied by the same constant, but, as has been said, the working sub- 
mergence is frequently unknown in advance. 



It is important that the pipes be proportioned to the flow, because 
with too large a pipe the air rises through the water without doing 
all the work it should, while, with too small a pipe, the results are 
undue friction loss and inefficient e.xpansion of the air bubbles. 
According to the Ingersoll-Rand Co., the proper dimensions and 
capacities of the arrangement shown in Fig. 22, which is the most 
economical and should be used when the well is sufficiently large, 
are as given in Table 12. 

Table 12. — Dimensions and Capacities of Air-lift Pumps of 
the Type Shown in Fig. 22 









Maximum econom- 


Air pipe, ins. 


Water pipe, ins. 


Size of well, ins. 


ical capacity on 

moderate lift, gals 

per min. 


1 


I 


3 


7 


i 


i| 


4 


20 


I 


2 


4i 


35 


I 


2i 


S 


60 


li 


3 


6 


90 


A 


3^ 


7 


120 


ih 


4 


8 


160 


ij 


S 


9 


250 


2 


6 


10 


350 



In case of necessity these capacities can be increased 20 to 40 per 
cent, but at a decreased efficiency. 

The arrangement shown in Fig. 23 is used to obtain the greatest 
possible output from a given size of well casing. It is not always 




12 14 16 18 20 22 24 26 28 30 32 
Revolutions of Compressor, per Minute 



20 22 24 26 28 30 32 34 36 38 40 
Revolutions of Compressor, per Minute 




12 14 16 18 20 22 24 26 28 30 32 
Revolutions of Compressor, per Minute 



20 22 24 26 28 30 32 34 36 38 40 
Revolutions of Compressor, per Minute 




12 14 16 18 20 22 24 26 28 30 32 
Revolutions of Compressor, per Minute 



i 1 1 ~1 i i '■ 

Volume of Water Kaisdd Pe* Minute 



..38 & 3^ 



■wirliinstosftK 

























Wei 


:.^o, 


^ 


EfBciency 














l^'^^o 


'tit's 




















"^^[^e. 













































22 24 26 28 30 32 34 36 38 40 
Revolutions of Compressor, per Minute 



10 15 20 25 30 35 40 45 50 55 60 
Revolutions of Compressor, per Minute 



6 
5 

'^ A 
O 

2 3 















C-'-i' 


.^' 












, 6P' 


f 


V 












-^0^ 


>^ 














Vol 


ume 


of A 
of 


r us( 
Wate 


3dPe 
r Ha 


rCu 
sed 


)ic F 


oot 













































10 15 20 25 30 35 40 45 50 55 60 
Revolutions of Compressor, per Minute 



1200 
^1000 

,3 800 

S 600 

p« 

§400 
U 200 









«, 


w 


)rUinB to 


jette 


r 






J^ 




B^ 


• 


^ 










<f 


k 


^ 


















Voh 


ime 


)f Wi 


.ter : 


iaise 


i Pe 


•Mi: 


lute 













































10 15 20 25 30 35 40 45 60 55 60 
Revolutions of Compressor, per Minute ' 



Fig. 26. Fig. 27. 

Figs. 26 to 28. — Results of air lift pump tests by Mr. 



Fig. 28. 



KeUy. 



COMPRESSED AIR 



463 



as economical as the arrangement shown in Fig. 22 but may be used 
when the well is very strong and a great deal of water is wanted from 
a few wells. According to the Ingersoll-Rand Co., the proper size 
of air pipe for the different sized casings and the capacities to be 
expected are about as given in Table 13. 

Table 13. — Dimensions and Capacities of Air-lift Pumps of 
THE Type Shown in Fig. 23 



Casing, ins. 


Air pipe, ins. 


Capacity, gals, per min. 


3i 


li 


80 to 100 


4 


I* 


100 to 150 


5 


2 


150 to 250 


6 


2 


275 to 37S 


8 


2i 


SCO to 650 


10 


2\ 


775 to 1000 



The efficiency of the air-lift pump is not high, but for many uses 
this is more than offset by its remarkable simplicity and consequent 
freedom from derangement and by its capacity to deliver all the 
water a well will supply — a capacity which is shared by no other 
deep-well pump. 



Th*e most complete set of test data known to the author are those of 
James Kelly {Proc. I. C. E. 1906). Two arrangements, shown in 
Figs. 24 and 25 were tested, the dimensions of the pumps being given 
in Table 14, while the results are given in Figs. 26, 27 and 28. When 
consulting these data it should be remembered that the water was 
measured in British gallons. The figures for efficiency give the ratio 
of the work done in raising water to the work indicated in the air 
cylinders of the compressor. The compressor used was a compound 
Ingersoll-Rand having air cylinders of 285 and 16 j in. diameter by 
24-in. stroke. 



Table 14. — Dimensions of Air-lift Pumps Tested 
BY Mr. Kelly 



Number 
of well 


Depth, 
ft. 


Diam- 
eter, ins. 


Area of 

delivery, 

sq. in. 


Effection 
area of air- 
tube, sq. 
ins. 


Depth of 

delivery, 

ft. 


Distance 

from 
comp. ft. 


38 
39 

40 


350 
350 
350 


12 
12 
12 


19-63 
12.56 
12.56 


18.75 
16.2 
3-1416 


339-5 
347-0 
326.5 


600 

820 

5400 



MECHANICS 



The mechanical advantage or iticrease of force due to the "me- 
chanical powers" — lever, pulley, wheel and axle, inclined plane, 
wedge and screw — whether used singly or in combination, is th.e in- 
verse ratio of the velocity of the applied force (power) and of the 
resisting force (weight). To determine the mechanical advantage 
it is only necessary to determine the velocities at the beginning and 
ending of the train of mechanism, when: 

power velocity of weight 

weight velocity of power 

Such calculations assume ideal conditions, of course, that is, they 
ignore the losses due to friction. 

Differential mechanisms are seldom successful because of a feature 
that is commonly overlooked. This feature was first pointed out 
by Geo. B. Grant, (Amer. Mack., Sept. lo, 1895). Mr. Grant 
discussed differential gearing only, but the cause of failure of such 
gearing appears to be general and to operate against the success of 
most applications of the differential principle. The cause of failure 
of differential gears, as pointed out by Mr. Grant, is that the teeth 
operate under a combination of the heavy pressure of the driven gear 
and the high speed of the driving gear, this combination leading to 
destructive wear and to such low efficiency that the mechanical ad- 
vantage for which differential mechanisms are usually designed is 
not realized. 

If the reader will reflect a moment he will see that this combina- 
tion of heavy pressure and high speed is common to all differential 

Table i. — Velocities Due to Heights of Fall 
From Clark's Manual of Rules, Tables and Data 

This table gives also the spouting velocities of water — the column for height, 
being read as head. 



Height, 

ft. 



Velocity, 

ft. per 

sec. 



.02 

• 03 
.04 

• OS 

.06 
.07 
.08 
.09 



.7 
.8 
.9 



1.4 
1.6 



2.50 

2.7s 



.803 
1. 14 
1.39 
1.61 
1.80 

1.97 
2. 12 
2.27 
2.41 
2.54 

3-20 

4.40 

3-07 
5-68 
6. 22 

6.71 
7.18 
7.61 
8.03 
8.79 

9-50 
10. IS 
10.77 
11-35 
12.04 

12.69 
13-31 



Height, 

ft. 



13 

14 
IS 

16 

17 



Velocity, 

ft. per 

sec. 



13.90 
iS-oi 
16.05 
17.03 
17.99 

18.82 
19.66 

20. 46 

21. 23 
21.97 

22.69 
23.40 
24.07 
24-73 
25-38 

26-62 
27 .80 
28.93 
30.03 
31.08 

32. 10 
33-09 
34-05 
34-98 
35-89 

36-77 
37-64 



Height, 
ft. 



23 
24 
25 
26 

27 

28 
29 
30 
31 
32 

33 

34 
35 
36 
37 

38 
39 

40 
41 
42 

43 
44 
45 
46 
47 



Velocity, 

ft. per 

sec. 



Height, 

ft. 



Velocity 

ft. per 

sec. 



38.49 
39-31 
40- 12 
40-92 
41-70 

42-47 

43-22 

43-95 
44-68 
45-39 

46. 10 

46-79 

47-47 
48 -IS 
48-81 

49-47 
SO. II 
50-75 
51-38 
52.01 

52.62 
53-23 
53-83 
54-43 
55-02 

55.60 
56.17 



50 
100 
150 
200 
300 

400 
500 
600 
700 
800 

900 
1000 
1500 
2000 
2500 

3000 
3500 
4000 
4500 
5000 

6000 
7000 
8000 
9000 

lOOOO 



56-74 
80.2s 
98.28 

1I3-S 

139 

160.5 
179-9 
196.6 
212.3 
226.9 

240.7 
253-8 
310.8 
358.9 
401.2 

439 -S 
474-7 
507-S 
538-3 
567.4 

621 .6 

671.4 
717-8 
761-3 
802-S 



mechanisms, of which about the only successful example is the dif- 
ferential pulley block. The exception of this construction to the 
general law is apparently due to the fact that the bearings subject 
to the combined heavy loads and high speed are of a type which per- 
mits them to be made large enough for the service. 

Table 2. — Heights of Fall Due to Velocities 
From Clark's Manual of Rules, Tables and Data 

This table gives also the heads necessary to produce given spouting velocities 
of water — the column for height being read as head. 



Velocity, 

ft. per 

sec. 


Height, 
ft. 


Velocity, 

ft. per 

sec. 


Height, 
ft. 


Velocity, 

ft. per 

sec. 


Height, 
ft. 


Velocity, 

ft. per 

sec. 


Height, 

ft. 


•25 


.0010 


19 


S-61 


46 


32-9 


73 


82.7 


-SO 


.0039 


20 


6-21 


47 


34-3 


74 


85 


■ 75 


.0087 


21 


6-85 


48 


35-8 


75 


88.4 


1. 00 


.016 


22 


7-52 


49 


37-3 


80 


99.4 


1.25 


.024 


23 


8.21 


SO 


38.8 


85 


112. 2 


1.50 


.035 


24 


8.94 


SI 


40.4 


90 


125.8 


I.7S 


.048 


25 


9-71 


52 


42 


95 


140. 1 


2 


.062 


26 


lo-s 


53 


43-6 


100 


155-3 


2-5 


.097 


27 


II. 3 


54 


45-3 


105 


I7I^2 


3 


.140 


28 


12. 1 


55 


47 


no 


187^9 


3-5 


. 190 


29 


13-1 


S6 


48.7 


115 


205.4 


4 


.248 


30 


14 


57 


SO. 4 


120 


223.6 


4-5 


• 314 


31 


14.9 


58 


52.2 


130 


262.4 


5 


• 388 


32 


lS-9 


59 


54-1 


140 


304.3 


6 


.559 


33 


16.9 


60 


SS-9 


ISO 


349 4 


7 


.761 


34 


17.9 


61 


57.8 


175 


475-5 


8 


.994 


35 


19 


62 


S9-7 


200 


621 


9 


1.26 


36 


20.1 


63 


61.6 


300 


1397 


10 


1-S5 


37 


21.3 


64 


63-6 


400 


2484 


II 


1.88 


38 


22.4 


65 


65.6 


500 


3882 


12 


2.24 


39 


23.6 


66 


67.6 


600 


5890 


13 


2.62 


40 


24.9 


67 


69.7 


700 


7609 


14 


3.04 


41 


26. I 


68 


71.8 


800 


9938 


15 


3.49 


42 


27.4 


69 


73-9 


900 


12578 


16 


3.98 


43 


28.7 


70 


76-1 


1000 


15528 


17 


4.49 


44 


30.1 


71 


78.3 






18 


5.03 


45 


31.4 


72 


80. 5 







Table 3. — Heights of Fall and Velocities Due to Time 
From Clark's Manual of Rules, Tables and Data 



Time, 
sec. 



13 
14 
IS 
16 



Height, 
ft. 



16. 1 

64.4 

144.9 

257.6 

402. s 
S79.6 
788.9 
1030 

1304 
1610 
1948 
2318 

2721 
3156 
3623 
4122 



Velocity, 
ft. per sec. 



32.2 

64.4 

96.6 

128.8 



161 
193-2 
225.4 
257.6 

289.8 
322 
354-2 
386.4 

418.6 
450.8 
483 
515-2 



Time, 
sec. 



17 
18 
19 



23 

24 

25 

26 

27 
28 

29 
30 
31 
32 



Height, 
ft. 



4.653 
5,217 
S.812 
6,440 

7,100 
7,792 
8,517 
9.273 

10,062 
10,884 
11,737 
12,622 

13.540 
14.490 
15.473 
16,487 



Velocity, 
ft. per sec, 



547.4 
579-6 
6X1. 8 
644 

676.2 
708.4 
740-6 
772-8 

805 
837-2 
869.4 
901.6 

933-8 
966 
998.2 
1030 



464 



MECHANICS 



465 



The thrust of a toggle joint may be determined graphically as in Figs. 
I and 2, the force F being applied at A, Fig. i, and the thrust at B 
or C. 

In the diagram, Fig. 2, make the perpendicular PQ of such length 
as to represent the applied force F and draw PR parallel to ^C and 
QR parallel to AB. The lengths of PR and QR will then represent 




Fig. I. 

FlC3. 2. 

Figs, i and 2. — Forces in a toggle joint. 



If the joint has equal arms this becomes 



R = - 



2 tan a 



The laws of falling bodies, starting from a state of rest, are ex- 
pressed by the equations: 

V =32.2 t 



V = \/64.4 h 

= S \/ h nearly 
^=16.1 t^ 

V 



t=- 



32.2 



t = 



= i\/~Fnearly 
thestressesin the links 4 C and ^5, respectively, while the horizontal in which » = acquired velocity, ft. per sec. 



SR represents the thrust. 
Trigonometrically we have: 



R=F- 



i = time of fall, sec, 
h = height of fall, ft. 
These relations of velocity, height and time are tabulated in Tables 
I, 2 and 3. 




500 600 700 800 900 1000 

Revolutions per Minute 
Trace vertically from r.p.m. to the curve and then to the left and multiply the quantity found by the diameter, ins. and by the 

weight, lbs. 

Fig. 4. — Centrifugal force. 
30 



466 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



The laws of motion of bodies acted upon by uniform accelerating 
forces are expressed by the equations: 

P 

V2=Vl+i2.2-^t 
P 

in which Vi = final velocity, ft. per sec. 

Di= initial velocity, if any, ft. per sec. (if the body starts 

from rest J)! = o) , 
P = force acting, lbs., 
G = weight of body, lbs., 
i = time during which force acts, sec, 
5 = space passed through, ft. 
If the force is a retarding force these become: 

P 

V2=Vi—32.2^t 
P 









A 

/O B 








/ s 








A ^ 






N 

A 


/ 




E 




y 


/ 






J' 


I E 


/ 









Fig. 3. — Graphical solution of problems in accelerated motion. 

Problems involving the laws of uniformly accelerated motion, as 
falling bodies, may be solved by drawing a diagram similar to Fig. 3. 
On the vertical line A Y lay off equal distances representing seconds, 
AB being unity. Make BC equal to the acceleration — 32.2 ft. per 
sec. for gravity — and draw AX. Then, after five seconds, for 
example, Z,F = velocity, area ALF= the distance traversed and area 
LFEN = distance traversed during the last second. 

If the acceleration is not uniform and its law is known, AX may 
be drawn to represent it, the construction being otherwise the same. 

The energy stored in moving bodies is expressed by the equation: 

Gv^ 



£ = 



64.4 



in which £ = stored energy, ft. lbs., 
G = weight of body, lbs., 
V = velocity of body, ft. per sec. 
The additional energy stored by an increase of velocity of a moving 
body is expressed by the equation: 

^ = 6^4^^^^-^^^) ■ 

in which £ = energy stored by the increase of velocity, ft.-lbs., 
G = weight of body, lbs., 
2)1 = initial velocity of body, ft. per sec, 
112 = final velocity of body, ft. per sec. 
The energy given out by a reduction of velocity of a moving body 
is expressed by the same equation with the notation of vi and fa 
inverted. 

The centrifugal force of revolving bodies is expressed by the equation: 
P = .ooo3399W^Gr 
in which P — centrifugal force, lbs. , 

w = revolutions per minute., 
G = weight of body, lbs., 

r = radius of gyration or with sufficient accuracy for most 
purposes radius of the center of gravity of the body, ft. 



Calculations of centrifugal force may be abridged by the use of 
Fig. 4, by N. J. Hopkins {Amer. Mach., Feb. 10, 1898). 

For the stress in a revolving ring due to centrifugal force, see 
Fly-wheel. 

The center of gravity of many plane figures is obvious at a glance, 
being the same as the center of area. Following are formulas for 
some other figures of common occurrence, the point c being the center 
of gravity in all cases. 






Circular arc: oc = 
Simicircle: 



chord X radius 





arc 
oc = .6366 X radius. 
For tabulated lengths of circular arcs, see 
Circular Arcs. 

Triangular area: Bisect two sides and con- 
nect the dvisioin points with the opposite 
angles. The intersection c is the center of 
gravity. Dropping a perpendicular ca to any 
side ca = JXaltitude be. 

Any quadrilateral: Draw the diagonals ab, 
de; bisect ab at /; make eg = di; join / and 
g and divide fg into three equal parts; c is the 
center of gravity. 




Circular sector: oc = 



2 X chord 



X radius. 



3Xarc 

Semicircle: oc = .4 244 X radius. 

Quadrant: oc = .6002 X radius. 

For tabulated lengths of circular arcs see Cir- 
cular Arcs. 



Circular segment: oc = ; 



chord^ 



12 X area 

For tabulated areas of segments see Areas of 
Circular Segments. 

A portion of a circular ring 

outer arc 



lr^-r^\ 
W-ri') 



X 



outer chord 



t 

I ( 
I / 
I / 



1.1' 



For tabulated lengths of circular arcs, see Cir- 
cular Arcs. 

Aia 

Circular crescent oc = —t — 
A2 

in which A = area of segment bounded by arc 

of smaller radius and common 

chord, 

/1 1 = area of segment bounded by arc 

of larger radius and common 

chord, 

/1 2 = area of crescent = A—Ai 

a = distance between centers of the 

two arcs. 

For tabulated values of areas of segments, 

see Areas of Circular Segments. 



The center of gravity of irregular figures may be found experimentally 
by the method shown in Fig. 5 as follows: 

Trace the figure upon heavy paper or card-board and cut it out. 
Suspend the figure thus made from a pin placed near the edge of the 
figure at A , allowing it to hang freely in a vertical plane. Suspend 
a plumb-line from the pin and draw upon the figure a line coincident 
with the position of this plumb-line. Suspend the figure from another 
point B and find a similar line. Where the two lines intersect is the 
center of gravity of the surface. 

The center of gravity of irregular figures may be found graphically 
by the method shown in Fig. 6 by F. H. Hummel {Proc. Brit. C. &* 
M. E. S., 1900). 



MECHANICS 



467 



The problem usually takes the form of finding a line in a given 
direction passing through the center of gravity of a given section. 
Let the direction be OY. Draw a line OF in this direction and touch- 
ing the base of the figure. If the figure is curved at the bottom the 
line OY must be a tangent at the lowest point of the figure. Next 
draw an axis OX at right angles to this. In most practical problems 
the section will be symmetrical, and in this case the line OX is 
naturally taken along the axis of symmetry, and the construction 




Fig. 5. — The center of gravity of irregular figures. 
X 




Fig. 6. — The center of gravity of irregular figures. 

has then to be made for one-half of the figure only; if it is not sym- 
metrical the axis OX can be drawn in any convenient position, and 
the following construction must be applied to each side of the figure. 
Draw a line MN parallel to OY about half-way up the figure, and at 
some even distance d from OY. 

Next draw a series of lines PR parallel to OY. In straight parts 
of the section, such as the web in the figure, these can be wide apart, 
but where the section changes rapidly, they must be drawn closer 
together in order that the final curve may be quite definite. At the 
point P, where one of these lines cuts the section, draw a line PQ 
parallel to OX, intersecting MN at Q. Join O and Q and produce to 
R on the line PR originally drawn. Ria a. point on the curve we are 
finding. Repeat this for each of the series of lines and connect 
points R so found by a curve (dotted in the figure). 

Then if the area of the original figure, most conveniently found by 
a planimeter, equals A , and the area of the new dotted figure equals 
G, the distance of the center of gravity from OY along OX is 



G. 

A 

If OX is not an axis of symmetry the area G must be taken as the 
sum of the areas of the new curves obtained for both sides, and A 
must be the area of the entire section. 

The moments of inertia of irregular sections may be obtained from 
Fig. 7 by O. A. Thelin (Amer. Mack., Aug. 15, 1907). 

The chart can be used in computing the moments of inertia of 
simple sections by dividing them into rectangles and computing each 
one separately. For more complicated sections, irregular in shape, 
divide into a number of equivalent rectangles and compute each one 
separately. The curve of the chart has been constructed to represent 
the formula for the moment of inertiax)f a rectangle about its side: 

~ 3 
in which I = moment of inertia, 
6 = breadth, 
h = height. 



h- 


-2H- 


--» 












--f- 






• 




t Id 

A. 


-X 



'>—in'^ 



-rj 



11 






i-y_ 



Fig. 8. 



Fig, 9. 



Fig. 10. 



"T 

4 
t 



'--/- 



w 



lA. 




- — iK^^ — ►; 
Fig. II. 



Figs. 8 to 12. — Illustrations of the use of the chart for computing 
moments of inertia. 

In the chart the rectangle has been considered of unit width, or 
I in. The hight above the axis OX, Fig. 8, is measured on the left- 
hand vertical scale and varies from -^s in. to 6 ins. The horizontal 
scale gives the value of / for each -jj-in. increase in the height of the 
unit section. The curve has been drawn in steps in order to make 
the horizontal or / values more definite. Should the vertical dimen- 
sions of a section run into sixty-fourths of an in., a middle point 
between the values for thirty-seconds can easily be read ofi on the 
curve. 

As the values of I for the unit section become very small below 
I in. height, the curve is enlarged 10 times for vertical dimensions 
between i in. and 5 in., and 100 times for such dimensions below 
5 in. in order to give accurate results. For large sections, where the 
distance from the neutral axis to the extreme fiber exceeds 6 ins., 
but is less than 12 ins., the chart may be used with following modifi- 
cations: Divide aU height dimensions of the section to be figured by 
2. Calculate the moment of inertia from the chart, write these new 
dimensions and multiply the result by 8, which will give I for the 
original section. 



468 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



A few examples will best show the method of using the chart and 
the advantage it has over formulas in the saving of time. 

For the section, Fig. 8 : 

From the chart for ^^ ins. on the vertical scale the corresponding 
value on the horizontal scale is 12.11. 
Then 7 = 2.5X12.11=30.27. 

For the section, Fig. g: 

-=1.875X2.67 = 5. 
/ = io. • 
For the section, Fig. 10: 



- = 4-75 (49-99-34-33)+-SX34-33=9i-SS' 

7 = 183.1. 
For the section, Fig. 11: 

7=1.625 (.92+.oi35) + i.62S (.248-.oi3s)=.964. 
For the section, Fig. 12: 

-=K-248-.o82) + 2(.248-.i23) + (.248-.04i5) + 

(.248— .0345) + (.107-. 0175) + (.082-. 0052) +.059+ 
.0314+.0224. 
7 = .2788. 



114.33 
108 
102 
96 

90 
84 
78 
72 
66 
60 
54 
48 



10 



9-7- 



10 4 



-14 



11 



10 



U 



10 



10 



10 



42 



36 



30 



24 



18 



12 





.03 



04 



Moments 
Fig. 7. — Moments of inertia of irregular figures. 



MECHANICS 



469 



Another graphical method of finding the movements of inertia of 
irregular figures is shown in Fig. 13, by F. H. Hummel {Proc. Brit. 
C. b- M. E. S., 1900). 

OY is the axis about which the moment of inertia is required, and 
XOXi is a Kne at right angles, if possible an axis of symmetry for 
the same reason as in the construction of Fig. 6. Above and below 
OY two lines, AB and CD are drawn parallel to it and situated at some 
exact distance d from it. The axis OY usually divides the length of 
the section KL unequally, and about the best value for d is the length 
of the shorter of these segments OK in the figure. These lines AB 
and CD may or may not cut the section. Next draw a series of lines 
JG parallel to OF across the section, and in each case set off CE = OJ. 
Join EF, and produce if necessary to cut CD in H. Join H to 0, 
cutting JG in G. (In this case OH is produced.) Then G is a point 
on the required curve. 

Repeat this for each of the lines drawn across the section for lines 
below OY using the lower parallel AB, and join all the points so 
found by a curve (dotted in the figure). 

Let the total area of this dotted curve = .4; then moment of inertia 
of section about OY = Ad^. 

The areas of irregular figures may be found with suiBcient accuracy 
for many purposes by the method shown in Figs. 14 and 15, by 
F. Howkin's {Amer. Mach., Nov. 14, 1905). 

To find the area of the figure shown in Fig. 14, make a tracing on 
thin paper and fold it along the line 2 — 2, adjusting it so that the 
areas on each side balance one another, the position when folded 
being shown in Fig. 15, in which, as nearly as may be, area a = area b. 
Next open the tracing and fold it along the line 3 — 3, again adjusting 
it so that the excess and deficiency areas of the lower half balance 
one another, the result being that each section of the lower half rep- 
resents one quarter of the original area and it only remains to find 
the area of one of these sections and multiply it by four to obtain 
the total area. This can readily be done by adopting the same prin- 
ciple and folding the paper on the line 4 — 4, making area c = area 
d-{-e and giving the equivalent triangle. 

In the example given the two sides of the triangle measure respec- 
tively 1.85 and 1.30 ins. Instead of multiplying the area of this 
triangle by four we multiply the two dimensions together and then 
multiply by two, which will, of course, give the same result: 

1.85X1.30=2.4050 
2.4050+2 =4.81 

= area of original figure. 

The areas of irregular figures may be found by Simpson's rule 
which, considering its simplicity and accuracy, deserves to be more 
widely used than is the case. The rule is thus expressed by Antonio 
Llano (Amer. Mach., March. 4, 1909): 

Divide the base into an even number of equal parts, and determine 
the ordinates at the points of division. Form the sum of the end 
ordinates, four times the intermediate even ordinates, and twice the 
intermediate odd ordinates. Multiply this sum by one-third the 
distance between two consecutive ordinates. The result will be the 
required area. 

The indicator card, Fig. 16, will serve as an illustration of the use 
of the rule. The length, 3.5 ins., has been divided into 14 equal 
parts and the intercepted or enclosed ordinates, when measured, 
have been found to have the lengths (ins.) marked in the figure, 
the value of the right-hand end ordinate being zero. The bredth 

of each space is therefore ^- and one-third of this, as called for by 

3-5 
the rule, is 7^777. The area of the figure is therefore: 



3-5 



X40-92 



3X14" 



3-S 



^^ [-9+0. -F4( i.4oH-i.6i-fi.46-l-.97 + -69+.5o+.32) +2 ( 1.59-!- 



3X14 
= 3.409 sq. ins. 

The mean ordinate is equal to this area divided by the length of the 
diagram, that is: 

3-409 



3-5 



= .974 in. 





X 










E 


\ 






L 
.7 


-^ 


\i^ ^)g 














C 






Vh 


D 






f 


■3 





/ 1 




. r 




, 




' ' 




U 




'3 


K 




^^^^^^^\^ 








' 




A 






B 




Xx 











Fig. 13. — The moment of inertia of irregular figures. 




Fig. is 

Figs. 14 and 15. — The area of irregular figures. 




1.52-1-1. 20-H.81 + . 59-1- .4o))-] = 



3-5 
3X14 



(.9-1-4X6.95 + 2X6.11) 



Fig. 16. — Simpson's rule applied to finding the area of indicator 

cards. 

Since in the above we have multiplied the value of the quantity 
within the parenthesis, 40.92, by 3.5 when getting the area, only to 



470 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



divide by it again in getting the mean ordinate, we may omit both 
multiplication and division and find the mean ordinate thus: 

40.92 . 

——, — = .974 ms., as before. 
3X14 
If the scale of the indicator spring is 30 lbs. per in., the mean 
pressure is .974X30=29.22 lbs. per sq. in. 

In the practical application of the rule, it is neither convenient nor 
accurate, when the ordinates have to be determined by actual 



measurement, to measure every ordinate separately. The best way 
to proceed is to mark on the edge of a long piece of paper, in succes- 
sion, all the even ordinates, then measure the distance between the 
first and the last mark, and multiply the result by 4. Similarly 
for the first and last ordinate, and for the odd ordinates. 

A more common method of finding the areas of indicator cards is to 
divide the card into strips (usually 10) as with Simpson's rule, 
estimate the mean ordinates by the eye, add these mean ordinates 




c 
•-3 

d 
ya 

•a 
o 



o 



MECHANICS 



471 



by a strip of paper as described, and divide the sum by lo. The 
most accurate method of locating the mean ordinate is to lay a thread 
horizontally across the top of each strip, equalizing the triangular 
areas between thread and curve and thus locating the mean 
ordinate. 

A. E. Wiener's graphical method of finding the areas of irregular 
figures, which when published {Amer. Mach., May 19, 1898) was 
received with the warmest expressions of approval, is illustrated in 
Figs. 17-22. The complete explanation in connection with Fig. 17 
leads to a multiplicity of lines and to an apparent complexity 
which is apparent only. The real simplicity of the method will be 
apparent from a glance at Figs. 18 and 19 which include all the lines 
drawn in actual applications. 

A horizontal line AB, Fig. 17, is drawn at a convenient distance 
from the area to be measured, and a vertical line CD is placed at 
such a point as to approximately bisect that area (this is not to say 
that the line CD shall be about half way between the ends of the 
given figure, but that it shaU divide it into two parts of nearly equal 
contents). Next a number of vertical lines are laid across the figure, 
dose together in regions where the width of the figure undergoes 
rapid changes, and farther apart where it varies but gradually. At 
the right of CD these lines are drawn from the horizontal AB upward, 
while on the left hand they are extended some distance below it. 
The width CD is then laid ofi from o (the point of intersection of 
CD with AB) to both the right and the left, giving two points o' 
upon AB, for both of which the condition 00' = CD is fulfilled. For 
all ordinates at the right-hand side of CD, the respective widths of 
the irregular figure are similarly laid off on AB to the right, and for 
the ordinates on the left hand of CD they are swung over to the left; 
thus, for instance, ii' = EF; gg' =GH, etc. In this manner the 

points, 1', 2', 3' 15' upon AB are obtained. With o' as 

center, an arc is drawn from o to half-way between ordinates o and i ; 
this arc is continued with i' as center to midway between ordinates 
I and 2, at which point the center is again changed to 3', and so on 
■ to the last ordinate 8, when the point (8) is obtained, the length 8, 
(8) being the end value marked Zj. The latter is combined with the 
other end value 15, (15) marked Zi constructed in the same manner, 
into a right-angled triangle (8), 8, M. From M a line is drawn at an 
angleof 45 deg. with AB, and MK made equal to M (8). The areaof 
the square KLMN thus found, having sides of 1^ ins. length, is 
2.54 sq. ins., while the given plane, accurately measured by means 



of a planimeter, was found to contain 2.55 sq. ins., these figures for 
the planimeter being the average of 5 careful measurements. The 
result obtained by the graphical method, consequently, differs from 
the actual area by an amount of only 4-10 of i per cent., and could 
have been approached still closer if a greater number of ordinates had 
been taken. 

While Fig. 17 has all the construction-lines dotted in for sake of 
explanation, in practice neither the quadrants determining the 
centers i', 2', etc., nor the radii of the arcs forming the auxilliary 
curve need be shown, all that is necessary being to draw the ordinates, 
mark off the corresponding centers by means of a pair of dividers, and 
by the use of a compass find directly the lengths of the end ordinates, 
the geometrical mean of which (found by transforming the right- 
angled triangle having the end-values as sides into an isosceles right- 
angled triangle having the same hypothenuse) is the side of the 
required square. 

In Figs. 18 and 19 two examples are executed in this manner, 
showing more strikingly the simplicity of the method. In Fig. 18 
the horizontal axis cuts across the given figure, being placed through 
the point in which the bisecting ordinate leaves the figure; and in 
Fig. 19 advantage is taken of the shape of the area, and the axis 
placed in one of its boundaries. Figs. 18 and 19 also show the differ- 
ence in effect of putting the bisecting ordinate to the right, or left, 
respectively, of its accurate position, in the former case the left-end 
value being greater than the right and the square found being some 
distance to the right of the given figure, while in the latter case the 
right-end value is greater than the left end, and the irregular figure 
is overlapped by its square of equal area. 

Trial diagrams composed of straight lines, semi-circles, and quad- 
rants which could be easily checked by calculation, as in Fig. 20, 
have been treated by this method with the result that the error 
seldom equals | per cent. The error due to the displacement of the 
middle ordinate depends on the shape of the curve. If its height is 
uniform for some distance each side of the bisecting ordinate, as in 
Fig. 21, the error due to displacement of that ordinate is slight, 
while if a large change in this height occurs at this point, as in Fig. 22, 
the error is greater. The error due to the displacement of the meet- 
ing points of the arcs also varies with circumstances. If these arcs 
meet at a considerable angle, the error due to displacement of the 
meeting point is considerable, but if the arcs are more nearly tangent, 
this error is less. 



STRENGTH OF MACHINE PARTS 



For the strength of shafts, see Shafts. 

For the strength of springs, see Springs. 

For the strength of steam boilers, see Steam Boilers. 

In a large percentage of cases the formulas for the strength of 
parts have but an indirect application in machine design. 

In the design of a bridge, a roof or a warehouse floor, the ability of 
the structure to carry the load is the chief requirement, and to insure 
that it shall do this with safety, even under accidental strains, a 
factor of safety is introduced; and although the name has been often 
criticised, it nevertheless represents with a fair degree of accuracy 
the state of the designer's mind in making the calculations. 

In treatises on machine design the same term is used to express 
the ratio between the actual working strain and the strain which 
would produce ruptiu'e, although there is and can be no such con- 
cepdon in the machine designer's mind in making the calculations. 
In such parts, for instance, as connecting-rod bolts, straps and keys, 
the stresses under the working loads will often be found to run down 
to 3000 lbs. per sq. in., while in engine frames the stresses seldom 
exceed 500 lbs., and will frequently run down to 300 lbs. per sq. in. 
With steel of 60,000 lbs. tensile strength, the figure for connecting- 
rod parts is equivalent to a factor of safety of 20, while for engine 
frames, cast-iron being assumed to have 20,000 lbs. tensile strength, 
this goes up to 40 and 70 for the two stresses named. Now, it is 
certain that no designer of such parts has any conception of a factor 
of safety, as that term is commonly understood, in his mind when 
he proportions these parts for such stresses, and the term "factor 
of safety" in this connection is absurd. 

The purpose of the designer in introducing these low stresses is 
not to provide a surplus of strength for accidental stresses, but to 
provide such a degree of stiffness that the parts will not yield unduly 
under the regular loads of everyday work. He has, in fact, very 
little thought of strength in the sense of ability to resisit rupture, 
his whole thought being to make the structure so rigid that the de- 
flection under the working load shall be inappreciable, or at any 
rate so small as to do no harm. From this point of view the great 
surplus of strength is rational and understandable, while from the 
factor of safety standpoint it can not be defended. 

A strictly scientific method of machine design would base the 
dimensions on the formulas for deflection rather than on those for 
the ultimate strength of the parts. In using the formulas for strength 
as he does, the designer practically converts them, in a rough and 
ready way, into formulas fbr stiffness, which is but the reciprocal 
for deflection, and so far as methods go, this is probably as far as 
we shall ever get or as it is practicable to get in most cases. That 
the allowable deflections under any considerable number of the in- 
finite variety of conditions prevailing in machine construction will 
ever be determined is not to be expected. 



Beams 

The standard formulas for the strength of beams and for the usual 
section factors, as arranged by the Carnegie Steel Co. are given below : 
Let A =area of section, sq. ins., 
/ = length of span, ins.. 



Table i. — Strength of the Chief Materials of Machine 
Construction 



Material 


Modulus of elasticity 


Ultimate strength 


Elastic 
strength 


Tension 
compression 


Torsion 


Tension 


Shear 


Ten- 
sion 


Shear 


Cast-iron (Cu- 
pola). 

Cast-iron (Air- 
furnace) . 

Wrought-iron . . 


/ 10,700.000 
1 15,000,000 


4,000,000 
6,000,000 


16,000 
20,000 
30,000 
40,000 
/ 47,000 
\ 57.000 
60,000 

70,000 
100,000 

/ 50,000 

\ 100,000 

22,000 

/ 28,500 

\ 33.000 

22,000 

28,500 

42,800 
57,000 


16,000 \ 
20,000 / 


8,000 


8000 


I:::::::::: 










28,000,000 

30,000,000 

30,000,000 
31,000,000 

30,600,000 

12,000,000 
16,000,000 

11.400.000 
12,800,000 

15,400,000 


11,000,000 

ii,.8oo,ooo 

11,800,000 
12,100,000 

11,800,000 


35.000 
43.000 
45.000 

52,000 

30,000 
60,000 










Steel .15 carbon 

Steel .25 carbon 

Crucible steel 

(high carbon). 

Steel castings. . 


40,000 

45,000 
75.000 








Copper castings 
Copper, rolled. 


6,000 
















Brass cast., yel. 
Brass cast., yel., 
red. 
Gun-metal 
























Phosphor- 
bronze. 






24,000 











The ultimate strength of cast-iron in compression is 90,000 to 
100,000 lbs. per sq. in. Its elastic strength in compression can be 
assumed as 25,000 lbs. per sq. in. The ultimate compressive 
strengths of the other materials can be taken as equal to their ulti- 
mate tensile strengths without appreciable error. 

Table 2. — Shrinkage of Castings 

Aluminum, pure 2031 in. per ft. 

Aluminum, nickel alloy 1875 in. per ft. 

Aluminum, special 17 1 8 in. per ft. 

Iron, small cylinders 0625 in. per ft. 

Iron pipes 12S in. per ft. 

Iron girders and beams 100 in. per ft. 

Iron large cylinders, contraction of diameter at top 623 in. per ft. 

Iron large cylinders, contraction of diameter at bottom . . .083 in. per ft. 

Iron large cylinders, contraction of length 094 in. per ft. 

Brass, thin 167 in. per ft. 

Brass, thick ISO in. per ft. 

Copper 187s in. per ft. 

Bismuth 1563 in. per ft. 

Lead 3125 in. per ft. 

Zinc 312s in. per ft. 

PF = load uniformly distributed, lbs., 
Af = bending moment, lbs. ins., 
/( = height of cross-section, out to out, ins., 
« = distance of center of gravity of section, from top or 

from bottom, ins., 
/= stress, lbs. per sq. in. in extreme fibers of beam, either 

top or bottom, according as n relates to distance from 

top or from bottom of section, 
Z) = maximum deflection, ins., 
/ = moment of inertia of section, neutral axis through center 

of gravity, 
7' = moment of inertia of section, neutral axis parallel to 

above, but not through center of gravity, 
d = distance between these neutral axeSj 
5 = section modulus, 
r = radius of gyration, ins., 
£ = modulus of elasticity, for steel 30,000,000; 



472 



STRENGTH OF MACHINE PARTS 



473 



Then: S-- 



n 



/= 



W = 



Mn M 



8/7 ^8/, 

In I 

Win Wl 



S, 



S—^ = 



87 8 5 



r=^I-\-Ad^, 
SWI^ 



D 



D- 



loaded. 



for beam supported at both ends and uniformly 



for beam supported at both ends and loaded 



4SEI 
with a single load F at middle, 

Table 3. — Bending Moments and Deflections of Beams Under 
Various Systems of Loading 



1^ = total load 
Z = length of beam 



(i) Beam fixed at one end and 
loaded at the other. 



Safe load = 5 that given in tables. 

Maximum bending moment at 
point of support = Wl. 

Maximum shear at point of sup- 
port =T7. 

Wl^ 
Deflection = —jirf • 
3EI 



(3) Beam supported at both ends, 
single load in the middle. 



_; 



^ 



Safe load = 5 that given in tables. 
Maximum bending moment at 

Wl 

middle of beam = 

4 _ 
Maximum shear at points of sup- 
port =^17. 

^ n • WP 

Deflection = -^-^r^- 
4&EI 



(s) Beam supported at both ends 
single unsymmetrical load. 

Q , 



Safe load = that given in tables 

^ 8ab' 

Maximum bending moment under 

, ^ Wab 
load = — ^ — 

Maximum shears; at support near 

Wb , Wa 

o = -7-; at other support =— ^■ 

Max. ^ Wab(2l-a) ^/^a{2l-a) 
deflec. qEII 



I = moment of inertia 
E = modulus of elasticity 



(2) Beam fixed at one end and 
uniformly loaded. 



^aaoaaaifei 



Safe load = J that given in tables. 

Maximum bending moment at 

Wl 
point of support = — ■ 

Maximum shear at point of sup- 
port = IF. 

WP 
Deflection = o rr 

o lLI 



(4) Beam supported at both ends 
and uniformly loaded. 



Safe load = that given in tables. 
Maximum bending moment at 

Wl 
middle of beam = -5-- 

o 

Maximum shear at points of sup- 
port =^PF. 

-r. n ■ WP 

Deflection = - 



76.8 EI 



(6) Beam supported at both ends 
two symmetrical loads. 



Safe load = that given in tables 
J_ 
4a 
Maximum bending moment be- 
tween loads = iWa. 
Maximum shear between load 
and nearer support = iW. 

Max. deflec. = -^{3P-4a^). 



WP 
D = „„j for beam fixed at one end and unsupported at 

the other and uniformly loaded, 

PP 
D = — ^r^ for beam fixed at one end and unsupported at 

other and loaded with a single load P at the latter end. 



Explanation of Tables of Safe Loads for I-beams 

Table 7 for I-beams gives the loads which a beam will carry safely 
(distributed uniformly over its length) for. the distances between 
supports indicated. These loads include the weight of the beam, 
which must be deducted in order to arrive at the net load which the 
beam will carry. 

For beams of heavier sections than those calculated in the tables, 
a separate column of corrections is given for each size, stating the 
proper increase of safe load for every additional pound in the weight 
per foot of beam. The values given are based on a maximum fiber 
stress of 16,000 lbs. per sq. in. 

It has been assumed in these tables that proper provision is made 
for preventing the compression flanges of the beams from deflecting 
sideways. They should be held in position at distances not exceed- 



Table 4. — Moment of Inertia, I, and Section Modulus, S, foe 
Usual Sections 

For methods of finding moments of inertia of irregular section see Moment 
of Inertia. 



Sections 



U.&-J 




r-fr 




^•-^••*| 




bh^ 



/' = 



bh^ 



7 = 



6Af 
36 



7' 



b¥ 



64 
= .0491 d^ 

¥-b'h'^ 



I=- 



7 =.0491 {d^d"^) 



7 = 



b'n^+bn'^-{b-b')a^ 



I=- 



b¥-2b'h'^ 



bj^ 
6 



Min. = 



bj^ 
24 



■Kd^ 

= .0982 d^. 



I 



i^'-'i) 



Min. =- 



I 

■Sh' 



XX denotes position of neutral axis. 



474 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 5. — Moments of Inertia of Rectangles. 




Axis 



Depth 






Width of rectangl 


; in ins. 






in ins. 


i 


A 


t 


A 


i 


A 


f 


6 


4-50 


5 63 


6-7S 


7.88 


9.00 


10. 13 


11. 25 


7 


7.15 


8-93 


10-72 


12.51 


14.29 


16.08 


17.86 


8 


10.67 


13-33 


16.00 


18.67 


21.33 


24.00 


26.67 


9 


15.19 


18.98 


22.78 


26.58 


30.38 


34-17 


37-97 


10 


20.83 


26.64 


31-25 


36.46 


41.67 


46.87 


52.08 


II 


27-73 


34-66 


41-59 


48.53 


55-46 


62.39 


69-32 


12 


36.00 


45.00 


54-00 


63.00 


72.00 


81.00 


90.00 


13 


45-77 


S7-21 


68.66 


80. 10 


91-54 


102.98 


114-43 


14 


57- 17 


71.46 


85-75 


100-04 


114-33 


128.63 


142-92 


IS 


70.31 


87.89 


IOS.47 


123.05 


140-63 


158.20 


175.78 


16 


85-33 


106.67 


128.00 


149.33 


170.67 


192.00 


213-33 


17 


102.35 


127-94 


153-53 


179-12 


204.71 


230.30 


255-89 


18 


121.50 


151-88 


182.25 


212.63 


243 . 00 


273.38 


303-75 


19 


142.90 


178.62 


214.34 


250.07 


285.79 


321-52 


357-24 


20 


166.67 


208.33 


250.00 


291.67 


333-33 


375-00 


416.67 


21 


192.94 


241.17 


289.41 


337-64 


385.88 


434-11 


482.34 


22 


221.83 


277-29 


332.75 


388.21 


443.67 


499 . 13 


554-58 


23 


253.48 


316.85 


380.22 


443-59 


S06.96 


570.33 


633-70 


24 


288.00 


360.00 


432.00 


504.00 


576.00 


648 . 00 


720.00 


25 


325.52 


406-90 


488.28 


569-66 


651.04 


732.42 


813-80 


26 


366.17 


457-71 


549-25 


640.79 


732.33 


823 88 


915-43 


27 


410.06 


512.58 


615.09 


717.61 


820. 13 


922.64 


1025.16 


28 


457.33 


571.67 


686.00 


800.33 


914-67 


1029.00 


1143-33 


29 


S08. 10 


635-13 


762.16 


889.18 


1016.21 


1143.23 


1270.26 


30 


562.50 


703.13 


843-75 


984-38 


1125.00 


1265-63 


1406.25 


31 


620.6s 


77S-8I 


930.97 


1086.13 


1241.30 


1396.46 


1551-62 


32 


682,67 


853.33 


1024.00 


1194.67 


1365-33 


1536.00 


1706.67 


33 


748.69 


935.86 


1123.03 


1310.20 


1497.38 


1684.55 


1871.72 


34 


818.83 


1023.54 


1228.25 


1432.96 


1637-67 


1842.38 


2047 - 08 


35 


893.23 


1116.54 


1339.84 


1563-15 


1786.46 


2009.76 


2233.07 


36 


972.00 


1215.00 


1458.00 


1701.00 


1944.00 


2187.00 


2430.00 


37 


1055.27 


1319.09 


1582.90 


1846.72 


2110.54 


2374-35 


2638.17 


38 


1143.17 


1428.96 


1714.75 


2000.54 


2286.33 


2572.13 


2857.92 


39 


1235.81 


1544-77 


1853.72 


2162.67 


2471.62 


2780.58 


3089.53 


40 


1333.33 


1666.67 


2000.00 


2333-33 


2666.67 


3000.00 


3333.33 



ing twenty times the width of the flange, otherwise the stress allowed 
should be reduced as per Table 6. 

Table 6. — Beams Without Lateral Support 



Length of beam 


Proportion of tabular load forming 
greatest safe load 


20 times flange width 


Whole tabular load 


30 times flange width 


X7 tabular load 


40 times flange width 


tV tabular load 


50 times flange width 


tV tabular load 


60 times flange width 


T% tabular load 


70 times flange width 


j^ tabular load 



In some instances, deflection rather than absolute strength may 
become the governing consideration in determining the size of beam 
to be used. 

Table 8 gives the deflections of Carnegie beams. 

The standard test specimens of the American Society for Testing 
Materials are shown in Fig. i. 

The strength of I-beams with reinforcing plates may be determined 
by the use of Table 9 or Fig. 2, by C. F. Blake (Amer. Mach., May 
30, 1901). Table 9 gives the section factors of various thicknesses 
of cover plates when applied to different sizes of beams. The heavy 
figures are for plates on the compression flanges, and the light figures 



o 
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■ o . 

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a ^ 

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o ■§ 

ph .5 

-O -S 

w - 

H ^ 

S 'Sj 

" (-1 



z 



a 
< 
o 
I-! 
w 
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STRENGTH OF MACHINE PARTS 



475 



Table 8. — Deflection Coefficients for 
64THS of an Inch 



I-beams Given in 



Figures given opposite C. S. and C S. are the deflection coefficients for 
steel shapes, subject to transverse strain for varying spans under their maxi- 
mum uniformly distributed safe loads, derived fram a fiber stress of 16,000 
and 12,500 respectively; the modulus of elasticity being taken at 29,000,000. 

To find the deflection of any symmetrical shape used as a beam under its 
corresponding safe load, divide the coefficients given in the above tables by 
the depth of the beam. This applies to such shapes as I-beams, channels, 
Z-bars, etc. 

Example: Required the deflection of a 12-in. I-beam, 31.5 lbs., 20 ft. 
span, under its maximum uniformly distributed safe load of 9.59 tons, as 
given in Table 7. The above tables give 423.7 as the deflection coefficient; 
dividing this by 12 gives 35.3 as the required deflection in 64ths of an in. 

For deflections due to different systems of loading see Table 3. 



Coefficient index 


Distance between supports in ft. 




6 1 8 1 10 1 12 1 14 1 16 1 18 20 1 22 


c. s. 

C.'S. 


38.1 
29.8 


67.8 
53-0 


105.9 
82.8 


152.5 

119. 2 


207.6 
162.2 


271-2 
21I-8 


343.2 
268.1 


423.7 512.7 
331-0400.5 




Coefficient index 


Distance between supports in ft. 




24 1 26 1 28 1 30 1 32 1 34 1 36 1 38 1 40 


C. S. 
C.'S. 


610.2 

476.6 


716. 1 
559.4 


830. S 
648.8 


953-4 
744-8 


108S-0 

847-4 


1225.0 
956.6 


1373-0 
1073-0 


1530 
1193 


169s 
1324 



for plates on the tension flanges, the area of two jf-in. rivet holes 
having been deducted from the area of the latter plate. The section 
factors from the table axe to be added to the section factor of the 
beam, and the sum to be multiplied by the allowable fiber stress to 
obtain the safe bending moment in lb-ins. for the girder. 

Example: A is-in. 42-lb. I-beam is to be reinforced with a |-in. 
plate on each flange. What will be the safe bending moment in 
lb. -ins. to allow upon the girder, the fiber stress to be 12,500 lbs. 
per sq. in.? 

From Table 10 of properties of I-beams, the section 

factor or modulus of a 15-in. 42-lb. beam is 58.9 

From Table g the section factor of the f -in. compression 

plate for a is-in., 42-lb. beam is 25 . 91 

and for the tension flange 18 . 58 

Total section factor for girder 103 . 39 

Then 103.39X12,500=1,292,375 lb. -ins. for the allowable bending 
moment upon the girder. 

The chart. Fig. 2, applies to beams with or without cover plates 
and for any bending moment and fiber stress. 

The small chart in the upper corner is to be used with the short 
row of bending moments at the left. The letters a, b, c and d denote 
the position of the plates, whether on the tension or compression 
flange, according to the figure in the lower corner. 

Example: A bending moment of 1,292,375 lb. -ins. is to be taken 
by a beam at a fiber stress of 12,500 lbs. per sq. in. Required the 
size of beam and cover plates. The nearest bending moment on the 
chart is 1,300,000. Follow the line from this to the diagonal line 
for 12,500, thence up, and read the size of beam and plates as 15- 
in., 42-lb. beam with two f-in. cover plates. 

Neither Mr. Blake's table nor chart take into account the necessity 
for supporting long beams against lateral deflection. For the allow- 
ances to be made in such cases, see Table 6 for beams without lateral 
support. 

Explanation of Tables 

On the Properties of Carnegie Standard and Special I-beams 
Table 10 on I-beams, is calculated for all weights to which each 

pattern is rolled. 

Columns 12 and 13 give coefficients by the help of which the safe, 

uniformly distributed load may be readily and quickly determined. 



Table 9. — 


Section Factors 


OF Reinforcing Plates for I-beams 








Section factors 




Width of 


Depth 


Weight 


Light figures for tension members 


plate in 


of 
beam 


of 
beam 


Heavy figures for compression 


members 


ins. 


Thickness of plate 




fin. 


Jin. 


f in. ■ 


fin. 


4l 


10 


25-30 


8.67 

S.64 


11.64 
7-53 


14.54 
9.41 




S 


10 


3S-40 


9.42 
6.32 


12.55 

8.43 


15-72 

10.57 




S 


12 


3li-35 


II .32 

7.57 


15. OS 

10.58- 


18.20 

12. 12 




Si 


12 


40 


11.82 

8.17 


15.85 
10.89 


18.30 

14.02 




si 


12 


4S 


12.14 

8.72 


16.15 

II .29 


20.35 

14.08 




si 


12 


SO 


12-37 

8-80 


16.55 

11.76 


20.71 

14-52 




si 


12 


SS 


12.68 

9.02 


16.8s 

12.04 


21.35 

15.08 




SJ 


IS 


42-4S 


IS 03 

10.91 


20.65 

14-59 


25-91 

18.58 




si 


15 


SO 


15.12 

11.27 


21.25 

15-04 


26.60 

18.83 




si 


IS 


55-60 


15.22 

11.57 


21.45 

15-49 


27. 11 

19-73 




6 


IS 


6S-70 


16.92 

12.32 


22.56 

16.44 


28.37 

20.59 




6i 


IS 


75 


17.60 

13-17 


23-71 

17-65 


29.32 

22 .09 


35 38 

26.51 


6J 


IS 


80-85 


18.33 

13.72 


24-47 

18.35 


30.63 

23.00 


36.82 

27-57 


6f 


IS 


90-95 


18.63 

14.52 


24-87 

18.80 


31-23 

23.80 


37-48 

28.48 


6J 


IS 


100 


20.28 

14-57 


25 -32 

19-25 


31-74 

24.10 


37-74 

29.08 


6 


18 


55-65 




27.06 

19.74 


33-87 

24-59 


40.07 

29.65 


6i 


18 


70 




28.46 

20.85 


35-23 

26.09 


42.44 
31-36 


6i 


20 


65-70 






39 13 

29.00 


47-09 

34-86 


(>i 


20 


75-80 






40.83 

30.59 


48.92 

36.77 


7 


20 


85-90 






43 90 

33-76 


52.7s 

40.44 


7i 


20 


95-100 






45-40 
35-61 


54-76 

42.80 


7 


24 


80-85 








63.25 

48.44 


7i 


24 


90-95 








64.45 

49-70 


7i 


24 


100 








6S.56 

51.38 



To do this, it is only necessary to divide the coeflScient given by the 
span or distance between supports in feet. 

If a section is to be selected (as will usually be the case) intended 
to carry a certain load for a length of span already determined on, 
it will only be necessary to ascertain the coefficient which this load 
and span will require and refer to the table for a section having a 
coefficient of this value. The coefficient is obtained by multiplying 
the load in pounds uniformly distributed by the span length in feet. 

In case the load is not uniformly distributed, but is concentrated 
at the middle of the span, multiply the load by 2 and then consider 
it as uniformly distributed. The deflection will be t'o of the deflec- 
tion for the latter load. 

For other cases of loading, obtain the bending moment in Ib.-ft. 
(the most common cases are given in the table of bending moments 
and deflections). This multiplied by 8 will give the coefficient 
required. 

If the loads are quiescent, the coefficients for fiber stress of 16,000 
lbs. per sq. in. for steel may be used; but if moving loads are to be 
provided for, the coefficient for 12,500 lbs. should be taken. Inas- 
much as the effects of impact may be very considerable (the stresses 
produced in an unyielding, inelastic material by a load suddenly 

{.Continued on page 478, first column) 



476 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 































's; 


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400 000 












/ 


/ 




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2,800,000 








#/ 


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2,700,000 






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/ 
















/ 


200 000 






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/ 


' / , 


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/ 






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2,100,000 i 


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1,700,000 


















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/ 






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/ 








/ 
























/ 












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/ 


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1,600,000 
















/ 






/ 






/ 










/ 






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1,500,000 














/ 






/ 
















/ 








yy 
















/ 






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/ 


/ 






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1,400,000 












/ 














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y 








y 
















/ 






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/ 






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1,300,000 










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/ 














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/ 


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/ 


/ 






/ 


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600,000 


/ / 


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/ 






/ 


y 
























/ / 


/ 
/ 


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/ / . 


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/ 


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/ 




























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/ / 


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400,000 


/ / 


/ y 


/ 


































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1 



Fig. 2. — Strength of reinforced I-beams. 



STRENGTH OF MACHINE PARTS 



477 



Table io. — Properties oe I-beams 
By the Carnegie Steel Co. 

Weights in heavy print are standard, others are special. 



Depth 

of beam, 

ins. 



Weight 

per ft., 

lbs. 



Area of 
section, 
sq. ins. 



Thick- 
ness of 
web, ins. 



Width 
of flange, 



Mom. of. 
inertia 
neutral 
axis per- 
pendicu- 
lar to 
web 
at center 
/ 



Mom. of 
inertia 
neutral 
axis co- 
incident 
with 
center 
line of 
web 
/' 



Radius of 
gyration 

neutral 
axis per- 
pendicular 

to web 
at center 



10 



Radius of 
gyration 
neutral 
axis coin- 
cident 
with cen- 
ter line of 
web 



Section 
modulus 

neutral 
axis per- 
pendicular 

to web 
at center 



Coefficient 

of strength 
for fiber 
stress of 

16,000 lbs. 

per sq. in. 
Used for 
buildings 



13 



Coefficient 
of strength 

for fiber 

stress of 
12,500 lbs. 

per sq. in. 

Used for 
bridges 

c" 



14 



Distance 
center to 
center re- 
quired to 
make radii 
of gyration 
equal 



Section 
index 



18 



IS 



I 00 . 00 
95.00 
90. 00 
85.00 
80.00 

100.00 
95.00 
90. 00 
85.00 
80.00 
75.00 
70.00 
6S.00 
70. 00 
65.00 
60.00 
55. 00 

100.00 
95.00 
90. 00 

85.00 
80.00 
75.00 
70.00 
65.00 
60.00 
55.00 
50.00 
45-00 
42.00 
55-00 
SO. 00 
45-00 
40. 00 
35-00 
31-50 
40 -00 
35-00 
30.00 
25 -00 
35-00 
30-00 
25-00 
21 -OO 

25-50 

23.00 
20.50 
18.00 

20. 00 

17 -SO 
15 -00 

17-25 

14-75 

12.25 

14-75 

12.25 

9-75 
10.50 
9.50 
8. SO 
7-50 
7 -SO 
6.50 
5-50 



29-41 


-754 


27.94 


.692 


26.47 


•.631 


25-00 


.570 


23-32 


.500 


29-41 


.884 


27-94 


.810 


26.47 


.737 


25.00 


.663 


23-73 


.600 


22. 06 


.649 


20.59 


• 575 


19.08 


.500 


20.59 


.719 


19. 12 


.637 


17.6s 


.555 


15-93 


.460 


29-41 


1. 184 


27-94 


1.085 


26.47 


.987 


25.00 


.889 


23.81 


.810 


22.06 


.882 


20.59 


.784 


19- 12 


.686 


17.67 


■ 590 


16-18 


.656 


14-71 


.558 


13-24 


.460 


12.48 


.410 


16.18 


.822 


14-71 


.699 


13-24 


.576 


11-84 


.460 


1O.29 


.436 


9-26 


.350 


11-76 


.749 


10.29 


.602 


8.82 


.455 


7-37 


.310 


1O-29 


.732 


8.82 


.569 


7-35 


.406 


6.31 


.290 


7.50 


.541 


6-76 


• 449 


6.03 


.357 


S-33 


.270 


5-88 


.458 


S.15 


.353 


4.42 


.250 


5-07 


• 475 


4-34 


.352 


3-61 


.230 


4-34 


.504 


3.60 


.357 


2.87 


.210 


3-09 


.410 


2.79 


.337 


2.50 


.263 


2.21 


.190 


2. 21 


.361 


1-91 


.263 


1-63 


.170 1 



7-254 


2380.3 


48.56 


7.192 


2309-6 


47-10 


7. 131 


2239-1 


45-70 


7.070 


2168.6 


44-35 


7.000 


2087.9 


42.86 


7.284 


1655-8 


52.6s 


7.210 


1606.8 


SO. 78 


7-137 


1557.8 


48.98 


7.063 


1508.7 


47-25 


7.000 


1466.5 


45-81 


6.399 


1268.9 


30.25 


6.32s 


1219.9 


29.04 


6.250 


1169.6 


27.86 


6.259 


921-3 


24.62 


6.177 


88I-S 


23-47 


6.095 


841-8 


22.38 


6.000 


795-6 


21.19 


6.774 


900.5 


SO. 98 


6.675 


872.9 


48-37 


6-577 


845-4 


45.91 


6-479 


817.8 


43-57 


6-400 


795-5 


41-76 


6. 292 


691 -2 


30.68 


6.194 


663-6 


29.00 


6.096 


636-0 


27-42 


6.000 


609.0 


25-96 


S.746 


5II-O 


17.06 


5. 648 


483-4 


16.04 


5-550 


455-8 


15-00 


5-500 


441-7 


14-62 


S.612 


32I-O 


17.46 


5.489 


303-3 


16.12 


5- 366 


285-7 


14.89 


5-2SO 


268.9 


13-81 


5. 086 


228.3 


10-07 


5- 000 


21S-8 


9-50 


5.099 


158-7 


9-50 


4-952 


146-4 


8.52 


4.80s 


134-2 


7-65 


4.660 


122. 1 


6-89 


4.772 


III. 8 


7.31 


4.609 


101.9 


6.42 


4.446 


91-9 


5.6s 


4-330 


84-9 


S.16 


4.271 


68.4 


4-75 


4-179 


64-5 


4-39 


4.087 


60.6 


4-07 


4-000 


56.9 


3-78 


3.868 


42.2 


3.24 


3.763 


39.2 


2.94 


3.660 


36.2 


2.67 


3-575 


26. 2 


2.36 


3-452 


24.0 


2 -09 


3-330 


21.8 


1-85 


3-294 


IS-2 


1.70 


3-147 


13-6 


1-45 


3.000 


12.1 


1.23 


2.880 


7.1 


1. 01 


2.807 


6.7 


.93 


2.733 


6-4 


.85 


3 . 660 


6-0 


.77 


2.521 


2-9 


.60 


2.423 


2.7 


.53 


2.330 


2.5 


-46 



9.00 
9.09 
9.20 
9-31 
9.46 
7.50 
7. 58 
7.67 
7.77 
7.86 
7.58 
7.70 
7.83 
6.69 
6.79 
6-91 
7-07 
5.53 
S.59 
5. 65 
S-72 
5.78 
5.60 
5.68 
S.77 
5. 87 
5. 62 
S.73 
5. 87 
S.95 
4.4s 
4-54 
4.6s 
4.77 
4.71 
4.83 
3.67 
3.77 
3.90 
07 
29 
40 
54 
67 
02 



3.09 
3.17 
3-27 
2.68 
2.76 
2.86 
2. 27 
2.35 
2.46 
1.87 
1-94 
2.05 
1.52 
1.55 
1-59 
1.64 
I. IS 
I. 19 
1-23 



1.28 
1.30 
I. 31 
1-33 
1-36 
1-34 
1-35 
1-36 
1-37 
1-39 
I-I7 
1.19 

1.21 
1.09 
1- II 
1-13 

I -15 
I-31 
1-32 
1-32 
1-32 
1-32 
I-I8 
I -19 
1 . 20 
1.21 
1.02 
1 .04 
1.07 
1.08 
1.04 
I. OS 
1.06 
1.08 
-99 

1 -01 

-90 
-91 
.93 
■97 
-84 
-85 
-88 

■ 90 
-80 
.81 
.82 
-84 
-74 
-76 
-78 
.68 
.69 
.72 
.63 
.63 
-65 
• 57 
.58 
.58 
.59 
.52 
-52 

■ 53 



198 

192 

186 

180 

174 

165 

160 

155-8 

150.9 

146 

126 

122 

117 

102 
97 
93 
88 

120 

116 

112 

109.0 

106. 1 
92.2 
88.5 
84.8 
81.2 
68.1 
64- 5 
60-8 
58-9 
53-5 
50.6 
47-6 
44-8 
38-0 
36-0 
31-7 
29-3 
26-8 
24.4 
24-8 
22.6 
20.4 
18.9 
I7-I 
16. 1 
IS-I 
14.2 

12. 1 

11. 2 
10.4 

8.7 
8.0 
7.3 
6.1 
5.4 
4.8 
3-6 
3-4 
3-2 
3-0 



1.7 



2,115,800 

2,052,900 

1,990,300 

1,927,600 

1.855.900 

1,766,100 

1,713,900 

1,661,600 

1,609,300 

1.564,300 

1,353.500 

1,301,200 

1,247,600 

1,091,900 

1,044,800 

997.700 

943,000 

1,280,700 

1,241,500 

1,202,300 

1,163,000 

1,131,300 

983,000 

943,800 

904,600 

866,100 

726,800 

687,500 

648,200 

628,300 

570,600 

539,200 

507,900 

478,100 

405,800 

383.700 

338,500 

312,400 

286,300 

260.500 

265,000 

241,500 

217,900 

201,300 

182,500 

172,000 

161,600 

151,700 

128,600 

119,400 

110,400 

93,100 

85.300 

77.500 

64,600 

58,100 

51,600 

38,100 

36,000 

33.900 

31,800 

20,700 

19.100 

17,600 



1,653,000 

1,603,900 

1,554.900 

1.505,900 

1,449,900 

1,379,800 

1,339,000 

1,298,100 

1,257,200 

1,222,100 

1,057,400 

1,016,600 

974,700 

853.000 

816,200 

779. Soo 

736,700 

1,000,600 

969,900 

939.300 

908,600 

883,900 

768,000 

737,400 

706,760 

676,600 

567,800 

537.100 

506,400 

490,800 

445.800 

421,300 

396,800 

373.500 

317,000 

299,700 

264,500 

244,100 

223,600 

203,500 

207,000 

188,700 

170,300 

157,300 

142,600 

134,400 

126,200 

118,500 

100,400 

93.300 

86,300 

72,800 

66,660 

60,500 

50,500 

45.400 

40,300 

29,800 

28,100 

26,500 

24,900 

16,200 

15.000 

13,800 



17.82 
17-99 
18.21 
18.43 
18.72 
14-76 
14.92 
15. 10 
15.30 
15-47 
14.98 
15-21 
15-47 
13-20 
13-40 
13-63 
13-95 
10.75 
10.86 
10.99 
II- 13 

11-25 

10-95 
II - II 
11-29 

11.49 

11.05 

II. 27 

11-54 

11-70 

.8-65 

8-83 

9.06 

9-29 

9- 21 



9- 

7- 

7- 

7- 

7- 

6. 

7. 

6. 

7. 

5- 

5- 

6. 12 

6.32 

S-IS 

5-31 

5 -50 

4-33 

4-49 

4-70 



-45 

. 12 
-32 

-57 
-91 
-36 
.58 
.86 
. 12 
.82 
-96 



B I 



B 2 



B 3 



B80 



B 4 



B 5 



B 7 



B 8 



B 9 



Bii 



B13 



Bis 



B17 



B19 



B21 



B23 



B77 



478 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



applied being double those produced by the same load in a quiescent 
state), it will sometimes be advisable to use still smaller fiber stresses 
than those given in the tables. In such cases the coefficients can 
readily be determined by proportion. Thus, for a fiber stress of 
8,000 lbs. per sq. in. the coefficient will equal the coefficient for 16,000 
lbs. fiber stress divided by 2. 

The section moduli are used to determine the fiber stress per sq. 
in. in a beam or other shape, subjected to bending or transverse 
stresses, by simply dividing the same into the bending moment ex- 
pressed in Ib.-ins. 

Column 14 gives the distance c.t.c. of beams, making the radii 
of gyration equal for both axes. 

These tables have all been prepared with great care. No approxi- 
mations have entered into any of the calculations, so that the figures 
given may be relied upon as accurate. 

Example: What section of I-beam will be required to carry 
40,000 lbs. uniformly distributed, including its own weight, over a 
span of 16 ft. between supports, allowing a fiber stress of 16,000 lbs. 
per sq. in.? 

Answer : The coefficient required = 40.000 X 1 6 = 640,000. 

In Table 10 of Properties of I-beams, look in column 12 for the 
nearest number corresponding to 640,000 which is 648,200. There- 
fore the beam to be used is 15 in. 45 lbs. 



-^y»'~- 



M: 








10 Pitch, U.S,form of Thread - 

Steel, Tensile 



ur- 



.7 Pitch, n.S.form of Thread 





Grey Iron, Tensile 



t-<— 2 to 8--»i i^—sa — ^ 

B = Width of pull section, should not 
be in excess of 2H" 
t = Thickness ot pull section 
Note: -5 X t should not exceed 1 sq.in. 
for the ordinary tensile testing 
machine. 

Flat Strips, Tensile 



o =., 



.0037 
flolJxJLr 






,0937 
Hole 



-571692- 



-5.1692- 



J3H 



Alternate Stress 
Fig. I. — A. S. T. M. standard test specimens. 



Beams of uniform strength for stakes of rivetting machines and 
similar structures may be laid out by the aid of Figs. 3, 4 and 5, 
which were originally developed at the Bement-Miles works of the 
Niles-Bement-Pond Co. (Amer. Mach., Feb. 21, 1901). The charts 
were designed especially for steel castings in which the compressive 
strength is about six-fifths of the tensile strength and, except in the 
case of beams of circular cross-section, the dimensions obtained from 
them always provide stresses in this ratio. 

Instructions for use: 

For full circular sections with a fiber stress of 10,000 lbs.: Read 
the load in tons of 2,000 lbs. on the right or left-hand scale and the 
length of the lever in ft. on the top or bottom scale of Fig. 3. Follow 



Table ii. — Safe Loads Uniformly Distributed for Rectan- 
gular Spruce or White Pine Beams i In. Thick 
By the Carnegie Steel Co. 

To obtain the safe load for any thickness: Multiply values for i in. by 
thickness of beam. 

To obtain the required thickness for any load: Divide by safe load for 
I in. 

This table has been calculated for extreme fiber stresses of 750 lbs. per sq. 
in. corresponding to the following values for Moduli of Rupture recom- 
mended by Prof. Lanza, viz.: 

Spruce and white pine 3900 lbs. 

Oak 4000 lbs. 

Yellow pine 5000 lbs. 

For oak increase values in table by i. For yellow pine increase values in 
table by |. 

The safe load for any other values per sq. in. is found by increasing or 
decreasing the loads given in the table in the same proportion as the 
increased or decreased fiber stress. 



Span in 


Depth of beam 


ft. 


6" 


7" 


8" 


9" 


10" 


11" 


12" 


13" 


14" 


I5"| 16" 


5 


600 


820 


1070 


1350 


1670 


2020 


2400 


2820 


3270 


3750 


4270 


6 


Soo 


680 


890 


II20 


1390 


1680 


2000 


2350 


2730 


3120 


3560 


7 


430 


S8o 


760 


960 


1 190 


1440 


1710 


2010 


2330 


2680 


3050 


8 


380 


SIO 


670 


840 


1040 


1260 


1500 


1760 


2040 


2340 


2670 


9 


330 


460 


590 


750 


930 


II20 


1330 


1560 


1810 


2080 


2370 


10 


300 


410 


S30 


670 


830 


lOIO 


1200 


1410 


1630 


1880 


2130 


II 


270 


370 


490 


610 


760 


920 


1090 


1280 


1490 


1710 


1940 


12 


2S0 


340 


440 


560 


690 


840 


1000 


1180 


1360 


1560 


1780 


13 


230 


310 


410 


520 


640 


780 


930 


1080 


1260 


1440 


1640 


14 


210 


290 


380 


480 


590 


720 


860 


lOIO 


1170 


1340 


1530 


IS 


200 


270 


360 


450 


560 


670 


800 


940 


1090 


1250 


1420 


16 


190 


260 


330 


420 


520 


630 


750 


880 


1020 


II80 


133c 


17 


180 


240 


310 


400 


490 


S90 


710 


830 


960 


IIOO 


1260 


18 


170 


230 


290 


370 


460 


560 


670 


780 


910 


1040 


1 190 


19 


160 


210 


280 


360 


440 


530 


630 


740 


860 


990 


1130 


20 


ISO 


200 


270 


340 


420 


510 


600 


710 


820 


■940 


1070 


21 


140 


190 


260 


320 


390 


480 


570 


670 


780 


890 


1020 


22 


140 


190 


240 


310 


380 


460 


540 


640 


740 


850 


970 


23 


130 


180 


230 


290 


360 


440 


520 


610 


710 


810 


920 


24 


130 


170 


220 


280 


350 


420 


500 


590 


680 


780 


890 


25 


120 


160 


210 


270 


330 


410 


480 


560 


660 


750 


860 


26 


IIO 


160 


210 


260 


320 


390 


460 


540 


630 


720 


820 


27 


no 


ISO 


200 


250 


310 


370 


440 


520 


610 


690 


790 


28 


IIO 


140 


190 


240 


300 


360 


430 


500 


580 


670 


760 


29 


IIO 


140 


180 


230 


290 


350 


410 


490 


560 


640 


740 



the lines from these readings to their intersection and find the re- 
quired diameter of the section on the diagonals. 

For any section of Fig. 4 with a tensile fiber stress of 10,000 and 
a compressive fiber stress of 12,000 lbs.: Multiply either load in 
tons or length of lever in ft. by the value of factor X for the section 
as given in Fig. 4 and proceed as before. The result given by Fig. 
3 is the diameter D of the various sections of Fig. 4. The section 
may then be laid out by the proportional figures for the section se- 
lected. The value of D and the cross-section are to be determined 
for a sufficient number of points on the stake, the same proportional 
figures being used throughout the length of the stake, except that 
it should be noted that Fig. 4 will give the cross-sections at different 
points in the length of the stake or beam strictly according to the 
law of the cubic parabola. When nearing the top of the stake it is 
desirable to use heavier sections from Fig. 4 in order to reduce the 
diameters at these sections and also to avoid thin ribs which could 
not be cast in steel. 

For any other tensile fiber stress than 10,000 lbs.: Multiply 
either the load in tons or the length of lever in ft. by 10,000 and 
divide by the desired tensile fiber stress and proceed as before. In 
the resulting beam the compressive fiber stress will always be equal 
to six-fifths of the tensile stress. 

Example: Find the dimensions of section 3 of Fig. 4 for a riveter 
stake at a point 8 ft. below the dies. The pressure on the dies is 



STRENGTH OF MACHINE PARTS 



479 



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Feet 



Fig. 3. — Beams of uniform strength. 



20 



25 30 35 40 46 50 60 70 80 90 100 



70 tons and the tensile fiber stress is not to exceed 8,000 lbs. The 

value of section factor X for section 3 is and, performing the 

multiplication, gives: 



100 I 0000 



117 



Finding this load at the right and the length, 8 ft., at the top of 
Fig. 3, we find at the intersection ot the lines through these points, 
28 1 ins. as the value of D for the section. 

The use of Fig. 5 for various methods of support and of loading is 
self-explanatory. 

The Hodgkinson section of cast-iron beams with heavy tension 
and light compression flanges proportioned in accordance with the 
widely differing ultimate strengths of the material in compression and 
tension, is now believed by many machine constructors to be funda- 



mentally wrong when applied to machine parts. The case against 
it is well made out by Jas. Christie {Proc. Engrs. Club of Phila., 
1907) as follows: 

This form of beam was largely adopted, and took precedence as 
long as cast-iron was used for beams in structures. We find that 
the same method of reasoning influenced the machine designer in 
disposing of cast-iron to seeming advantage in the construction of 
machines, massing the metal to resisit tension, and permitting high 
unit stress on metal in compression; and especially is this observed 
in machines of the open-jaw or gap type, such as presses, punching 
and shearing machines, etc. 

I believe that usually the unit stresses should be little, if any, 
higher in compression than in tension, for the following reasons: 
In machinery rigidity or stiffness is usually the chief consideration; 
many machines do not fulfil the intended purpose properly, not by 



480 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 




No. 1. Section, X=-|^, No. 2 Section, X= ^ No,3.Section,X=y? No. 4 Section, X=^^ No. 5 Section, X=J-^j_ No. 6 Section, X= |^. No. 



No. 10 Section, X=*^ 




No.USection,X=l°i No. 12 Section,X'= ^° No. 13 Section,X=102 No. 14 Section,X=J|2 No.15 Section,X=aiO No.16 Section,X=152 No. 17 Section,X=|™ No. 




No. 21 Sectlon,X=-i22 
63.7- 



No. 22 Section,X=-i50 No. 

34- 



FiG. 4. — Sections of beams of 

Table 12 — Ultimate Strength of Hollow Round and Hollow 

Rectangular Cast-iron Columns 
By the Carnegie Steel Co. 

Ultimate Strength in Lbs. per Sq. In.: 
Round Columns Rectangular columns 

Square bear- Pin and 

ing square Pin bearing 



No. 

uniform strength. 

i 



No. 30 Section,X=-15» 
71.6- 



Take Full Load, If 



Beam kept at one End, Load at otlier End 



I 



1 + 



mg 
80000 

(I2Z)2 



square 
80000 



Pin bearing 
80000 



^+ i6ood2 1+ 4ooo(2 



^^ 



Take H of the Load, if 



Beam kept at one End, Load equally distrib'd 



80000 



80000 



80000 



8ood 

I =Length of column in ft. 
d= External diameter or least side of rectangle in ins. 



3(i2/)2 9(i2/)g 3(12;;'' 

^+ 3zoo<i2 ^+ 6400^2 ^+ i6oo(i2 



■ Take K of the Load, if 



Beam supported at both Ends, Load Central 
Take H of the Load, if 







;iound columns 


Rectangular columns 




Ultimate strength in lbs. 


Ultimate strength 


in lbs. 


d 




per sq. in. 






per sq. m. 




Square 


Pin and 


Pin 


Square 


Pin and 


Pin 




bearing 


square 


bearing 


bearing 


square 


bearing 


I.O 


67,800 


62,990 


58,820 


70,480 


66,520 


62,990 


1. 1 


65,690 


60,300 


55,730 


68,790 


64,260 


60,300 


1.2 


63,530 


57,600 


52,690 


67,000 


61,940 


57.600 


1-3 


61,340 


54.930 


49.740 


65,140 


59,600 


54.960 


1.4 


59,140 


52,3:0 


46,900 


63,260 


57.270 


52,320 


1.5 


56,940 


49,770 


44,200 


61,350 


54.960 


49,760 


1.6 


54.760 


47,300 


41.630 


59,450 


52,680 


47,300 


1.7 


52,620 


44.940 


39.210 


57,550 


50,460 


44,960 


1.8 


50,530 


42,670 


36,930 


55,670 


48,300 


42,670 


1.9 


48,490 


40,510 


34.790 


53.800 


46,230 


40,510 


2.0 


46,510 


38,460 


32,790 


51.940 


44,200 


38,460 


2. 1 


44,600 


36,520 


30,920 


50,160 


42,260 


36,520 


2. 2 


42,750 


34.680 


29,180 


48,400 


40,400 


34.680 


2.3 


40,980 


32,940 


27,540 


46,670 


38,630 


32,950 


2.4 


39,280 


31.310 


26,030 


44.990 


36,930 


31,310 


2.S 


37,650 


29.770 


24,620 


43.390 


35.310 


29,760 


2.6 


36,090 


28,320 


23,300 


41,820 


33.770 


28,320 


2.7 


34.600 


26,950 


22,070 


40,320 


32,310 


26,950 


2.8 


33,180 


25.670 


20,930 


38,870 


30,920 


25,670 


2.9 


31.820 


24,460 


19,860 


37.470 


29,600 


24,460 


3.0 


30,530 


23,320 


18,870 


36,120 


28,340 


23,320 


3.1 


29,310 


22,250 


17,940 


34.830 


27,150 


22,250 


3.2 


28,140 


21,250 


17.070 


33.580 


26,030 


21,250 


3.3 


27,030 


20,300 


16,260 


32,390 


24,660 


20,300 


3.4 


25,970 


19,410 


15,500 


31,240 


23,940 


19,410 



M. 



w 



Beam supported at both Ends, Load equ'y distrib'd 
_| ^Take » of the Load, if 



Beam rigidly kept both Ends, Load Central 
^^^^^^^Take /,, of the Load, if 



^ 



Beam rigidly kept both Ends, Load equ'y distrib'd 

Fig. 5. — Loading and supporting beams. 



failure through fracture, but by a want of sufficient stiffness. Deflec- 
tion has to be limited, and when that is done, breaking from excessive 
tension is sufficiently guarded. Remembering that cast-iron yields to 
compression, as much as with the same unit stresses it yields to ten- 
sion, it follows that the compressive stress should not exceed the ten- 
sile strength per unit of section if it is desired to dispose a given mass 
of metal with least deflection. It is believed that rupture sometimes 
occurs in a machine apparently through tension, where the origin 
of the weakness could be traced to a want of material to sufficiently 
resist compression, the improperly supported tension side severing 
by cross-bending or transverse stress. 

Taking for illustration an open-gap machine with frame as illus- 
trated in Fig. 6, tension at T and compression at C, if the section 
is so shaped that compressive unit stress is six times that of the tensile 
unit stress, then, elastic moduli being equal, the frame will yield at 
C six times as much by compression as it does by tension at T. 
This permits an oscillation of the mass at T around its center. If 
this oscillation becomes dangerous, by extent or frequency, the frame 
will break by cross-bending at the mass T, giving the impression 



STRENGTH OF MACHINE PARTS 



481 



Table 13. — Safe Loads in Tons of 2000 Lbs. for Hollow Round Cast-iron Columns 

By the Carnegie Steel Co. 



Outside diam., 


Thickness 
of metal 


Length of columns in ft. 


Sec- 
tional 
area, 
ins. 


Weight, 

lbs. of 

columns per 

ft. of 


ins. 


8 


10 


12 


14 


16 


18 


20 


22 


24 




Tons 


Tons 


Tons 


Tons 


Tons 


Tons 


Tons 


Tons 


Tons 


length 


5 


1 


26.2 


23.0 
33.0 
37.6 
41.9 
46.0 

43.1 


20. 1 


17. 5 
25.0 
28.5 


15.2 
21 . 7 


13.2 
18.9 
21. 5 


11.5 






8.6 


26.95 
38.59 
43.96 
49.01 
53.76 

45.96 


37. S 

42.7 
47.6 
52. 2 

47.7 


28.8 


16.5 






12.4 
14. 1 
IS. 7 
17.2 

14-7 


1 


32.8 
36.5 
40. 1 

38.5 


24.7 
27.6 


18.8 






6 
6 


a 
I 


31.8 


24.0 
26.3 


21.0 







I 1- 


34.8 
34-3 


30.2 
30.4 


23 . 






7 


i 


26.9 


23-9 


21.2 


18.9 


7 


I 


61. 1 


55.2 


49.3 


43.8 


38.9 


34.4 


30.6 


27.1 


24. 2 


18.9 


58.90 


7 


li 


67.2 


60.8 


54.3 


48.3 


42.8 


37.9 


33.7 


29.9 


26.7 


20.8 


64.77 


8 


i 


57-9 


53.3 


48.6 


44.1 


39.7 


3S.8 


32.2 


28.9 


26.1 


17. 1 


53.29 


8 


I 


74-6 


68.7 


62. 5 


56.7 


SI. I 


46.0 


41.4 


37.3 


33.6 


22.0 


68.64 


8 


^i. 


89.9 


82.8 


75. 5 


68.4 


61.7 


55.5 


49.9 


44.9 


40.5 


26. s 


82.71 


9 


i 


68.1 


63.6 


58.9 


54.2 


49.6 


45.2 


41.2 


37. S 


34-1 


19.4 


60.65 


9 


I 


88.0 


82.3 


76.2 


70.0 


64. I 


58.4 


53.2 


48.4 


44-1 


25.1 


78.40 


9 


li 


106.6 


99.6 


92.2 


84.8 


77.6 


70.8 


64.4 


58. 7 


S3-4 


30.4 


94.94 


9 


ij 


123.8 


IIS. 7 


107. 1 


98. S 


90. 1 


82.2 


74.8 


68.1 


62.0 


3S.3 


110. 26 


9 


i| 


139.6 


130. 5 


120.8 


III. I 


I0I.6 


92.7 


84.4 


76.8 


69.9 


39.9 


124.36 


10 


I 


I0I.4 


9S.9 


89.8 


83.6 


77.4 


71. 5 


65.8 


60. 5 


55.5 


28.3 


88.23 


10 


li 


123.3 


116. 5 


109. 1 


I0I.6 


94-1 


86.8 


79.9 


73-4 


67.5 


34.4 


107.23 


10 


li 


143.7 


135.8 


127.3 


118. 5 


109.7 


101.2 


93.2 


85.6 


78.7 


40.1 


124.99 


10 


ij 


162.7 


153.8 


144.1 


134. 1 


124.2 


114.6 


105.5 


97-0 


89.1 


45. 4 


141.65 


II 


I 


114. 8 


109.4 


103. S 


97.3 


91.0 


84.8 


80.2 


73.1 


67.7 


31.4 


98.03 


11 


li 


139.9 


133-3 


126. 1 


118. 6 


110.9 


103.3 


97.8 


89.4 


82.5 


38.3 


119 . 46 


II 


i^ 


163. S 


155-9 


147.5 


138.6 


128.7 


120.8 


114. 3 


104. 1 


96.4 


44.8 


139.68 


II 


ii 


18S.7 


177-I 


167.5 


157. 5 


147.3 


137-2 


129.8 


118. 3 


109.5 


SO. 9 


158.68 


" 


2 


206.6 


196.9 


186.3 


175. 1 


163.8 


152-6 


144.4 


131. 5 


121. 8 


56.6 


176.44 


12 


I 


128.0 


122.9 


117. 2 


III.O 


104.7 


98.4 


92.2 


86.1 


80.4 


34-6 


107.51 


12 


li 


IS6.4 


150. 1 


143. I 


135.7 


127.9 


120.2 


112.6 


105. 2 


98.2 


42.2 


131.41 


12 


15- 


183.3 


175.9 


167.7 


159.0 


149.9 


140.9 


132.0 


123.3 


115. 1 


49.5 


154.10 


12 


li- 


208.7 


200.4 


191.0 


181. 1 


170.7 


. 160.4 


ISO. 3 


140.5 


131.1 


56.4 


175.53 


12 


2 


232.7 


223.4 


213.0 


201.9 


190.4 


178.9 


167.6 


156.6 


146. 1 


62.8 


195.75 


13 


I 


141. 2 


136.3 


130.7 


124.7 


118. 5 


112. 1 


105.8 


99-5 


93.5 


37.7 , 


117.53 


13 


li 


172.8 


166.8 


160.0 


152.7 


14s. 


137.2 


129.4 


121.8 


114.4 


46. 1 


143 . 86 


13 


li 


203.0 


195.9 


187.9 


179.3 


170.3 


161. 1 


152.0 


143.1 


134.3 


54. 2 


168 . 98 


13 


ij 


231.6 


223.6 


214-S 


204.7 


194.4 


183.9 


173.5 


163.3 


153.3 


61.9 


192 . 88 


13 


2 


258.9 


249.9 


239-7 


228.7 


217.3 


205.5 


193-9 


182.5 


171. 3 


69. 1 


215.56 


14 


I 


154-3 


149.6 


144-3 


138. 5 


132.3 


125.9 


II9-S 


113. 1 


106.8 


40.8 


127.60 


14 


li 


189.2 


183.4 


176.9 


169.7 


162.2 


154-4 


146.5 


138.6 


131.0 


SO. I 


156.31 


14 


li 


222.6 


215.8 


208.1 


199.7 


190.8 


181. 7 


172.3 


163. 1 


IS4.I 


58.9 


183.67 


14 


li 


254-4 


246.7 


237.9 


228.3 


218.1 


207.6 


197.0 


186.5 


176.2 


67.4 


210.00 


14 


2 


284.8 


276.2 


266.4 


255-6 


244.2 


232.4 


220.6 


208.8 


197.2 


75. 4 


235-12 


IS 


I 


167.4 


162.9 


157.8 


152. 1 


146.0 


139.7 


133.3 


126.8 


120.4 


44-0 


137-28 


IS 


li 


205.5 


200.0 


193.7 


186.7 


179.3 


171. 5 


163.6 


IS5.7 


147.9 


54-0 


168.48 


IS 


li 


242. 1 


235.7 


228.2 


220.0 


211. 2 


202. 1 


192.8 


183.5 


174-2 


63.6 


198.74 


IS 


I i 


277.2 


269.8 


261.3 


251.9 


241.9 


231.4 


220.7 


210. 1 


199-5 


72.9 


227.4s 


IS 


2 


310.8 


302. 5 


293.0 


282.5 


271. 2 


259.5 


247.5 


235. S 


223-6 


81.7 


254.90 




Fig. 6. — The case against the Hodgkinson section. 
31 



that more material is needed to resist tension, whereas the fact may- 
be that more material should be placed at C to prevent excessive 
yield by compression. 

The parabolic outlines of beams of uniform strength are seldom 
used. In forged beams the outlines are difScult to produce while 
cast beams are usually flanged, to which construction the theoretical 
outline has no application. There is, however, an approximate out- 
line which is easily produced in forged material and leads to most of 
the economy of material of the theoretical outline, while it is very 
satisfactory in appearance. 

Calculate and lay down the section a b, Fig. 7, for the mid length 
of the beam and suitable for the load, make cd = }^ce and draw df and 
dg, when fghi is the required approximate outline, the thickness being 
uniform as shown in the end view. The dotted parabola shows the 
theoretical outline to which the outline found is an approximation. 
The same construction is to be followed for a beam having a straight 



482 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



^^d 




Fig. 7. — Approximate beam of uniform strength. 

Table 14. — Properties of Column Sections 
A = area of section. R = radius of gyration 



A=bh 

R = .2 



y^b-^ 



5" A=h^ 

_iL i? = . 28876 




^=.7854/)^ 




.4 = .78546/! 




4=.866Z)2 
R = .2635D 



A —hihi — hiht 




= 2{bi+hi-2t)t 



H-firM 




A=bi^-b2^ 
= 4ih-t)t 



R = .2887\/bi^+b2^ 



^ = .7854(Z>i^-W) 
= 3.i4i6(Z?i-0/ 





A=.78s4ibihi—b2h2) 
t^t^ =i-S7o8(6i+/!i- 2/)/ 



h-61 ■*! 




^=.82841)2 

R = .2S7D 



h- ^-^ 



upper face, that is approximating the half parabola below the center 
line. 

Arms, brackets, etc., dimensioned by intuition and judgment may 
be laid out in the same manner and the appearance will be satisfactory 
because appropriate. 

Columns 

The properties of a cross-section that determine the elasticity and 
strength of a beam are the moment of inertia and the section modulus 
and, similarly, the properties of a section that determine the strength 
of a column or strut are the area of the section and the radius of gyra- 
tion, which latter property is equal to the square root of the quotient 
of the moment of inertia divided by the area of the cross-section. 
Formulas for both properties for all common and some uncommon 
machine sections are given in Table 14. 



Among the most satisfactory experiments on the strength of 
wrought-iron and steel columns are those of James Christie (Trans. 
A. S. C. E., 1884) which have been arranged for convenient use by 
the National Tube Company in Table 15. The safe working loads 
in this table were obtained from Mr. Christie's experimental crippling 
valves by the application of the following safety factors: 





Fixed 


Hinged 




Fixed 


Hinged 




Fixed 


Hinged 


I 


and 


and 


I 


and 


and 


I 


and 


and 


R 


flat 


round 


R 


flat 


round 


R 


flat 


round 




ends 


ends 




ends 


ends 




ends 


ends 


20 


4.8 


4-9S 


120 


6.3 


7. 20 


220 


7.8 


9.4s 


40 


S. I 


5-40 


140 


6.6 


7-65 


240 


8. I 


9.90 


60 


5.4 


5.8s 


160 


6.9 


8. 10 


260 


8.4 


10. 35 


80 


5-7 


6.30 


180 


7.2 


8. 55 


280 


8.7 


10.80 


100 


6.0 


6. 75 


200 


7-5 


9.00 


300 


9.0 


II. 2S 



It will be observed that the safety factors increase with an increase 
in the slenderness ratio, because of the greater inabiUty of a long 
column to resist the bending actions in practice due to causes other 
than axial loading^ and for similar reasons, columns having hinged 
or round ends should have greater safety factors than those having 
fixed or flat ends. 

The safe working load corresponding to any other safety factor 
may be obtained by multiplying the tabular value by the corre- 



Table 15.- 



-Safe Working Loads of Solid and Tubular Steel 
Columns 



^^ 


9^ 


i 


Case I 
umn or strut with 


^^B 


f » Case ] 
Column or s 


I 




Col 








1;rut with 




fixed 


ends and loaded 






flat 


ends and loaded 




axially 








axially 






I = length of column 
in inches 

R = least radius of gy- 
ration in inches 






I 
R 


slenderness ratio of 
column 




-^m. 








Working loads, lbs. per 




Working loads, lbs. per 




sq. in. 


of cross-section 


R 


sq. m. 


of cross-section 


R 


Steel of 


Steel of 


Steel of 


Steel of 


Steel of 


Steel of 




50,000 


70,000 


100,000 




50,000 


70,000 


100,000 




lbs., t. s. 


lbs., t. s. 


lbs., t. s. 




lbs., t. s. 


lbs., t. s. 


lbs., t.s. 


20 


9,590 


14.570 


20,800 


20 


9.590 


14.570 


20,800 


30 


8,690 


10,300 


14.940 


30 


8,690 


10,300 


14,940 


40 


7,870 


9,040 


12,820 


40 


7,870 


9.040 


12,820 


50 


7,210 


8,380 


11,590 


50 


7,2 10 


8,380 


11,590 


60 


6,650 


7,770 


10,730 


60 


6,650 


7.770 


10,730 


70 


6, 120 


7,190 


9,970 


70 


6, 120 


7,190 


9,970 


80 


5,690 


6,670 


9,220 


80 


5,650 


6,670 


9,220 


90 


5,330 


6,180 


8,520 


90 


5,250 


6,160 


8,450 


100 


5,000 


5.760 


7,880 


100 


4,870 


S.670 


7,700 


no 


4.700 


5.380 


7.290 


no 


4.520 


5.210 


6,990 


120 


4,410 


5,000 


6,700 


120 


4,180 


4.760 


6,310 


130 


4.130 


4.640 


6,140 


130 


3.850 


4.340 


5,690 


140 


3.850 


4.290 


5.6 10 


140 


3.520 


3.950 


5.100 


150 


3.580 


3.950 


5.130 


150 


3.200 


3.550 


4.550 


160 


3.310 


3,620 


4,660 


160 


2,900 


3.190 


4,060 


170 


3,050 


3.290 


4,220 


170 


2,610 


2,830 


3.610 


180 


2,790 


2,970 


3.800 


180 


2,350 


2,500 


3,200 


110 


2,540 


2,650 


3.400 


190 


2,130 


2,2 10 


2,850 


200 


2,310 


2,370 


3.030 


200 


1,940 


1,970 


2,530 


210 


2, 100 


2, 130 


2,680 


210 


1,780 


1,780 


2,250 


220 


1,910 


1,920 


2,380 


220 


1,630 


1,630 


2,0 10 


240 


1,600 


1,600 


1,910 


240 


1,380 


1,380 


1,650 


260 


1.350 


1.350 


1.580 


260 


1,170 


I, 170 


1.360 


300 


970 


970 


1. 130 


300 


800 


800 


950 



STRENGTH OF MACHINE PARTS 



483 



Table 15. — Safe Working Loads of Solid and Tubular 
Steel Columns {Continued). 



Table 16. — Safe Loads for Rectangular Wooden Pillars 
(Seasoned). By the Carnegie Steel Co. 



c 


3 < 


-1 Case III 


mm. 


Case IV 




Column or strut with 






Column or strut with 






hinged ends and loaded 






round 


ends and loaded 






axially. 






axially. 








I = length of column 






I 










in inches 






■p = slenderness 


ratio of 


c 


3 


R = least radius of gy- 
_J ration in inches 


M 


^ column. 






Working loads, lbs. per 




Working loads, lbs. per 


R 


sq. in. of cross-section 


I 

R 


sq. m. 


of cross-section 


Steel or 


Steel of 


Steel of 


Steel of 


Steel of 


Steel of 




50,000 


70,000 


100,000 




50,000 


70,000 


100,000 




lbs.,t.s. 


lbs.,t.s. 


lbs., t. s. 




lbs., t. s. 


lbs., t. s. 


lbs., t. s. 


20 


9,290 


14, 170 


20, 100 


20 


8,890 


13,540 


19.340 


30 


8,330 


9,880 


14,260 


30 


7,780 


9.230 


13,400 


40 


7.490 


8,520 


11,950 


40 


6,770 


7,760 


10,480 


50 


6,780 


7,810 


10,800 


50 


5,950 


6,880 


9,390 


60 


6,150 


7,160 


9,930 


60 


5,200 


6,090 


8,430' 


70 


5,560 


5,530 


9,040 


70 


4,530 


S.370 


7,470 


80 


5,020 


5,910 


8,170 


80 


3,970 


4,710 


6,550 


90 


4.560 


5,310 


7,300 


90 


3.490 


4,080 


5,620 


100 


4. ISO 


4,760 


6,490 


100 


3,040 


3.520 


4,810 


no 


3.7SO 


4,280 


5,760 


no 


2,650 


3.030 


4,090 


120 


3,370 


3,850 


5,120 


120 


2,290 


2,600 


3,480 


130 


3,020 


3,430 


4.510 


130 


1,970 


2,2 10 


2,900 


140 


2,670 


3.030 


3,920 


140 


1,670 


1,870 


2,390 


ISO 


2,3S0 


2,620 


3,370 


150 


1,410 


1,570 


1,990 


160 


2,050 


2,240 


2,860 


160 


1,180 


1,320 


1.670 


170 


1,780 


1,910 


2,440 


170 


1,0X0 


1,110 


1,420 


180 


1,540 


1,630 


2,090 


180 


870 


930 


1,210 


190 


1,350 


1.390 


1,800 


190 


760 


780 


1,030 


200 


1,190 


1,200 


1,560 


200 


670 


670 


870 


210 


1,060 


1,060 


I.3S0 


210 


590 


590 


750 


220 


940 


940 


1,160 


220 


530 


530 


650 


240 


760 


760 


910 


240 


440 


440 


520 


260 


630 


630 


730 


260 


370 


370 


420 


300 


450 


450 


520 


300 


250 


250 


290 



spending safety factor above given, and then dividing the result by 
the desired safety factor. 

In case a column or strut is entirely free from bending actions due 
to causes other than a constant axial loading, then a constant safety 
factor of from four to six may ordinarily be used. 

Flat Plates 

The strength of fiat plates in accordance with the formulas of Gra- 
shof and the experimental researches of Professor Bach may be 
determined from Table 17 by Eugene Messner (Amer. Mach., 
Nov. 25, 1909). In these formulas 

£ = modulus of elasticity. 

^ = uniformly distributed load, lbs. per sq. in., 
P = total load acting at a point or over an indicated area, lbs., 
5 = fiber stress due to bending, lbs. per sq. in., 
(f = deflection, ins. 

dimensions of plates in ins. 

Regarding the accuracy of the formulas Professor Bach's tests 
have demonstrated that the strength of the plates depends much 
on the fastening or support at the edges, the spacing of the bolts 
(for flanges, etc.), the forces exerted by those bolts (making a more 
or less elastic joint), the gaskets, the character of the tightening 
surfaces, etc. 

The formulas assume the supports to be as shown — a rigid support 



Yellow pine (southern) 

II2S 
^"*' 1100(^2 


White oak 
925 


White pine and spruce 
800 


' IXOO<f2 



I = length of pillar ins. 

d = width of smallest side ins. 

These formulse give safe loads of one-fourth the ultimate strength 
for short pillars decreasing to one-fifth the ultimate for long pillars. 



Ratio of length 
to Ipfl-St sirlf* 


Safe load in lbs. per sq. in. 


of section 


d 


Yellow pine 
(southern) 


White oak 


White pine 
and spruce 


12 


995 


818 


707 


14 


955 


78s 


679 


16 


913 


7SO 


649 


18 


869 


715 


618 


20 


82s 


678 


587 


22 


781 


642 


556 - 


24 


738 • 


607 


525 


26 


697 


575 


495 


28 


6S7 


541 


467 


30 


619 


509 


440 


32 


583 


479 


414 


34 


549 


451 


390 


36 


S16 


425 


367 


38 


487 


400 


346 


40 


458 


377 


326 



being truly rigid and the plate rigidly fixed to it — conditions that 
seldom obtain in practice. 

These formulas hold good within the limit of elasticity only. The 
rupturing loads cannot be found from them, this being doubly true 
in the case of ductile materials, such as boiler plate. With such 
materials the bulging under pressure leads to the formation of spher- 
ical surfaces and the destruction of the fundamental conditions on 
which the formulas are based. The formulas have been unjustly 
criticised because they do not agree with the results of tests carried 
to destruction, but such criticisms are based on a fundamental 
misunderstanding of the formulas. 

Plates of much size made of ductile materials begin to bulge under 
moderate loads, leading to a change in the fundamental conditions 
even within the elastic limit, and, with such materials, the formulas 
have less application as the diameter increases. The formulas are 
most applicable to brittle materials (cast-iron) for which there is no 
reason to suppose that they give other than the true fiber stresses 
within the elastic limit. If, with such materials, they lead, as they 
often do, to apparently excessive thicknesses, that should be taken 
as an indication that crowned and not flat surfaces should be used 
whenever possible and that, when flat surfaces must be used, they 
should be ribbed. 

Ribbing should be done judiciously, as otherwise it may do harm 
and not good. With narrow, and especially with shallow, ribs the 
concentration of stress on the edges of the ribs may be an added 
source of danger. Also, with cast-iron plates the ribs should, if 
possible, be on the compression side in order to take advantage of the 
greater strength of that material in compression than in tension. 

Calculation of the strength of ribbed plates is scarcely possible, 
judgment and precedent being the chief guides and a free use of 
material the only safe course. ' 



484 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 
Table i7.^Strength and Deflection of Flat Plates 



Shape and fastening of 
plate 



Maximum fiber stress due to bending 



Coefficients 



Deflection in center 
of plate 



Coefficients 



Remarks 



R-<^Wy^ 






!-2i2'l W- 



-M 1*2'-^ 






1-2A-1 






C2 






-s-^ 



I— B-H 












P 7? 

i = 0.438plog — 






^=*(^-I^)iT 






, 8 + 4«!±3»« f 
'3 + 2»^ + 3«*"«' 



,0' f 
^2 i + n2 



^'°'''3 + 2«2+3M4"i2 






£2 + 62^(2 






^262 ^ 



,_Bb_ P 



Cast-iron <^ = . 8 
Steel ■^=•45— -S 






d= .22 



«3 £ 



Cast-iron (j>= 1.2 
Steel ^=.67— .75 






Cast-iron ^ = i . 44 



i = (i 



J3 E 



Cast-iron <^ = 2 . 26 
Steel 1^ — i.io 



Cast-iron <^ => . 76 



Cast-iron 1^ = i . 34 
Steel <^= .86 



Cast-iron ^ = . 85 



Cast-iron ^= .38 
Steel </>= .24 



Cast-iron </> = 2 . 63 



Cast-iron <^= .S7 
Steel ^= .36 



Cast-iron ^ = 30 



Cast iron ^ = 0.17 



For cast-iron max- 
imums in center, 
for steel at circum- 
ference. 



Use Naperian loga- 
rithm for ^. 



Cast iron ^ = . 6 



Maximum ^ in cen- 
ter. 



Cast-iron ^=.4 — .s 



<^ for steel esti- 
mated. 



for steel esti- 
mated. 



<l> for steel- esti- 
mated. 



^ for steel esti- 
mated. 



P ^fc 



hB--. 



H £ 



t^BH 



n ? 








R2 



B2. 



Q23 



i-2iei 



Cast-iron <^= .19 
Steel <^= .12 



Cast-iron <^ = 1 . 3 2 



Cast-iron ^= .28 
Steel <^= .18 



Cast-iron •^=1.3 



d =.0284-^X2 



5 = /! 



r+^'( ^-^<^+y 



Cast- J 4, = .& 
iron \ 01 = .8 



Steel 



1 '^■i^ 



• S 

.33 -.38 



4) for steel esti- 
mated. 



i independent of 
B. Deflection 
only varies. 



4i for steel esti- 
mated. 



X independent of 
B. Deflection 
only varies. 



Stayed plate. The 
formula is for one 
field. 



According to stiff 
riveted joint. 



nes3 of cylinder or 



Flat boiler head 
with round edges. 



STRENGTH OF MACHINE PARTS 



485 



Table i8. — Safe Loads in Tons of 2000 Lbs. for Square Wooden 
Pillars by the Carnegie Steel Co. 



Unsupported 

length of 

column in 

ft. 



Size of pillar in ins. 



6X6 I 8X8 I 9X9 |ioXio|i2Xi2|i4Xi4|i6Xi6 









White pine or spruce 




6 


12.80 
11.70 
10.60 

9-54 
8.46 
7.38 














8 


22.7 
21.3 
19.8 
18.4 
17.0 

iS-S 
14. 1 


29.6 
28.0 
26.3 
24.7 
23.1 
21. s 
19.8 
18.2 










10 


3S-S 
33-7 
31-9 
30.1 
28.3 
26. s 
24.7 
22.9 








12 


49.0 
46.8 

44-7 
42.5 
40.3 
38.2 






14 
16 
18 
20 
22 • 


69.6 
67.0 

64-5 
62.0 

59-5 
S7.0 


91.0 
88.0 
8S.2 
82.3 
79-4 


24 





















White oak 






6 


14. 80 

13-50 
12.20 

II. CO 

9-73 
8.64 












•• 


8 


26.2 
24. 6 
22.7 
21. 1 

,19.5 
17.8 

16.3 


34-0 
32.4 
30-4 
28.4 
26. S 
24.7 
22.7 
21. 1 










10 


41 . 
39-1 
36-7 
34-6 
32.4 

30. s 

28.2 
26.4 










59-1 
56.9 
54-0 
Si-i 
49-° 
46.1 

43-9 






14 
16 
18 
20 

22 


80.4 
77.8 
74- S 
71-3 
68.3 
65.5 


105.0 

102.0 

98.5 

94-7 

90.9 


?4 



















Yellow pine (southern) 




6 


18.0 
16.4 
14.9 

13-3 
II. 9 

10.4 














8 


32.0 
29.9 
27.8 
25.8 

23-7 
21.8 
19.8 


41.6 
39-4 
36.9 

34-7 
32.3 
30.0 
27.8 
25-7 










10 


50.0 
47.6 
44-7 
42.3 
395 
37-0 
34-6 
32.2 








12 


72 .0 
69.1 

6S-S 
62.6 
59-8 
56.2 
53-3 






14 
16 
18 
20 
22 


98.0 
94.6 
90.7 
86.9 
83-6 
80.0 


132 
128 
124 
120 
115 


2/1 






III 









Combined Tension and Shear 

The combination of direct tension or compression with shearing 
stresses may be made by means of Fig. 8, by E. R. Douglass {Amer. 
Mack., July 10, 1902). 

Among the cases covered by the chart are those of shafts transmit- 
ting power and at the same time carrying heavy weights or acted on 
by overhung cranks and the like. The actual maximum stress at 
any point will be greater than that due to either the torsion or the 
bending alone, and wiU be exerted in a direction different from either. 

Suppose the stresses to be combined are a tension T acting per- 
pendicularly to the plane of a shearing stress S, these values express- 
ing intensities, such as lbs. to the sq. in. For convenience the 

left-hand half of the diagram is plotted for yp> to be used when 5 is 

T 
less than T, and the right-hand half is plotted for -^> to be used when 

5 is greater than T. Then, P being the maximum resultant tensile 
stress and Q being the maximum resultant compressive stress, at 

P Q P Q 

right angles to P, the values of the ratios „ and t; or -^ and ^' and of 

the tangent of angle x between P and T may be read at once in terms 

, S T 
of Y or-^- 

Had the original stress T been a compression instead of a tension. 



P would have been a compression and Q a tension, x still being the 
angle between P and T. 

As an example of the use of this chart, suppose that in some case, 
as that of the shaft mentioned above, there is found to exist at a 
certain point a tensile stress of 8000 lbs. per sq. in. and a shearing 
stress of 3600 lbs. per sq. in. in a plane perpendicular to the tension. 
S 3600 P 

-■ — .45. Consulting the diagram we find -j, = 

Q 



The ratio ^ is 5 ' or 

2 8000 



I-I7S, 



.175, and tan x = .38. Then the maximum resultant 



tension P=i. 175X8000 = 9400 lbs. per sq. in., and its direction 
makes an angle whose tangent is .38, or 20° 50' with the original 
tension, while there also exists a compression Q, normal to P, of 
value .175X8000 = 1400 lbs. per sq. in. 



1.5 



Wh 





^''^•^ 


y ^^ _,x 


— I ^ ^^y^ 


?'-</' V'^tT 


^^ ^^eT 


'ti^'^ "^■^ - 


<tu ^s ofc:^ 


^t,^ ^<^ 


\f- V sH-n rn-l 


''^^ ^^ 


^i ?^ 


,^'' j£^5] 


-■=--- — — ^%^ 


.,.:m^- 


,..^■^+5^' 


^5C:^-=- 


Tpl^^l.^ 


d^a:^-rp 




CL --'''' 


5ii ^-^ ,^ 


-iff"}:^' / 


— ,<>'-=' / 


<<^ ^ /, 


T^' yW 


4C. ^ , ^ n^S. .^ _ 


1^ ^ ^t^ 


^/ 77g«^ 


'V -^ J^ 


A~ A -^ 


/iL iU-'_&^ ' 


'7 .'^iL. _lL ^ -. 



L5 



1.0 



©lOQ 



Value of-^ 
T 



1.0 
1.0 



Value of - 



Fig. 8. — The combination of direct and shearing stresses. 



The material must safely stand a stress of 9400 lbs. per sq. in. in 
the direction found. 

Had the original tension been, for instance, 3000 lbs. per sq. in. 
and the shearing stress 7500 lbs. per sq. in., we would have taken 
T 3000 . S . 

•^ = = .4, instead of ^ as in the former case. Corresponding 

T P 

to "F = .4 we find, in the right-hand side of the diagram, -^ = 1.22, 

e 

S' 

Whence P = i. 22X7500 = 9150 lbs. per sq. in. maximum tension, 
making an angle of 39° 20' with the original tension. = 75ooX.82 
= 6150 lbs. per sq. in. compression at right angles to P. In this 
case the resultant tension is more than three times the original one. 



Pvinch and Shear Frames 

The strength of cast-iron frames for punching and shearing machines 
formed the subject of experiments by Prof. A. L. Jenkins {Trans. 
A.S. M. E., Vol. 32). Model frames were made and tested to destruc- 
tion, test bars being cast with and as part of the frame castings in 
order to avoid assumptions regarding the strength of the iron. 

Although the experiments are not sufficiently exhaustive to 
justify rigid conclusions, they seem to indicate that the following 
statements are approximately true: 

(a) There is no rational method for predicting the strength of 
curved cast-iron beams suitable for punch and shear frames. 



486 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



(b) Of the three formulas suggested for the design of punch frames, 
the well-known beam formula, 



McW_^y 
' I ^A^ 



(■+S) 



in which 5 = unit tensile stress, 

M = bending moment = WL, 
c = distance from the center of gravity of the section to 
the most extreme fiber in tension, 
W = load applied, 
A = area of the section considered 
i = distance from the line of application of the load to the 

center of gravity of the section considered, 
/ = moment of inertia of the section, 
ir = radius of gyration, 
is the most accurate statement of the law of stress relations existing 
in such specimens. 




Fig. 9. — Correct and incorrect sections of punch and shear frames. 

(c) The stress behind the inner flange at the curved portion is an 
important consideration that should be recognized by the designer. 

(d) There seems to be no definite relation existing between the 
strength of a curved cast-iron beam and the transverse strength of a 
test bar cast with it. 

(e) The Resal and Pearson-Andrews formulas are unwieldy and 
awkward in their application and offer many chances for error. 

According to Wilfred Lewis, punch and shear frames, when made 
with the section shown at the left of Fig. 9, break on the line ab, 
whereas, when made with the section shown at the right, he has 
never known them to break. 

Hoisting Hooks and Lifting Eyes 

The dimensions of hoisting hooks of trapezoidal section may be deter- 
mined from Fig. 11 by Axel K. Pedersen, analytical expert, of the 
General Electric Co. {Amer. Mach., Dec. 26, 191 2). The charts 
are based on Bach's theory of curved beams. They impose one 
restriction, namely, that the section MN be a trapezoid. Most 
hook sections, especially those of large hooks, can be transformed 
into a trapezoid without serious error by the method shown in Fig. 10, 
which shows in full lines the actual shape of the important hook sec- 
tion, which then is transformed into the trapezoid shown in dotted 
lines. Only a very small reduction of the actual dimension Hi is 
required, it being sufficient that the area ffli-|-a2 is approximately 
equal to the area as, this being done by the eye of the observer with- 
out any refined measurement. The reduced dimension H and the 
increased dimension B are then used in the calculation. In designing 
new hooks, the theoretical dimension H is increased to Hi and B 
decreased to Bi. In calculating, the selected, or actual inner radius 
of the hook, may be used without regard to the change of H. 

The proposed capacity of the hook P in lbs, being given, we 
select the radius of the inside of the hook. Chart i. Fig. 11, ^, in ins. 
On the chart a table gives the practice of the Pawling & Harnisch- 
feger Co. for this dimension. From these data and the allowable 
maximum tensile stress, 5 in lbs. per sq. in., we can proceed in the 
following two ways in determining the dimensions of the important 
hook-section, MN. 



(i) Calculate the dimension B in ins. from 

B = .o225\/p (a) 

To facilitate this calculation, curve No. i, Chart 2, Fig. 10, was plotted, 
giving the values of B for different loads P. As the dimension Di, 
the shank of the hook, usually is calculated from 

Dl = .0225\/P 

which is identical with (a), we shall, after transformation, have the 
final dimension Bi smaller than Di, which is considered good practice, 
resulting in easy manufacture. Then select the ratio x = H-t-A 
from which then 

H = Ax (6) 

Suitable to most cases is a; = 2 to 3. 
Calculating the factor C from 

s 



C = BH 



P 



{c) 



we use Chart i as follows : Locating C on its scale we trace parallel 
to the ratio z-scale to the curve giving the proper ratio x, thence 
horizontally to the left to the z-scale and read the value 2. Then 

b=^Bz (d) 

The larger the ratio x is selected the smaller we will get b, which is 
preferable as it tends to keep the weight of the hook reasonably low. 
(2) The second method which can be employed, is the following: 
Select as before the ratio x which then gives 
H = Ax 

Then, calculate the ratio z from 

I 
z = — I — (e) 

i+x ^ ' 

this relation between 2 and x usually resulting in good proportions 

of the hook-section. It may, however, be especially noted, that the 

chart can be used for any value of z; in other words, that it is not based 

upon any fixed relation between z and x. Now, locate this value of 

z on the z-scale, trace horizontally to the right to the proper curve 

for the value of x, thence vertically down to the C-scale and read the 

factor C, then 

PC 

. ^=7h W 

and 

b=Bz (g) 

For determining the general dimensions of the hook, the following 
relations may serve as a guide: 

Z>i = .0225\/p as already stated. Z>2 = . 8752)1, PF = 1.5^, ^ = .75^' 
to .90^, 1,1 = 2.34 to 2.6A, L2 = 4.sA to 4.5A. 

The calculation of B according to equation (a) is, of course, not 
necessary; however, for the reason above stated, the method gives 
very practical results. In calculating the dimension B or Di from 
(a) or determining it from curve i, the nearest size of commercial 
available iron should be used, if the hook is to be forged from round 
bar iron. 

The material for a new hook should preferably be a high-grade of 
iron rather than steel. 

Most steel hooks, if overloaded, break without warning, giving no 
slow visible deflection as is the case with iron hooks, which open up 
gradually before ultimately breaking. 

For hooks made from a high grade of iron and properly heat- 
treated, a maximum tensile stress of 17,000 lbs. per sq. in. may safely 
be allowed. 

To check the capacity of existing hooks, measure the dimensions 
A, H, B and b, using the transformed section as already explained. 
Fig. 10. Then, calculate the ratios x = Z7-^4 and 2 = 6^5. Locate 
z on the z-scale of the chart, trace horizontally to the right to the 
curve giving the proper ratio x, thence vertically down to the C-scale 
and read the factor C, then 

BH 



P = - 



{h) 



STRENGTH OF MACHINE PARTS 



487 



if the capacity of the hook for a given maximum unit stress is required, 
and 

PC 



BE 



(0 



if the unit stress at a given load must be determined. Of course, 
the properties of the material from which the hook was made being 
unknown, the allowable maximum stress should be selected rather 
conservatively, an average value of 15,000 lbs. per sq. in. insuring 
reasonable safety. 




Examples: Design a hoisting hook of 50- tons capacity, when 
the radius of the inside of the hook A=s ins., and the maximum 
allowable tensile stress 5 = 17,000 lbs. per sq. in. 

According to the first method of calculating we would have from (a) 

5 = .G2 25\/ 100,000 = 7.115 ins. 

Say 5 = 7ins., which also could have been determined from curve i. 
Selecting x = 2.2 we get from (&) 

Zf = 2. 2X5 = 11 ins. 
Then from (c) 

17,000 

C = 7XiiX = 13.00 ins. 

' 100,000 "^ ^ 

Now using Chart i, we obtain z = .2$. Hence from {d) 

6 = . 25X7 = 1. 75 ins. 
Using the second method, we would for .t = 2.2 have H =xA=2.2y.5 
= 11 ins.; then from (e) 

I 

z = — ; = .^1 ins. 

1+2.2 -^ 

Hence from Chart i, C = 12.7; and then from (/) 
100,000 12.7 



B-- 



- V- 

17,000 II 



= 6.788 ins. 



Fig. 10. — Transformation of the actual hook section into a trapezoid. 

It wUl be observed that the chart gives solutions for all trapezoidal 
sections for values of z = o, that is, a triangle up to z = i., that is a 
rectangular section. The compression stress at the back of the hook 
at the dimension h in no case will exceed allowable limits, even for a 
triangular section In other words, the hook will fail only if too 
high tension stresses are allowed. 



and from {g) 

J = .3iX6. 788 = 2. 104 ins. 

Thus, the two methods do not give identical proportions of the 
hook section. Whichever is to be preferred depends entirely upon 
the individual judgment of the designer, the aim being to combine 
strength with lightness and good appearance. 

Determine the capacity of a hook of the following dimensions: 
Fi = 3.125 ins., 5i = i.75 ins., & = .5 in. and ^=1.25 ins. After 
{.Continued on page ^Zg^first column) 

Dimension D 1 , Ins. 
76543210 




13. 14. 15. 

Values of Constant C 



Fig. 1 1. -^Dimensions of hoisting hooks of trapezoidal section. 



488 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



s- 

9- 

i- 

s'-i 

ID 

c I 

0:f 



o 
o 



&5 



z- 



g 
Pi 



S'S- 



S*8- 



^- 



Z' 



2'- 



r- 



s*- 



^ 


9- 






-l-> 






1* 


W 




m 


«• 


m 




2 


6* 


o 


T 


u 




ni 








3 




fj 




^ 








U 




<H 


S'l 


O 




V( 




<u 




■*-> 




s 


--S 


CS 








Q 


s-s 




s 




S'8 




f 



(^ 



Cc) 



— rrr 


ri ■] r-i 


ooo o o 


1 1 


o\ o 


1 1 1 1 1 1 

o 


" 1 

o 


c^ 


f~^ 


ooo o o 


f~^ 


o \ o 


o 


o 


o 


o 


OO o o o 


o 


o \ o 


o 


lO 
















o 


us 


O 05 00 t-» CO 


lO 


'^ \o 


M 


r-l 


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STRENGTH OF MACHINE PARTS 



489 



transformation into a trapezoid, we measure H = s ins. and B = 2 
ins., hence 

3 



H 

X=-T= '■ =2.4 

A 1. 25 



and 



^=£=T = -^5 



Then using the chart we find C = 12.925, and allowing a stress of 
s = 16,000 lbs. per sq. in. we get from (h) 

2X3 



BH 

C 12.925 



X 16,000 = 7,427 lbs. 



If this same hook were to be used as a 4-ton hook or for P = 8,000 lbs., 
we would have the stress from (i) 

PC 8000X12.925 

— ^17,233 lbs. per sq. in. 





Fig. 12. — Hoisting hooks of circular cross section. 

The dimensions of hoisting hooks of circular cross-section may be 
determined from Fig. 13, also by Mr. Pedersen and, like the preced- 
ing chart, laid out in accordance with Bach's formula. The chart is 
applicable to any of the hooks shown in Fig. 12. Directions for use 
wiU be found below the chart. 

The dimensions of eye bolts and lifting eyes may be determined 
from Fig. 14, by Mr. Pedersen (Amer. Mach., May 18, 191 1). The 
chart is the outgrowth of experiments at the testing laboratory of 
the General Electric Co. 

The chart applies to the cases shown in Figs. 15, 16 and 17 and 
determines the dimension D in ins., having given A and (for Fig. 15) 
B in ins., P, the load, in lbs. and s, the maximum allowable tensile 
stress, in lbs. per sq. in. occurring in the eye. T, Fig. 15, is one-half 
the angle which includes the unyielding part of the eye. For Fig. 
17, the angle T = o and for Fig. 16, T = go deg. 

Calculate the factor: 

„ msA^ 

using the value ot = 2 which was deduced from the experimental tests. 
Locate F on the F-scale and trace parallel to the Z-scale to the 
proper curve among the curves for the sine of T and read the value 
of Z on the Z-scale. Then the dimension D is 

D = ZXA 

To employ the proper curve for the sine of T, the following rules 
must be observed: 

For Fig. 15 calculate the value of sine of T approximately from 
sine of ri = the ratio B-hA. 

For Fig. 17: Use the curve, sine of T = 0. 

For Fig. 16: Use the curve sine of r=i.o. 

Allowable stresses: 

For the eye: Maximum stress allowed =1 of the elastic limit of 
the iron used. 

For the shank: Maximum stress =| of the elastic limit of the iron 
used. 

Example: Assume an eyebolt for 60,000 lbs. load, having an inside 
diameter of eye of 6 ins. Elastic limit of iron used, 30,000 lbs. per 
sq. in. 



Allowing a stress in the shank = \ 30,000 

= 10, coo lbs. per sq. in., 

we find the diameter at the root of the thread to be 2. 77 ins., giving 

•B = 3i ins., U. S. Standard. 

Allowing a stress in the eye =f 30,000 

= 12,000 lbs. per sq. in., 

the factor F becomes: 

„ msA'^ 

^ = -p- 

2X12,000X6^ 

60,000 

= 14.4 

. ■ B 3.25 
Also sin ri = ^= — ^ = .54 

Now using the chart, the value of Z is found to be 
Z = .479 
and Z) = ZX^ = .479X6 = 2.874 
= 2I ins. 
Giving each of the three lifting tools, shown in Figs. 15,16 and 171 
the same dimensions, A=6 ins. and Z)=2|, as in above example, 
and denoting the factors Fi, F2 and F3 and the loads Pi, P2 and P3, 
respectively, the relative strength can be ascertained by locating 
Z = .479 on the Z-scale, and reversing the method of using the chart, 
determining the factors ^1 = 14.4 (for Fig. 15), i^2=i7 (for Fig. 17) 
and ^3= 11.45 (for Fig. 16). Then according to formula 

msA^ 



Fi-.Fi 



and for equal stresses, we have the proportion: 

I 
P 
or for P = 60,000 lbs. (for Fig. 15), we get 
fi 



^' Pi -Pi 



14.3 : 17.0 : 11.45 



P2 = ^Pi = 51 jooo lbs. 



(for Fig. 17), and 



Fi 



^i = 7S>Soo lbs. 



(for Fig. 16). 

In calculating the shank of the eyebolt, account should be taken 
of any bending action of the load. Even for straight lifts, that is, 
lifts in the direction of the shank, it is practically impossible to avoid 
this bending tendency; only a low stress should therefore be allowed. 
For straight lifts a maximum stress in the shank equal to one-third 
of the elastic limit of the iron may still be considered safe. 

If two or more eyebolts are used in connection with slings, the 
shank is subjected to heavy bending and shoulder eyebolts or a 
suitable spreader should be used, whenever possible. If shoulder 
eyebolts are employed, care should be taken to have the shoulder tight 
against the part to be lifted; this is often neglected. Generally, 
however, straight-shank eyebolts are used and, to avoid accidents, 
stronger eyebolts must be employed than for straight lifts. 

India Rubber 

The stress-strain relationship of india rubber, vulcanized for elas- 
ticity, which is unique among constructive materials, was investigated 
by Dr. R. H. Thurston and is presented in Fig. 18 {Science, 1898). 
It is a matter of common observation that, when this substance is 
subjected to a pull of steadily increasing intensity, its resistance 
increases, as does that of any elastic and ductile material; but that, 
at the end, instead of suddenly losing power of resistance, or even 
snapping without observable decrease of load, its resistance for a 
time rapidly and largely increases up to the point of rupture. This 
can be readily felt in even the breaking of one of the small bands of 
partially vulcanized rubber so universally employed for filing papers 
and other purposes. At the end of the period of extension the 

{Continued on page ^gi^first column) 



490 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 




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STRENGTH OF MACHINE PARTS 



491 



resistance rises so rapidly as to produce the sensation of bringing 
the hand up against a rigid obstacle, resisting further elongation. 

Fig. 1 8 shows the property as determined in a testing machine. 
The substance behaves precisely like other familiar materials, up to 
a point which, in this case, is found at a load of 30 per cent, of the 
maximum, the breaking load, and at an extension one-half the max- 
imum. At this point there exists a reversal of the line, and the cur- 
vature is thence maintained convex to the axis of X, up to the point 
of rupture; fracture taking place, at the end, sharply and without 
any indication of that method of flow of the mass which, in the case 



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12 3 4 5 6 7 

Stretch per Unit Length 

Fig. 18. — Stress strain diagram of india rubber. 

of the irons and softer steels, for example, permits a falling off of 
resistance after passing a point of maximum tenacity well within 
the breaking limit. The ratio of increase of load to increase of elon- 
gation steadily increases from the zero point, as with all substances, 
other than iron and steel, so far as known, up to this point of contrary 
flexure on the diagram, at which place the ratio is inverted and resist- 
ance increases in greater proportion than extension, finally assuming 
a comparatively high value. 

India rubber exhibits none of the phenomena giving the character- 
istic form of the diagrams of the irons and steels. Even when 
stretched to the point of rupture it restores itself very nearly to its 
original dimensions, and gradually recovers a part of the loss of 
form at that instant observable. Its almost complete stability of 
form when relieved from load, and especially when in the shape of 
springs such as are used on railway trucks, constitutes one of its 
most valuable properties. Like cork, when confined laterally it is 
practically incapable of distortion when used as a spring. The 
volume of the mass remains, so far as can be seen, constant, or 
nearly so. 

Materials and Constructions for Resisting Shock 

The former universally accepted dictum that the capacity of a 
material for resisting shock is measured by its toughness is now known 
to be erroneous although still believed by the poorly informed. Its 
fallacy was demonstrated by experience with steam hammers and 
rock drOls. 

The property which determines the capacity of a material to resist 
shock is resilience. Not toughness, not elongation, not reduction 
of area, not ultimate strength, not any possible property developed 
after the elastic limit has been passed, but resilience — the energy 
absorbed during the deformation of a piece of material when stressed 
to the elastic limit — is the property that measures the capacity of that 
material to resist shock without damage to itself. 

This property of resilience is nientioned in books on the strength 
of materials, but little more than mention is made of it, while tables 
of the resiliences of various materials are not to be found in either 



treatises or reference books. The importance of the property has 
been universally overlooked in engineering literature and information 
about it is correspondingly meager. For these reasons some ele- 
mentary facts about it are given here. 

Fig. 19 is a stress-strain diagram of a piece of steel. The vertical 
ordinates represent stresses and the horizontal ordinates elongations. 
The diagram is essentially of the same character as an indicator card, 
the area under the curve at any point representing the work done in 
stretching the piece te the elongation under that point. The area 
under the entire curve up to the breaking point represents the work 
done in breaking the piece, while the area of the triangle ending at 
the elastic limit represents the work done in stretching the piece to 
the elastic limit, this work being the resilience of the piece. ^ If the 
piece is of unit cross-section and of unit length — that is, of unit 
volume — the area of the little triangle becomes the measure or 
modulus of resilience of the material, and this modulus is the direct in- 
dex of the capacity of the material to resist shock. 

The only definite unit by which to measure the magnitude of the 
blow of a steam hammer, for example, is the unit of energy. A 
hammer head of a given weight moving at a given velocity contains 
an amount of energy corresponding to this weight and velocity. 
This energy is expended on the piece of work on which the hammer 
falls, and measures the magnitude of the blow. 

If the energy of a blow, or shock, expended on a piece does not 
deform the piece beyond the elastic limit, the energy is restored by 
the recovery of the piece, the action being like that of a spring under 
the influence of a force that does not produce a permanent set. In- 
deed, not only is the piece like a spring, it is a spring; a very stiff 



o70 
w 

g60. 



w20- 



5IO.OOO 



c 
b 



0.05 0.10 0.15 0.200.25 0.30 0.35 
Proportionate Elongation 




30,000,000 



Fig. 19. Fig. 20. 

A stress-strain diagram and magnified beginning of it. 

spring, indeed, but still a spring. If, however, the piece is deformed 
beyond the elastic limit, the case becomes one of a spring under a 
load that gives it a permanent set, and if this force be repeated a 
sufficient number of times, failure results. 

From this it should be clear that the material that will resist the 
greatest blow is one having the greatest resilience, which, as will be 
shown presently, is measured by the elastic limit. Since high-carbon 
steel has a higher elastic limit, and hence higher resilience, than low- 
carbon steel, the former is the superior material and, by the same 
token, tempered is superior to untempered steel, because the effect 
of tempering is to raise the elastic limit and, with it, the resilience. 

The most constant of all the properties of steel is the modulus of 
elasticity, which remains at about 30,000,000 for all percentages of 
carbon, for the tempered and untempered conditions, and for alloy 
steels alike. This means that — the horizontal and vertical scales 
of stress-strain diagrams being supposed constant — the angle cob, 
Fig. 19, is fixed for all grades of steel and that the area of the triangle 

1 Throughout this discussion the difference between the elastic limit and the 
yield point has been ignored. Strictly speaking, the argument applies to the 
elastic limit, but, being more definitely known, the yield point is used when 
comparing steels. The difference is unimportant. 



492 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



varies with and may be determined from the elastic limit alone. 
This, however, is not all of the story. The strength of a piece at the 
elastic limit is represented by the length of the line, while the resilience 
is represented by the area of the triangle. The angle cob being 
constant for all grades of steel, all triangles for stresses under the 
elastic limit are similar triangles. Since the areas of similar triangles 
are to each other as the squares of their like sides, it follows that the 
resiliences of two pieces of steel are to each other as the squares of 
their elastic limits — a steel having twice the elastic limit of another 
having four times the resilience, or shock-resisting power, from which 
we see at once the enormous value of high elastic limit for this pur- 
pose — a value that is far in excess of that for resisting simple static 
loads. 

The same reasoning that shows high-carbon to be superior to low- 
carbon steel in shock-resisting power shows also tempered to be 
superior to untempered steel. Tempered steel is, however, a very 
general term. Our knowledge of the variation of resilience with 
grades of temper is too limited to enable us to say what grade of 
temper has the greatest resilience. The general diSerences between 
tempered and untempered steel are increased elastic limit and ulti- 
mate strength and reduced elongation, with no change in the modulus 
of elasticity, but we do not know if the resilience continues to in- 
crease as the degree of temper increases. Experience with springs 
shows a comparatively mild temper to be necessary, and this is gen- 
erally believed to be due to the need of some toughness. Until 
we know more exactly the relation between temper and resilience it is 
an open question if reduced resilience at the higher tempers may not 
be the real explanation of the suitability of a mild temper to spring 
service. Under accidental overloads toughness may prevent break- 
age by substituting permanent set, but it is certainly difficult to see 
how a property that is not developed under proper working loads can 
have any value under such loads. Regardless of such considerations, 
however, it is reasonable and safe to infer that the temper best suited 
to springs is also best for shock resistance. 

While the determination of actual stresses due to shock is scarcely 
feasible, it is, nevertheless, easy to compare the relative values of 
materials to resist shock throxigh their moduli of resilience and to 
select the best. Tables of these moduli are not common, but their 
calculation, given the elastic limit and the modulus of elasticity, is 
a simple matter. 

When we say that the modulus of elasticity of steel is 30,000,000, 
we mean that a force of i lb. will stretch a piece of steel of i sq. in. 

cross-section by the part of its length. Let Fig. 20 repre- 

■' 30,000,000 ^ ° o jr 

sent the beginning of a greatly magnified stress-strain diagram of a 
piece of steel, the length of which is i in., the cross-section i sq. in. 
and the load upon it i lb., the stretch being, as has just been said, 

in. The work done by this pound in stretching the piece 

30,000,000 J r o r- 

is represented by the triangle abc, Fig. 20, being 



Work done = HXiX: 



-in. -lbs. 



"30,000,000 60,000,000 

The angle cah being fixed and the areas of triangles for other loads 
being as the squares of the loads, we have, for any other load P less 
than the elastic limit, the area of the triangle ade, or 

p2 



Work done=; 



-in. -lbs. 



60,000,000 

Placing for P the strength of the material per square inch of section 
at the elastic limit, we obtain the modulus of resilience. Thus, for 
an elastic limit of 30,000 lbs. we obtain: 

30,000^ 



Modulus of resilience = 



^15 in. -lbs. per cu. in. 



60,000,000 

There is a surprising lack of published information regarding the 
properties of steels as influenced by the carbon content above about 
50 points carbon. 

Dr. Henry M. Howe has generously placed at the author's dis- 



posal some unpublished data from which the yield points of Table 19 
have been taken and by calculation in accordance with the above 
formula the moduli of resilience also given in the table have been 
obtained. Strictly speaking, these moduli should be obtained from 
the elastic limits, but in their absence the figures obtained from the 
yield points are necessarily used and, for purposes of comparison, 
answer every requirement. 

Table 19. — Resiliences of Open-hearth Carbon Steels 



Percentage of carbon 


Yield point 


Modulus of resilience 


0. 10 


32,000 


17.0 


0. 20 


38,500 


24.7 


0.30 


45,500 


34-5 


0.40 


52,000 


45 -o 


0.50 


58,500 


57-0 


0.60 


65,000 


70.4 


0.70 


71,000 


84.0 


0.80 


76,000 


96.3 


0.90 


82,000 


112 . 



Of the desired properties of alloy steels exact knowledge is naturally 
more meager than of carbon steels. Under suitable heat treatment 
they give remarkable results and their superiority as relates to 
shock resistance has been amply demonstrated in rock drill work. 
The resiliences of two of these steels of the highest types are given 
in Table 20, the figures for Type D vanadium steel being for this 
material as heat treated for use in springs. It should be noted, 
however, that the American Vanadium Co. does not recommend such 
a steel as this for steam-hammer piston rods, its recommendation 
for that purpose being a steel having an elastic Hmit of 80,000 lbs. 
and an elongation of 20 per cent. 

To what extent this recommendation is the result of elimination 
tests of steels of higher resiUence, the author is not informed. It is 
even possible that it may indicate a survival of the old belief in 
toughness. It must be remembered that the property of resilience is 
almost ignored in the testing of materials. Transverse bending tests 
to destruction of pieces under alternating or revolving stresses can 
throw but indirect, if any, light on this property. Direct-shock 
resistance-testing machines would be easy to design and make, but 
they are not in use. 

Other alloy steels give equally promising indications of possibilities, 
and Table 20 includes also the properties of duplex-gear steel No. i, 
made by the Crucible Steel Co. of America. It will be understood 
that the different combinations of properties are obtained by varia- 
tions in the heat treatment. 

If these figures look extraordinary, it must be remembered that they 
are no more so than the working results. 

Table 20. — Resiliences of Alloy Steels 



„, f , , Elastic 
Class of steel ,. . , 
limit' 


Elongation in 
2 in., per cent. 


Modulus of 
resilience 


Type D, vanadium . . . 
Duplex Gear No. i . . . . 


1 170,000 
1 225,000 
r 125,000 
I 195,000 
220,000 


15 
10 
20 
13 
9 


481 
844 
260 

634 
■ 806 



Forms and Dimensions of Parts to Resist Shock 

Not only is the selection of a grade of steel for resisting shock based 
on different principles from the selection for resisting static loads, 
but the determination of the forms and dimensions is based on equally 
different principles. The determination of dimensions to resist 
static loads involves little more than proportioning the smallest 

' So given; in fact, probably the yield point. 



f 



STRENGTH OF MACHINE PARTS 



493 



section to the load, but such a procedure is but a beginning when 
shock is to be resisted. 

While the capacity of a given hammer to deliver blows can be 
measured in units of energy only, it is nevertheless true that the 
hammer-head exerts a certain pressure on the work which can be 
expressed in pounds. The amount of this pressure is not, however, a 
fixed quantity for any given hammer, its value being dependent on the 
space moved though during the destruction of the velocity of the 
hammer head. To find the energy when the force and the space 
moved through by it are given, we have the universal equation. 

Energy = force X space, 
and, similarly, if the energy and the space are given, the equation for 
the force becomes, 

energy 



Force = 



space 



from which it is clear that the force exerted by a hammer head con- 
taining a given amount of energy grows less in proportion as the space 
moved through in stopping the hammer is increased. The force 
found by the formula is, of course, the mean and not the maximum. 
This points out at once the universal expedient to be followed in 
designing parts to resist shock — make the space moved through in 
absorbing the shock as great as possible. To the extent that this space 
can be increased, to the same extent will the force resulting from the 
shock be reduced. To put it in another way, recalling the fact that a 
piece under shock is a spring, the aim should be to make it as flexible 
a spring as possible, since any device which increases the stretch dur- 
ing which the shock is absorbed reduces the force which the cross- 
section must resist and hence, if need be, the cross-section itself. 




Fig. 21. Fig. 22. 

Strengthening a bolt by removing some of its material. 

In Fig. 21 is represented a bolt suspended from above and sur- 
rounded by a ring weight a, the dropping of which produces a shock by 
its impact on the lower nut. The energj of the blow being fixed, its 
force is determined by the stretch of the body of the bolt, while it 
must be resisted by the section b at the root of the thread. The body 
of the bolt has a surplus of strength over that at b, and since the 
strength of a chain is determined by its weakest link, the surplus 
strength of the body does not add to the strength of the bolt. By 
reducing the diameter of the body to that at the root of the thread, 
Fig. 22, the static strength is unchanged, but the body of the bolt, 
considered as a spring, is made more flexible and the stretch under a 
given blow is increased. As has been explained, this reduces the 
force of the blow and, hence, the stress on the section b. Put in 
another way, the bolt of Fig. 22 will endure a heavier blow than the 
one of Fig. 21, and we have the apparently paradoxical, but never- 
theless sound, conclusion that in effect, and as against shock, the bolt 
has been made stronger by removing some of its material. One of Dr. 
Sweet's aphorisms expresses this conclusion in the briefest possible 
way — "Make the piece weaker where it doesn't break." 

The principle illustrated by Figs. 21 and 22 may be put in per- 
fectly general form as follows: Whatever the nature of the stress a 
body to resist shock shotdd have the form of uniform strength. 



To continue the spring analogy, this change in the diameter of body 
of the bolt is equivalent to making a helical spring of smaller wire. 
Such a spring may also be increased in flexibility by increasing 
the number of coils, that is, the total length. To this change in the 
spring there is, with the bolt, a perfectly analogous change, namely, 
to make the bolt longer. Such increase in the length of the bolt will 
obviously increase the stretch under a given blow and reduce the 
force of the blow, precisely as the reduction of the body diameter re- 
duced it. This principle may be put in brief form by saying that, in 
such a bolt and as against shock, increase of length is just as useful as 
increase of sectional area. 

This conclusion is in strict agreement with the meaning of resili- 
ence. The modulus of resilience is the number of units of energy 
absorbed per unit of volume of the material, the resilience of any piece 
being equal to the modulus for the material multiplied by the units of 
volume of the piece, regardless of any one of its linear dimensions. 




J^ 




Fig. 23. — The use of long bolts for resisting shock. 

There principles are illustrated and confirmed by successful and 
unsuccessful constructive details of rock driUs. In the early 
days of the rock drill industry, when the author was connected 
with it, one of the problems was to keep the front cylinder head on the 
cylinder. The head was at first fastened to the cylinder in essen- 
tially the same manner as steam-engine cylinder heads, the only 
difference being in the use of through bolts instead of studs, in order 
to facilitate the renewal of broken bolts. In the operation of a rock 
drill, it is impossible to avoid an occasional sharp blow of the piston on 
the front head instead of the intended rock, and the bolts described 
suffered in consequence. A rubber buffer ring was placed inside the 
cylinder to receive these blows, but apart from the destruction of the 
buffer by the oil, it did not prevent the frequent breakage of the bolts. 

The difficulty was overcome by the construction shown in Fig. 23. 
Instead of separate short bolts for the two cylinder heads, two long 
bolts, or side rods, a and b were carried down the sides of the cylinder, 
connecting the two heads and securing both to the cylinder in the 
manner shown. At their upper ends they were connected by a cross- 
piece, between which and the back head was placed the rubber buffer. 
With this construction the difficulty of broken bolts vanished, and 
after what has been said, the reason is clear. The increased stretch 
of the bolts due to their increased length reduced the stress to a figure 
which the bolts could carry. 

After the consolidation of the Ingersoll and Rand interests, the 
cross-piece and buffer construction was replaced by the steel springs 
c, as shown, but the breakage of the bolts was overcome by no other 
change than the increase in their length. 

Along with breakage there goes another effect of shock — the rattl- 
ing apart of fastenings, especially screws and nuts. While, of course, 
less serious than breakage, this action was an unmitigated nuisance 
during those pioneer days, and again the experience gained is of wide 
application. The solution of this is essentially the same as that of the 
breakage problem — make the parts elastic, give the resilience a chance 
to act. If the parts are to be bolted together, make the bolts as long 
as possible and no larger in diameter than necessary. If a bolt can- 
not be long, introduce a spring piece under the nut and subject to its 
pressure. 

The simplest and most generally useful of these spring pieces is 
the elastic lock-spring washer originally introduced for use on railroad- 
track bolts. Its security is not absolute and it should not be used 
where life depends on its action, but its great usefulness has been 
demonstrated in rock drill work. The frame by which the cylinder is 
supported and on which it moves forward for the feed has on each side 



494 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



a bolted on slide or gib a. Fig. 24. The bolts b must be short and in 
consequence the nuts constantly rattled ofif. This annoyance was 
reduced to a negligible amount by the spring washers c under the nuts. 
The author has repeatedly seen machines come back after such an 
amount of use as to require general repairs, with every bolt, nut and 




m 



Fig. 24. — Elastic constructions for resisting shock. 




Fig. 25. — Unsuccessful rigid construction. 

washer in place and so covered with mud and rust as to show that they 
had never been disturbed. 

One of the most annoying of the details of the early machines was 
the chuck — the device at the end of the piston rod for gripping the 
long piece of drill steel. A simple way to grip the steel would be by 
a setscrew through the body of the chuck, but such a construction 



would rattle loose almost as fast as it could be tightened because it 
lacked the element of elasticity, and device after device failed for the 
same reason. One such failure is shown in Fig. 25. 

The successful chuck — now in universal use — is shown in Fig. 26 
and, while a simple thing, it is based on sound philosophy. The shape 
and overhang of the U-bolt a combined with the elasticity of the key 
b, with its cut-away center and end bearings on the steel shank by 
which it acted as a spring piece overcame the difficulty. 




Fig. 26. — Successful elastic construction. 

Another illustration is the standard, d, Fig. 24, one of which 
placed on each side of the frame, the two being connected at their 
upper ends by a cross-piece carrying the upper end of the feed screw. 
Formerly the standards were bolted to the frame by a nut located 
immediately under the boss e but, so placed, the nuts constantly 
rattled off. The addition of the long (and slender) tail / and the 
location of the nut at g introduced the necessary elasticity and the 
trouble ceased. 



WEIGHTS AND MEASURES 



While the British continue to use certain units of measurement 
which Americans have discarded, notably the stone and the hundred- 
weight of 112 lbs., the fundamental units of length and weight and 
their chief derivatives are identical in Great Britain and the United 
States. Measures of capacity, unfortunately, differ. 

The base of American measures of capacity for liquids is the 
Winchester gallon of 231 cu. ins. and the corresponding base of the 
British measures is the Imperial gallon of 277.274 cu. ins._ The 
division of the two gallons into gills, pints and quarts follows the same 
scale. Following are the relations of the two gallons: 
I U. S. gallon = .833 Imperial gallon. 

I Imperial gallon =1.200 U. S. gallons. 

7.48 U. S. gallons =1 cu. ft. 

6.24 Imperial gallons =1 cu. ft. 

I U. S. gallon of water at 62 deg. Fahr. =8.34 lbs. 
I Imperial gallon of water at 62 deg. Fahr. = 10 lbs. 

The U. S. (Winchester) bushel contains 2150.42 cu. ins., while the 
Imperial bushel is based on the Imperial gallon, of which it contains 
8 or 2218.19 cu. ins. The division of the two bushels into pints, 
quarts and pecks follows the same scale. Following are the relations 
of the two bushels: 

I U. S. bushel = .969 Imperial bushel. 

I Imperial bushel =1.032 U. S. bushels. 

I U. S. bushel =1.244 cu. ft. 

I Imperial bushel = 1.284 cu. ft. 

The Metric System 

That monument to scientific zeal combined with ignorance of 
practical requirements — the metric system — is unfortunately present 
in the world and cannot be ignored. 

The claims for the ease of adoption and the wide use of the system 
have been shown by S. S. Dale and the author to be grotesquely 
false {Trans. A. S. M. E., Vols. 24 and 28 and The Metric Fallacy). 
The facts are that no nation has ever made serious progress toward 
the adoption of the system in trade and commerce except by the 
force of compulsory law, and that no nation has ever discarded its 
old units by force of compulsory or any other law. 

The case for France was officially summed up and confessed in a 
circular letter dated Paris, Apr. 11, 1906, from the French Minister 
of Commerce, Industry and Labor, to the presidents of French Cham- 
bers of Commerce, of which the folio-wing is a translation in part. 
The full text may be found in the Transactions of the A. S. M. E., 
Vol. 28: 

" My Department at different times has been called upon to give 
to the Department of Weights and Measures instructions for accom- 
plishing the total suppression of the measures and weights prohibited 
by the old law of July 4, 1837 by the seizure of the prohibited articles. 
The Department, in spite of all such efforts, has not succeeded in 
attaining the desired result. 

"I have learned that in certain industries the advertisements, 
prospectuses, catalogues, etc., used by the merchants among them- 
selves and also for sending to their customers contain the illegal 

expressions They thus continue to designate in lignes 

and inches all the articles they sell 

"I do not consider it worth while to enumerate here the industries 
and professions which have continued to employ the proscribed 
standards, but they are still numerous and most of them known to 
members of your organization." 



In the metric countries of western Europe, great industries, 
although selling their products by the metric system, make exclusive 
use of the old systems in the manufacture of those products. Thus 
in France the leading industry is the manufacture of silk fabrics, 
and this industry makes exclusive use of the aune and denier as its 
manufacturing units of length and weight respectively. Again, in 
Germany, the cotton industry is based exclusively on the British 
yard and pound and the woolen industry is similarly based on a 
great variety of old German ells and pounds. Throughout the metric 
and non-metric world lumber is sawn to the inch. 

The actual condition is diametrically the opposite of the imaginary 
one pictured by the metric party. For actual uniformity of measures 
in all industries and commerce and for actual simplicity of calcula- 
tions due to that uniformity we must turn to English-speaking 
countries, while, for actual diversity of measures and complexity of 
calculations due to the necessity for repeated conversions between 
incommensurate units, metric countries supply an example and a 
warning. 

Outside western Europe and contrary to oft-repeated but un- 
founded assertion, the system is used but little. Many countries, 
notably those of Spanish America, have "adopted" the system, 
but without compulsion, the result being that it has become an official 
government system used chiefly in the collection of customs duties 
and sometimes only partially there, while among the people it is 
used but little or not at all. 

In other countries which are frequently classed as metric (Japan, 
Russia, the treaty ports of China) the law goes no further than to 
make the system permissive exactly as in Great Britain and the 
United States. 

These conclusions are proven by an array of facts that is over- 
whelming. No serious attempt to answer them has ever been made, 
because such answer is impossible. 

There are but two possible explanations of these facts — either the 
advantages of the system are not sufficient to justify its adoption 
or its adoption is attended with so much difficulty as to be impractic- 
able. Either explanation is fatal to the pro-metric argument. 

The feature of the system on which most stress is laid by its 
advocates — its convenience in the reduction or conversion of units, 
due to the fact that it has the same base as our unfortunate system 
of arithmetical notation — overlooks the fact that in the affairs of 
every-day life such conversions are of too infrequent occurrence to 
lend importance to this feature. All customary calculations of the 
engineer or business man are made as readily in the British as 
in the metric system. 

Were it otherwise, the repeated conversions during the transition 
period of two systems of units used conjointly and bearing incom- 
mensurate ratios with one another, offset many times over even the 
claims made for economy of time in calculations by the metric system. 
Of the probable length of the transition period we may form some 
idea from the fact that as acknowledged by the Minister of Com- 
merce, Industry and Labor it is still far from complete in France. 

The metric system is, at best, a complete subordination of the 
greater to the lesser — of the function of measuring to that of calcula- 
tion. Its advocates forget that the chief function of a system of 
weights and measures is to weigh and measure, not to make calcu- 
lations." Because of this some of its units are ill adapted to many 
of the purposes of life, while the decimal division of units is far in- 
ferior to binary divisions for the purposes of commerce and 
manufacture. 

{Continued on page 499, first column) 



495 



496 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



British-American-Metric Conversion Factors 

(Except Capacity Measures which are American only) 
From The U. S. Bureau of Standards 















Leng 


ths 












Inches 


1 


Millimeters 


Inches 


1 


Centimeters 


Feet 


Meters 


U. S. yards | 


Meters 


U. S. miles 


1 


Kilometers 


.03937 

.07874 
.I1811 
.15748 


= 


I 
2 
3 

4 


.3937 

.7874 
I 
I . 1811 


= 


1 

2 

2.54001 

3 


1 = 

2 = 

3 = 
3.28083 = 


.304801 
.609601 
.914402 
I 


1 = 
I. 0936 I I = 

2 = 
2.187222 = 


.914402 
I 

1.828804 
2 


.62137 
I 

1.24274 
I. 8641 I 


= 


I 

1.60935 

2 

3 


.19685 
.23622 
.27559 
.31496 
.35433 


= 


5 
6 
7 
8 
9 


1.5748 

1.9685 

2 

2.3622 

2.7559 


= 


4 

5 

5 .08001 

6 

7 


4 = 

5 

6 

6.56167 = 

7 


I. 219202 

1.524003 

1.828804 

2 

2.133604 


3 = 
3.280833 = 

4 

4.374444 = 

S 


2.74320S 

3 

3.657607 

4 

4.572009 


2 

2.48548 

3 

3.1068s 

3.72822 


= 


3.21869 

4 

4.82804 

5 

6 


I 
2 
3 
4 


= 


25.4001 
50.8001 

76. 2002 

loi .6002 


3 

3.1496 
3.5433 
4 


= 


7.62002 
8 
9 
10.16002 


8 
9 

9.84250 = 
13.12333 = 


2.43840s 
2.74320s 
3 

4 


S. 468056 = 

6 

6.561667 = 

7 = 


5 

5. 48641 I 

6 

6.400813 


4 

4.34959 
4.97096 
5 


= 


6.43739 
7 
8 
8.04674 


S 
6 
7 
8 
9 


=. 


127.0003 
152.4003 
177.8004 
203 . 2004 
228.6005 


5 
6 
7 
8 
9 


= 


12.70003 
15.24003 
17.78004 
20.32004 

22. 86005 


16.40417 = 
19.68500 = 
22.96583 = 
26.24667 = 
29.52750 = 


5 
6 
7 
8 
9 


7.655278 = 

8 

8.748889 = 

9 

9.842500 = 


7 

7.315215 

8 

8.229616 

9 


5. 59233 

6 

7 

8 

9 


= 


9 

9.65608 
11.26543 
12.87478 
14.48412 



Areas 



Square 
inches 


Square 
millimeters 


Square 
inches 


Square 
centimeters 


Square feet 


Square meters 


Square yards 


Square meters 


Square miles 


Square 
kilometers 


.00155 = 1 
.00310 = 2 
.00465 = 3 
.00620 = 4 

.00775 = 5 
.00930 = 6 
.01085 = 7 
.01240 = 8 
.01395 = 9 

1 = 645.16 

2 = 1,290.33 

3 = 1,935.49 

4 ' = 2,580.65 

5 = 3.225.81 

6 = 3,870.98 
, 7 = 4,516.14 

8 = 5,161.30 

9 = s. 806. 46 


.1550 = I 
.3100 = 2 
.4650 = 3 
.6200 = 4 

.7750 = S 
. 9300 = 6 

1 = 6.452 
1.0850 = 7 

I . 2400 = 8 

1.3950 = 9 

2 = 12.903 

3 = 19.355 

4 = 25.807 

5 = 32.258 

6 = 38.710 

7 = 45.161 

8 = 51.613 

9 = 58.065 


1 = .09290 

2 = .18581 

3 = .27871 

4 = .37161 

5 = .46452 

6 = .55742 

7 = .65032 

8 = .74323 

9 = .83613 

10.764 = I 
21.528 = 2 
32.292 = 3 
43-055 = 4 

53.819 = 5 
64,583 = 6 
75.347 = 7 
86. Ill = 8 
96.87s = 9 


1 = .8361 
I. i960 = I 

2 = 1.6723 
2.3920 = 2 

3 = 2.5084 
3.5880 = 3 

4 = 3.344s 
4.7839 = 4 

5 = 4.1807 

5.9799 = 5 

6 = 5. 0168 

7 = 5.8529 
7.1759 = 6 

8 = 6,6890 
8.3719 = 7 

9 = 7.5252 
9.5679 = 8 

10.7639 = 9 


.3861 = I 
.7722 = 2 

1 = 2 . 5900 
1.1583 = 3 

I . 5444 = 4 
1.9305 = 5 

2 = S.180O 
2.3166 = 6 
2.7027 = 7 

3 = 7.7700 
3.0888 = 8 
3.4749 = 9 

4 = 10.3600 

5 = 12.9500 

6 = 15.5400 

7 = 18.1300 

8 = 20.7200 

9 = 23.3100 


Volumes 


Areas. — Continued 


Cubic 
inches 


Cubic 

millimeters 


Cubic 
i nches 


Cubic 
centimeters 


Cubic feet 


Cubic 
meters 


Cubic yards 


Cubic 
meters 


Acres 


Hectars 


.000061 = 1 
.000122 = 2 
.000183 = 3 
,000244 = 4 

.00030s = s 
.000366 = 6 
.000427 = 7 
.000488 = 8 
.000549 = 9 

1 = 16,387.2 

2 = 32,774-3 

3 = 49,161.5 

4 = 65,548.6 

5 ■ = 81,935.8 

6 = 98,323.0 

7 = 114,710.1 

8 = 131,097.3 

9 = 147,484.5 


.0610 = I 
.1220 = 2 
.1831 = 3 

.2441 = 4 

.3051 = 5 
.3661 = 6 
.4272 = 7 
.4882 = 8 
.5492 = 9 

1 = 16.3872 

2 = 32.7743 

3 = 49.1615 

4 = 65.5486 

5 = 81.9358 

6 = 98.3230 

7 = II4.7101 

8 = 131.0973 

9 = 147.4845 


1 = .02832 

2 = .05663 

3 = .08495 

4 = .11327 

5 = .14159 

6 = . 16990 

7 = .19822 

8 = .22654 

9 = .25485 

35.314 = 1 

70.629 = 2 
105.943 = 3 
141.258 = 4 

176.572 = 5 
211.887 = 6 
247.201 = 7 
282.516 = 8 
317.830 = 9 


1 = .7645 
1.3079 = 1 

2 = 1.5291 
2.6159 = 2 

3 = 2.2937 
3.9238 = 3 

4 = 3.0582 

5 =■ 3.8228 
5.2318 = 4 

6 = 4.S874 
6.5397 = S 

7 = 5.3519 
7.8477 ■= 6 

8 = 6.1165 

9 = 6.8810 
9.1556 = 7 

10.463s = 8 
11.7715 = 9 


1 = . 4047 

2 = . 8094 
2.471 = I 

3 = I. 2141 

4 = 1.6187 
4.942 = 2 

5 = 2.0234 

6 = 2.4281 

7 = 2.8328 

7.413 = 3 

8 = 3.237s 

9 = 3.6422 
9.884 = 4 

12.355 = 5 
14.826 = 6 
17.297 = 7 
19.768 = 8 
22.239 = 9 



WEIGHTS AND MEASURES 



497 



British-American-Metric Conversion Factors — (Continued) 

(Except Capacity Measures wmcH are American only) 

Capacities 



U. S. liquid 
quarts 


Liters 


U. S. liquid 
gallons 


Liters 


U. S. dry 

quarts 


Liters 


U. S. pecks 


Liters 


U. S. bushels 


Hectoliters 


I 

1.05668 = 

2 

2. I 1336 = 


= 


• 94636 

I 

1 . 89272 

2 


.26417 = 
.52834 = 
.79251 = 
1 


1 

2 
3 

3.78543 


.9081 

1 

1. 8162 

2 


= 


1 

1 . 1012 

2 

2. 2025 


.11331 
.22702 
.34053 
.45404 


= 


I 

2 

= 3 

= 4 


I 

2 

2.83774 

3 


= 


-35239 

.70479 
I 
I .05718 


3 

3-17005 = 

4 

4.22673 = 

5 


= 


2.83908 

3 

3.78543 

4 
4.73179 


1.05668 = 
1.32085 = 
1.58502 = 
I. 84919 = 
2 = 


4 
5 
6 
7 
7-57087 


2.7242 

3 

3.6323 

4 
4.5404 


= 


3 

3-3037 
4 

4-4049 
5 


.56755 
.68106 
.79457 
.90808 

I 


= 


= 5 
= 6 
= 7 
= 8 
= 8.80982 


4 

5 

5.67548 

6 

7 


= 


I 40957 
I .76196 
2 

2.11436 
2.46675 


5-28341 = 

6 

6.34009 = 

7 


= 


5 

5.6781S 
6 
6.62451 


2.11336 = 
2.37753 = 
3 

4 


8 

9 
11.35630 
15.14174 


5 

5.4485 
6 
6-3565 


= 


5.5061 
6 

6.6074 
7 


1.02157 
. 2 
3 

4 


= 


= 9 

17.61964 
26.42946 
35-23928 


8 

8.51323 
9 
11.35097 


= 


2.81914 
3 

3.17154 
4 


7.39677 = 

8 

8.45345 = 

9 

9.S1014 = 




7 

7.57088 

8 

8.51723 

9 


5 
6 
7 

8 = 

9 = 


18.92717 
22.71261 
26.49804 
30.28348 
34.06891 


7 

7 . 2646 

8 

8.1727 = 

9 


= 


7.7086 

8 

8.8098 

9 

9.9110 


5 
6 

7 
8 
9 


= 


44.04910 
52.85892 
61.66874 
70.47856 
79.28838 


14.18871 
17.0264s 
19.86420 
22.70194 
25.53968 




5 
6 
7 
8 
9 



Weights 



Grains 


Grams 


Avoirdupois 
ounces 


Grams 


Troy 
ounces 


Grams 


Avoirdupois 
pounds 


Kilograms 


Troy pounds 


Kilograms 


I 
2 
3 
4 


= 


.06480 
.12960 
. 19440 
-25920 


-03527 = 
-07055 = 
.10582 = 
.14110 = 


= 


I 
2 
3 
4 


-03215 = 
.06430 = 
.0964s = 
.12860 = 


I 
2 
3 

4 


I 

2 

2 . 20462 

3 


■45359 
.90718 

1 

1.36078 


I 

2 

2.67923 

3 


= 


.37324 
.74648 
1 
= 1.11973 


S 

6 

7 
8 
9 


= 


-32399 
-38879 
-45359 
-51839 
-58319 


-17637 = 
.21164 = 
.24692 = 
.28219 = 
-31747 = 


= 


5 
6 

7 
8 
9 


.16075 
.19290 = 
.22506 = 
.25721 = 
.28936 = 


5 
6 
7 
8 
9 


4 

4.40924 = 

5 

6 

6.61387 = 


1.81437 
2 

2 . 26796 
2.72155 
= 3 


4 

5 

5.35846 

6 

7 




1.49297 
= 1.86621 

2 
= 2.23945 

2.61269 


15.4324 
30.8647 
46.2971 
61.7294 


= 


I 

2 

3 

4 


I 

2 = 

3 

4 


= 


28.3495 

56.6991 

85.0486 

113-3981 


1 
2 
3 

4 


= 31.10348 
= 62.20696 
= 93.31044 
= 124.41392 


7 

8 

8.81849 = 

9 


- 3.17515 
3.62874 

= 4 

4.08233 


8 

8.03769 
9 
10. 7 1 69 1 


= 


= 2.98593 

= 3 

= 3.35918 

= 4 


77.1618 
92.5941 
108.0265 
123.4589 
138.8912 


= 


5 
6 
7 
8 
9 


5 
6 
7 
8 
9 


= 


141.7476 
170.0972 
198.4467 
226.7962 
255.1457 


5 
6 
7 
8 
9 


= 155.51740 
= 186.62088 
= 217.72437 
= 248.82785 
= 279.93133 


II.02311 
13.22773 = 
15.43236 = 
17.63698 = 
19.84160 = 


= 5 
= 6 
= 7 
= 8 
= 9 


13.39614 
16.07537 
18.75460 
21.43383 
24. 11306 


= 


= 5 
= 6 
= 7 
= 8 
= 9 







British-Metric and Metric-British Equivalents of 


Units of Length 






Unit 


In. 


Ft. 


Yd. 


Rod 


Furl. 


Mile 


Cm. 


Meter 


Km. 


Unit 


In 


I 

12 
36 
198 
7920 
63360 
.3937 
39.37 
39370 


.083 

1 

3 

16.5 

660 

5280 

.03281 

3.28083 

3280.83 


.027 

.3 

I 

5.5 

220 

1760 

.01094 

1.09361 

1093.61 


.0050 

.06 

.18 

1 

40 

320 

.001988 

.19884 

198.84 






2.54 

30.48 

91.4402 


.0254 

-3048 

.914402 

5. 029 

201. 17 

1609.35 

.01 

I 
1000 




In. 


Ft 


.0015 
.0045 

.025 

1 
8 


.0001893 
.0005681 
.003125 

.125 

1 




Ft. 


Yd 


.0009144 
.005029 
.20117 
1-60935 


Yd. 


Rod 


Rod 


Furl 




Furl. 


Mile 




Mile 


Cm 


I 

100 
100,000 


Cm. 


Meter 

Km 


-00497 
4.97096 


.0006214 
.62137 


.001 

I 


Meter 
Km. 



British-Metric and Metric-British Equivalents of Units of Weight 



Unit 


Grain 


Gram 


Oz. av. 


Lb. av. 


Kilog. 


Short 
cwt. 


Ton 


Unit 




Short 


Metric 


Long 






1 

15-43236 

437-5 

7000 

15432.36 


.0647989 

I 

28.3495 

453-592 

1000 


.0022857 

-035274 

I 

16 
35-27396 


.00014286 

.0022046 

.0625 

I 

2. 20462 

100 

2000 

2204.62 

2240 


.000064799 

.001 

.028349s 

-453592 

1 

45-3592 

907-185 

1000 

1016.05 












Gram 










Gram 


Oz. av 










Oz. av. 


Lb. av 

Kilog 


.01 

.0220462 
I 

20 
22.0462 

22.4 


.0005 
.0OH0231 

-OS 

1 

I. 10231 

I. 12 


.0004536 

.001 

.045359 
.907185 

I 
1.01605 


. 0004464 
.00098421 
. 0446429 
.8928571 
.984206 
1 


Lb. av. 
Kilog. 
Short cwt. 


Short cwt 


Short ton . . . . 








Short ton 


Metric ton 








Metric ton 


Long ton 








Long ton 





32 



498 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



British-American-Metric Conversion Factors for Fractional Dimensions of Length 
From the U. S. Bureau of Standards 
Binary Fractions of an Inch to Millimeters 



Vs 


i's 


8ths 


i6ths 


32nds 


64ths 


Milli- 
meters 


Decimals 
of an inch 


Inch 


¥s 


i's 


8ths 


i6ths 


32nds 


64ths 


Milli- 
meters 


Decimals 
of an inch 












I 


= .397 


.015625 














33 


= 13 . 097 


.515625 










I 


2 

3 


= .794 
= I. 191 


.03125 
.046875 












17 


34 
35 


= 13-494 
= 13-891 


-5312s 
•54687s 








I 


2 
3 


4 

S 
6 

7 


= I.S88 

= 1.984 
= 2.381 
= 2.778 


.0625 

.078125 

■09375 

-I0937S 










9 


18 
19 


36 

37 
38 
39 


= 14-288 

= 14.684 
= 15-081 
= 15-478 


-5625 

■578125 

•59375 

•60937s 






I 


2 


4 
5 


8 
9 

lO 

II 


= 3-175 

= 3-572 
= 3-969 
= 4-366 


- 1250 

. 140625 

-15625 

-171875 








5 


10 


20 
21 


40 

41 
42 
43 


= 15-875 

= 16- 272 
= 16.669 
= 17.066 


-625 

-64062s 

-65625 
-671875 








3 


6 


12 

13 

14 
15 


= 4-763 

= 5-159 
= 5-556 
= 5-953 


-1875 

.203125 

.21875 
-234375 










II 


22 
23 


44 

45 
46 
47 


= 17.463 

= 17.859 
= 18.256 
= 18.653 


-6875 

■70312s 

■7187s 

•734375 




I 


2 


4 


8 
9 


l6 

17 
i8 
19 


= 6.350 

= 6.747 
= 7-144 
= 7-S4I 


. 2500 

.265625 

.28125 

.296875 






3 


6 


12 


24 

25 


48 

49 
50 
51 


= 19.050 

= 19.447 
= 19 . 844 
= 20. 241 


-75 

•765625 

•78125 

•79687s 








5 


10 

II 


20 

21 
22 
23 


= 7-938 

= 8.334 
= 8.731 
= 9.128 


■ 312s 

.328125 

-34375 
-359375 










13 


26 

27 


52 

53 

54 
55 


= 20.638 

= 21.034 
= 21.431 
= 21.828 


-8125 

-828125 

.84375 

-85937s 






3 


6 


12 

13 


24 

25 
26 
27 


= 9.525 

= 9.922 
= 10.319 
= 10.716 


-3750 

-390625 

.40625 

.42187s 








7 


14 


28 
29 


S6 

57 
58 
59 


= 22.225 

= 22.622 
= 23.019 
= 23.416 


.87s 

•89062s 

•90625 

•921875 








7 


14 
IS 


28 

29 
30 
31 


= II. 113 

= 11.509 
= 11.906 
= 12.303 


-4375 

-453125 

-4687s 

■484375 










IS 


30 

31 


60 

61 
62 
63 


= 23.813 

= 24. 209 
= 24.606 
= 25.003 


• 9375 

-9S3I2S 

-96875 

-98437s 


I 


2 


4 


8 


i6 


32 


= 12. 700 


■ 5 


I 


2 


4 


8 


16 


32 


64 


= 25.400 


I .000 



Hundredths of an Inch to Millimeters 



Hundredths 






















of an 





I 


2 


3 


4 


5 


6 


7 


8 


9 


inch 



























.254 


.5O8 


.762 


I. 016 


1.270 


1-524 


1-778 


2-O32 


2.286 


10 


2.540 


2.794 


3-048 


3-302 


3 556 


3-810 


4.064 


4-318 


4-S72 


4.826 


20 


5. 080 


S.334 


5-588 


5-842 


6.096 


6-350 


6 . 604 


6.858 


7 . 112 


7-366 


30 


7.620 


7.874 


8.128 


8-382 


8.636 


8-890 


9-144 


9-398 


9-652 


9^906 


40 


10. 160 


10-4I4 


10.668 


10-922 


II. 176 


11.430 


11.684 


11.938 


12. 192 


12.446 


SO 


12.700 


12.954 


13.208 


13-462 


13-716 


13.970 


14.224 


14-478 


14.732 


14.986 


60 


15-240 


15-494 


15.748 


16 -002 


16. 256 


16.510 


16.764 


17-OI8 


17.272 


17.526 


70 


17.780 


18.034 


18.288 


18.542 


18.796 


19-050 


19.304 


19-558 


19- 812 


20.066 


80 


20.320 


20.574 


20.828 


21.082 


21.336 


21-590 


21. 844 


22. 098 


22-352 


22.606 


90 


22.860 


23.114 


23.368 


23.622 


23.876 


24.130 


24.384 


24-638 


24.892 


25.146 



Millimeters to Decimals of an Inch 



Millimeters 





I 


2 


3 


4 


5 


6 


7 


8 


9 







.03937 


-07874 


.11811 


.15748 


.19685 


.23622 


.27559 


.31496 


.35433 


10 


-39370 


.43307 


-47244 


•SIlSl 


.55118 


.59055 


.62992 


.66929 


.70866 


. 74803 


20 


.78740 


.82677 


.86614 


.90551 


.94488 


.98425 


1.02362 


1.06299 


I- 10236 


1.14173 


30 


I- 18110 


1.22047 


1.25984 


I. 29921 


1.33858 


1.37795 


1.41732 


1.45669 


I - 49606 


1.53543 


40 


1-57480 


I.61417 


1-65354 


1. 6929 1 


1.73228 


1.77165 


I. 81102 


1-85039 


1.88976 


r. 92913 


SO 


I-96850 


2.00787 


2.04724 


•2.08661 


2. 12598 


2.1653s 


2.20472 


2.24409 


2.28346 


2.32283 


60 


2.36220 


2.40157 


2.44094 


2.48031 


2.51968 


2.55905 


2.59842 


2.63779 


2.67716 


2.71653 


70 


2.75590 


2.79527 


2.83464 


2.87401 


2.91338 


2.95275 


2.99212 


3-03149 


3.07086 


3. 11023 


80 


3. 14960 


3-18897 


3.22834 


3.26771 


3 30708 


3.3464s 


3.38582 


3-42519 


3-46456 


3.50393 


90 


3.54330 


3.58267 


3.62204 


3.66141 


3^70078 


3.74015 


3.77952 


3-81889 


3.85826 


3.89763 



WEIGHTS AND MEASURES 



499 



British-Metric Conversion Factors for Compound Units 
From Clark's Manual of Rules, Tables and Data 





Metric-British 




British-Metric 


I kg per m. 


= 


{ 


.672 lb. per ft.. 
2.016 lbs. per yd. 


I lb. per ft. 
I lb. per yd. 


= 




1 . 488 kg. per m. 
. 496 kg. per m. 


I kg. per sq. cm. 


= 




14. 2232 lbs. per sq. ft. 


I lb. per sq. in. 


= 




.0703077 kg. per sq. cm. 


1.0335 kg. per sq. cm. 


= 




14.7 lbs. per sq. in. 


I lb. per sq. ft. 


= 




4.883 kg. per sq. m. 


(i atmosphere) 
















I kg. per sq. m. 


= 




. 205 lbs. per sq. ft. 


I in. of mercury 


= 




2 . 540 cm. of mercury 


I cm. of mercury 


= 




. 394 in. of mercury 


I lb. per sq. in. 


= 




5.170 cm. of mercury 


I cm. of mercury 


= 




. 193 lb. per sq. in. 


I lb. per cu. ft. 


= 




16.020 kg. per cu. m. 


I kg. per cu. m. 


= 




.0624 lb. per cu. ft. 


I cu. ft. per lb. 


= 




.0624 cu. m. per kg. 


I cu. m. per kg. 


= 




16.019 cu. ft. per lb. 


I ft.-lb. 


= 




.138 kgm. 


I kgm. 


= 




7 . 233 f t.-lbs. 


I British h.p. 


= 




i.0139 metric h.p. 


1 metric h.p. 


= 




.9863 British h.p. 


I lb. per British h.p. 


= 




.447 kg. per metric h.p. 


I kg. per metric h.p. 


= 




2. 23s lbs. per British h.p. 


I sq. ft British per h.p. 


= 




.0916 sq. m. per metric h.p. 


I sq. m. per metric h.p. 


= 




10.913 sq. ft. per British h.p. 


I B.t.u. 


= 




.252 carlorie 


I calorie 


= 




3.968 B.t.u.'s 


I ft. per sec. or per min. 


= 




. 305 m. per sec. or per min. 


I m. per sec. 


= 




3.281 ft. per sec. 
196.860 ft. per min. 
2. 236 miles per hr. 


I mile per hr. 


= 


{ 


.447 m. per sec. 
1 . 609 km. per hr. 


I km. per hr. 






.621 miles per hr. 











British-Metric Conversion Factors for Units of Pressure Metric-British Conversion Factors for Units of Pressure 



Lbs. 


Kgs. 


Lbs. 


Kgs. 


Lbs. 


Kgs. 


Lbs. 


Kgs. 


per 


per sq. 


per 


per sq. 


per 


per sq. 


per 


per sq. 


sq. m. 


centim. 


sq. m. 


centim 


sq. in. 


centim. 


sq. m. 


centim. 


I 


.0703 


26 


1.828 


51 


3-5857 


76 


5-3434 


2 


. 1406 


27 


1-8983 


52 


3-656 


77 


5.4138 


3 


. 2109 


28 


1.9686 


53 


3-7263 


78 


5-4841 


4 • 


.2812 


29 


2.0389 


54 


3-7966 


79 


5-5544 


S 


.3515 


30 


2. 1092 


55 


3.8669 


80 


5-6247 


6 


.4218 


31 


2.1795 


56 


3-9373 


81 


5-695 


7 


• 4921 


32 


2.2498 


57 


4.0076 


82 


5-7653 


8 


.5624 


33 


2.3202 


58 


4-0779 


83 


S-8356 


9 


.6327 


34 


2-3905 


59 


4-I482 


84 


5-9059 


10 


.70309 


3S 


2.4608 


60 


4-218S 


85 


5-9762 


II 


■ 7734 


36 


2.S3II 


61 


4.2888 


86 


6.046s 


12 


.8437 


37 


2.6014 


62 


4.3591 


87 


6. I 168 


13 


.9140 


38 


2-6717 


63 


4-4294 


88 


6.1872 


-14 


■9843 


39 


2.7420 


64 


4.4997 


89 


6.2S7S 


IS 


1.0546 


40 


2.8123 


65 


4.5700 


90 


6.3278 


16 


I. 1249 


41 


2.8826 


66 


4.6404 


91 


6.3981 


17 


I. 1952 


42 


2.9529 


67 


4.7107 


92 


6.4684 


18 


1.2655 


43 


3-0232 


68 


4.781 


93 


6-5387 


19 


1-3358 


44 


3.0936 


69 


4.8513 


94 


6.609 


20 


I .4062 


45 


3-1639 


70 


4.9216 


95 


6-6793 


21 


1.4765 


46 


3.2342 


71 


4-9919 


96 


6.7496 


22 


1.5468 


47 


3.3045 


72 


5.0622 


97 


6.8199 


23 


I.6171 


48 


3-3748 


73 


5-1325 


98 


6.8902 


24 


1.6874 


49 


3.4451 


74 


S-2028 


99 


6 . 9606 


25 


1-7577 


50 


3-5154 


75 


5-2731 


100 


7.0309 



It is for this latter reason that the millimeter is universally used as 
a measure of length in machinery manufacture, this little unit being 
multiplied because the decimal division of larger units has been found 
impracticable. It is for the former reason that units has been both 
dropped from and added to the original list. 

Those who do not know the above facts do not know enough about 
the subject to make their opinions regarding the wisdom of the adop- 
tion of the system of the slightest value. 

The customary tables are very misleading. It was inevitable 
that a schedule of units based on a rigid relationship should contain 
many that are redundant and fail to contain others required by 
considerations of convenience. The result is that the tables contain 
many units that are not used and they omit others which necessity 
or convenience has brought into use, while, of those given, they fail 
entirely to indicate those that are used and those that are not. 

The accompanying conversion tables are but an illustration of the 



Kgs. 

per sq. 

cen. 


Lbs. per 
sq. in. 


Kgs. 

per sq. 

cen. 


Lbs. per 
sq. in. 


Kgs. 

per sq. 

cen. 


Lbs. per 
sq. in. 


Kgs. 

per sq. 

cen. 


Lbs. per 
sq. in. 


I 


14-223 


3-6 


51-203 


6.2 


88.183 


8.8 


125. 162 


l.I 


15-64S 


3.7 


52-625 


6.3 


89.605 


8.9 


126. 5§5 


1.2 


17.068 


3.8 


54-047 


6.4 


91.027 


9 


128 . 007 


1.3 


18.490 


3.9 


55-470 


6.S 


92.450 


9.1 


129,429 


1.4 


19.912 


4 


56.892 


6.6 


93.872 


9.2 


130,852 


i.S 


21.335 


4.1 


58-314 


6.7 


95.294 


9.3 


132.274 


1.6 


22.757 


4.2 


59-737 


6.8 


96.716 


9.4 


133 696 


1.7 


24.179 


4.3 


61.159 


6.9 


98.139 


9.5 


135-119 


1.8 


25.601 


4.4 


62-581 


7 


99.561 


9.6 


136.541 


1.9 


27.024 


4.5 


64.004 


7.1 


100.983 


9.7 


137,963 


2 


28.446 


4.6 


65.426 


7.2 


102.406 


9.8 


139 -38s 


2.1 


29.868 


4.7 


67- 848 


7.3 


103.828 


9.9 


140,808 


2.2 


31-291 


4.8 


68.270 


7.4 


105.250 


10 


142 ,230 


2.3 


32.713 


4.9 


69-693 


7.5 


106.673 


10. 1 


143,652 


2.4 


34-135 


S 


7I-I15 


7.6 


108.095 


10.2 


145,074 


2.S 


35.558 


5.1 


72.537 


7.7 


109.517 


10.3 


146.497 


2.6 


36.980 


5.2 


73-960 


7.8 


110.939 


10.4 


147.919 


2.7 


38.402 


5.3 


75-382 


7.9 


112.362 


10.5 


149.341 


2.8 


39.824 


5.4 


76- 804 


8 


113.784 


10.6 


150.764 


2.9 


41.247 


5.5 


78.227' 


8.1 


115.206 


10.7 


152.186 


3 


42.669 


5.6 


79.649 


8.2 


116.629 


10.8 


153.608 


3.1 


44.091 


5.7 


81.071 


8.3 


118. 051 


10.9 


155-030 


3.2 


45.514 


5.8 


82.493 


8.4 


119-473 


11 


156-453 


3.3 


46.936 


5.9 


83-916 


8-5 


120,896 


II. I 


157-875 


3.4 


48.358 


6 


85-338 


8.6 


122.318 


II. 2 


159-297 


3-5 


49-781 


6.1 


86.760 


8-7 


123-740 


11.3 


160.720 



confusion which the system has already introduced, and every exten- 
sion of it adds to this confusion, for the dream that it would supplant 
the old systems has proven as vain as the dream of the millenium. 
The whole movement for its origin and spread must be regarded as 
unfortunate and pernicious. 

The use of the accompanying tables of equivalents is best shown 
by an example: Required the metric equivalent of 38.5 ins. From 
the proper table we find: 

ins. mm. 

= 762.002 
= 203 . 200 
;= 12.700. 



30 



38-5 987.902 



500 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



American-Metric Conversion Factors for Compound Units of Value 
From The U. S. Bureau of Standards 



Francs 
per 


Dollars 
per avoir. 


Francs 
per 


Dollars 
per 


Francs 
per 


Dollars 

per U.S. 

liquid 

gal. 


Francs 
per hec- 


Dollars 
per U.S. 


kilogram 


pouud 


meter 


yard 


liter 


toliter 


bushel 


I = .088 


I = .176 


I = .731 


I = .068 


2 = .175 


2 = .353 


2 =1.461 


2 = .136 


3 = .263 


3 = .529 


3 =2.192 


3 = .204 


4 = .350 


4 = .705 


4 =2.922 


4 = .272 


s = .438 


5 = .882 


5 =3.653 


s = .340 


6 = .525 


6 =1.058 


6 =4.384 


6 = .408 


7 = .613 


7 =1.234 


7 =5.114 


7 = .476 


8 = .700 


8 =1.411 


8 =5.844 


8 = .544 


9 = .788 


9 =1.587 


9 =6.575 


9 ' = .612 


11.423 = 1 


5.667 = 1 


1.369 = 1 


14.703 = 1 


22.846 = 2 


11.334 = 2 


2.738 = 2 


29.407 = 2 


34.269 = 3 


17.000 = 3 


4.106 = 3 


44.110 = 3 


45.691=4 


22.667=4 


5.475 = 4 


58.813 = 4 


S7.I1S = S 


28.334 = 5 


6.844 = 5 


73.517 = 5 


68.537 = 6 


34.001 = 6 


8.213 = 6 


88.220 = 6 


79.960 = 7 


39.668 = 7 


9.581 = 7 


102.923=7 


91.383 = 8 


45-334 = 8 


10.950 = 8 


117.627 = 8 


102.80 


6 = 9 


51.001 =9 


12.319 = 9 


132.330 = 9 



Marks 

per 

kilogram 


Dollars 

per avoir. 

pound 


Marks 

per 
meter 


Dollars 
per yard 


Marks 
per liter 


Dollars 

per U. S. 

liquid 

gal. 


Marks 
per hec- 
toliter 


Dollars 

per U. S. 

bushel 


I = .108 


I = .218 


I = .901 


I = .084 


2 = .216 


2 = .435 


2 =1.802 


2 = .168 


3 = .324 


3 = .653 


3 =2.703 


3 = -252 


4 = -432 


4 = .871 


4 =3-604 


4 = -335 


5 = .540 


5 =1.088 


5 =4-505 


5 = .419 


6 = .648 


6 =1.306 


6 = 5 . 406 


6 = -503 


7 = .756 


7 =1.523 


7 =6.307 


7 = -587 


8 = .864 


8 =1.741 


8 =7.207 


8 = .671 


9 = .972 


9 =1.959 


9 =8.108 


9 = -755 


9.263 = 1 


4-595 = 1 


1.110 = 1 


11-923 = 1 


18.526 = 2 


9. 190 = 2 


2.220 = 2 


23.847 = 2 


27.789 = 3 


13-785 = 3 


3-330 = 3 


35-770 = 3 


37.052=4 


18.380 = 4 


4.440 = 4 


47-693=4 


46.316 = 5 


22.975 = 5 


5.550 = 5 


59-6l6 = s 


55.579 = 6 


27.570 = 6 


6.660 = 6 


71-540 = 6 


64.842 = 7 


32.165 = 7 


7.770 = 7 


83.463 = 7 


74-105 = 8 


36.760 = 8 


8.880 = 8 


95-386 = 8 


83.368 = 9 


41-355=9 


9.990 = 9 


107.310 = 9 



Electrical Horse-power 
Amperes 



Volts 


I 


10 


20 


30 


40 


50 


60 


70 


80 


90 


100 


no 


I 


.00134 


-0134 


.0268 


.0402 


-0536 


.0670 


.0804 


-0938 


.1072 


.1206 


• 1341 


• 1475 


5 


.00670 


.0670 


-I34I 


.2011 


.2681 


.3351 


.4022 


.4692 


-5362 


.6032 


-6703 


-7373 


10 


.01341 


.1314 


.2681 


.4022 


-5362 


.6703 


.8043 


.9383 


1.072 


1.206 


I -341 


1-475 


15 


.02011 


.2011 


.4022 


.6032 


-8043 


1.005 


1 .206 


1.408 


1.609 


1.8x0 


2. on 


2.212 


20 


.02681 


.2681 


.5362 


.8043 


1-072 


1.340 


1 .609 


1.877 


2.14s 


2.413 


2.681 


2.949 


25 


- 03351 


.3351 


.6703 


1. 005 


1. 341 


1.676 


2. on 


2.346 


2.681 


3.016 


3.351 


3.686 


30 


-04022 


.4022 


-8043 


1.206 


1.609 


2 .011 


2-413 


2.815 


3.217 


3.619 


4.022 


4-424 


35 


.04692 


-4692 


.9384 


1.408 


1.877 


2.346 


2.815 


3.284 


3-753 


4.223 


4.692 


5. 161 


40 


.05362 


-5362 


1 .072 


1 .609 


2.145 


2.681 


3.217 


3.753 


4.290 


4.826 


5. 362 


5.898 


45 


.06032 


.6032 


1 . 206 


1.810 


2.413 


3.016 


3-6l9 


4223 


4.826 


5-439 


6.032' 


6.63s 


50 


.06703 


-6703 


1.341 


2 .011 


2.681 


3.351 


4.022 


4.692 


5.362 


6.032 


6.703 


7-373 


75 


.10054 


I -005 


2. on 


3.016 


4.021 


5.027 


6.032 


7.037 


8.043 


9.048 


10.05 


II .06 


100 


-13405 


I -341 


2.681 


4. 022 


5.362 


6.703 


8.043 


9.384 


10. 72 


12.06 


13.41 


14.75 


500 


-67025 


6.703 


13.41 


20. II 


26.81 


33.51 


40. 22 


46.92 


53.62 


60.32 


67.03 


73.73 


1,000 


1-3405 


13-41 


26.81 


40. 22 


53.62 


67.03 


80.43 


93.84 


107. 2 


120.6 


134- I 


147-5 


5.000 


6.7025 


67.03 


134- I 


201. 1 


268.1 


335.1 


402. 2 


469.2 


536.2 


603.2 


670.3 


737-3 


10,000 


13-405 


134- I 


268.1 


402 . 2 


536,2 


670.3 


804-3 


938.3 


1072. 


1206 


134I 


1475 



British-Metric and Metric-British Conversion Factors for 
Work and Power 





Horse-power 


Horse-power 


Foot-pounds 


Kilogrammeters 




Metric to 


British to 


to kilogram- 


to 




British 


Metric 


meters 


foot-pounds 


I 


.986 


1. 014 


.1383 


7.2329 


2 


1-973 


2.028 


.2765 


14.4659 


3 


2.959 


3.042 


.4148 


21 .6988 


4 


3.945 


4.056 


-5530 


28.9317 


5 


4-932 


5.069 


.6913 


36. 1646 


6 


5-918 


6.083 


.8295 


43.3976 


7 


6.904 


7-097 


.9678 


50.6305 


8 


7-890 


8,111 


I . 1061 


57.8634 


9 


8.877 


9.125 


1.2443 


6s . 0963 



MATHEMATICAL TABLES 



The range of arithmetical tables may he greatly extended by an under- 
standing of a few principles. 

Areas of circles of fractional diameters may be obtained from 
tables of areas of circles whose diameters are whole numbers, by 
putting the diameter in the form of a decimal. For example, find 
the area of a circle of .97 in. diameter. The area of 97. is 7389. 
Point off twice as many decimal places as are in the diameter, and 
we have .7389 the area. Or take diameter .01 in. The area of 
I is .7854; add four decimals and we have .00007854 in. Or again, 
take diameter 34.7 ins. The area of 347 is 94,569, and pointing 
o£f two decimals gives 945.69 for the area belonging to diameter 34.7. 



It is often required to find the square or cube root of numbers 
larger than are given directly in the table. Suppose the square root 
of 12.850 is desired. Look in the column of squares for the nearest 
number, and it will be found that the square of 113, which is 12,769, 
is the nearest, but is too small, and the square root will be a fraction 
more than 113. To get one decimal place in the root will require 
two in the number; hence it would make a total of seven figures. 
Look down the column of squares to where there are seven figures 
and find the nearest to 12,850 (considering the two right-hand figures 
out of the seven as decimals), and the nearest number is 12,859.56, 
and the root is 113. 4. With the usual table going up to 1,600 this 



Table i. — Factors and Relations of t: 



3. 1416 divided by 



2 = 1 


.5708 


68= .0462 


561 = .0056 


3 = 1 


.0472 


77 = .0408 


616= .0051 


4 = 


.7854 


84 =.0374 


7 14 =.0044 


6 = 


.5236 


88=. 0357 


748= .0042 


7 = 


.4488 


102 = .0308 


924 =.0034 


8 = 


■ 3927 


119 =.0264 


952= .0033 


II = 


.2856 


132 = .0238 


1,122 = .0028 


12 = 


.2618 


136= .0231 


1,309= .0024 


14 = 


.2244 


154= .0204 


1,428 = .0022 


17 = 


.1848 


168= .0187 


1,496= .0021 


21 =^ 


.1496 


187 = .0168 


1,848= .0017 


22 = 


.1428 


204= .0154 


2,244= .0014 


24 = 


.1309 


231 = .0136 


2,618= .0012 


28 = 


. 1122 


238= .0132 


2,856= .0011 


33 = 


.0952 


264= .0119 


3,927 = .0008 


34 = 


.0924 


308= .0102 


4,488= .0007 


42 = 


.0748 


357 = .0088 


5,236= .0006 


44 = 


.0714 


374= .0084 


7,854= .0004 


Si = 


.0616 


408= .0077 


10,472= .0003 


56 = 


.0561 


462 = .0068 


15,708= .0002 


66 = 


.0476 


476= .0066 


5,280= .000595 



The reason for doubhng the number of decimal places of the diam- 
eter comes from- the fact that to find the area of a circle, the diam- 
eter is first multiplied by itself, or squared; hence there must be 
twice as many decimal places in the product, to conform to the rule 
for multiplication of decimal numbers. 

Sometimes it is required to find the area of a circle larger than is 
in the table. The range of the table may be doubled by taking the 
area for half of the desired diameter and multiplying it by 4. For 
example: Required the area for 996 diameter: half of this is 498, 
the area of which is 194,782, and this multiplied by 4 = 779,128, the 
area required. 

Referring to the table of squares, cubes and roots of numbers, 
which usually gives the squares and cubes of whole numbers only, 
it is sometimes required to know the square or cube of a fractional 
number. To find the square of .9 take the square of 9 and point 
off two decimal places, giving .81; or the cube, and point off three, 
giving .729, as all cubed numbers must have three times and all 
squared numbers two times as many decimals places as there are 
in the number to be cubed or squared. Finding the square or cube 
of a whole number and fraction is done the same way. To find the 
square of 71, -take the square of 725 = 525,625, and pointing off four 
decimals gives 52.5625; or the cube of ^\ =381.078125. 



.7854 divided by 



2=.3927 


34= .0231 


238= .0033 


3=.26l8 


42 = .0187 


357 = .0022 


6= .1309 


SI = .0IS4 


374= .0021 


7= .1122 


66= .0119 


462= .0017 


11= .0714 


77=. 0102 


561 = .0014 


14= .0561 


102= .0077 


714= .0011 


17= .0462 


119= .0066 


1,122= .0007 


21= .0374 


154= .0051 


1,309= .0006 


22 = .0357 


187 = .0042 


2,618= .0003 


33= .0238 


231 = .0034 


3,927 = .0002 



Log. ?r = . 4971499 

1 

Tt 
I 

\/7r = 1.7724538 

I 

\/2 = 1.4142136 
"^ =9.8696044 

rJ =31.0062767 



-= .3183099 

.1013212 
■7724538 
-= .5641896 



V'^ = 1-4645919 
7r\/2 = 4.4428829 



\/2 



= 2.2214415 
= .4501582 



^^ = 1.2533141 
•\/^= .7978846 



method is available only for finding the square root with one decimal, 
of numbers between 1600 and 25,600. 

By the use of the column of cubes of numbers in the same manner, 
the cube root with two decimal places may be found for numbers 
from 1,600 to 4,088; or the root with one decimal place for numbers 
from 4,096 up to 4,088,324. For example: Find the cube root of 
3,504; the nearest number in the column of cubes is 3,375, the cube 
of 15. As there are to be two decimals in the cube root there must 
be three times this = 6 added to the number of figures which makes 
10. Looking in the column of cubes we find 3,504.881359 (using 
the six right-hand figures as decimals), and the root is 15.19. 

Always be careful to keep in mind that in finding square roots 
there must be twice as many decimal places in the number as in 
the root, and in finding cube roots there must be three times the 
decimal places of the root. 

The value 0} tt to eight places of decimals in 3.14159265. The 
355 
ratio reduced to decimals is 3.1415929, which is far more nearly 

the true value than 3.1416 which is customarily used. Doubling 

710 
both numerator and dominator gives — 7 which may be found with- 
out estimation on the C- and Z)-scales of an ordinary slide rule. 



501 



502 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Ratios in vulgar fraction form are necessary when calculating gear 
trains for cutting diametral pitch worms and racks. Following are 
such values arranged in the order of accuracy: 



355 
"3" 

22 

7 " 



=3.1415929 



= 3.1429 



69 



= 3-1364 



12^ 

IS 



=3-1333 



For tabulated change gears for cutting diametral pitch worms 
see Cutting Diametral Pitch Worms. 

. The value 3.1416 has many exact factors, as it is the product of 
2X3X4X7X.11X.17. Table i gives various factors and other 
relations of ?r. 



Table 2. — Logarithms 

The supplementary table at the right gives proportional parts without calculation. Thus to find log 2985, opposite 29 and under 8 read .4742 and in 
the same line of the supplementry table under 5 read 7 which added to .4742 gives .4749, the log required. 








I 


2 


3 


4 


S 


6 


7 


8 


9 


I 


2 


3 


4 


5 


6 


7 


8 


9 


10 


0000 


0043 


0086 


0128 


0170 


0212 


0253 


0294 


0334 


0374 


4 


8 


12 


17 


21 


25 


29 


33 


37 


II 


0414 


0453 


0492 


0S3I 


0569 


0607 


0645 


0682 


0719 


0755 


4 


8 


II 


15 


19 


23 


26 


30 


34 


12 


0792 


0828 


0864 


0899 


0934 


0969 


1004 


1038 


1072 


II06 


3 


7 


10 


14 


17 


21 


24 


28 


31 


13 


II39 


1173 


1206 


1239 


1271 


1303 


I33S 


1367 


1399 


1430 


3 


6 


10 


13 


16 


19 


23 


26 


29 


14 


I46I 


1492 


1523 


1553 


1584 


1614 


1644 


1673 


1703 


1732 


3 


6 


9 


12 


15 


18 


21 


24 


27 


IS 


I76I 


1790 


1818 


1847 


187s 


1903 


1931 


1959 


1987 


2014 


3 


6 


8 


II 


14 


17 


20 


22 


25 


16 


2041 


2068 


209S 


2122 


2148 


2I7S 


2201 


2227 


2253 


2279 


3 


5 


8 


II 


13 


16 


18 


21 


24 


17 


2304 


2330 


2355 


2380 


240s 


2430 


2455 


2480 


2504 


2529 


2 


5 


7 


10 


12 


15 


17 


20 


22 


18 


2553 


2577 


2601 


2625 


2648 


2672 


269s 


2718 


2742 


2765 


2 


5 


7 


9 


12 


14 


16 


19 


21 


19 


2788 


2810 


2833 


2856 


2878 


2900 


2923 


2945 


2967 


2989 


2 


4 


7 


9 


II 


13 


16 


18 


20 


20 


3010 


3032 


3054 


307s 


3096 


3118 


3139 


3160 


3I8I 


3201 


2 


4 


6 


8 


II 


13 


IS 


17 


19 


21 


3222 


3243 


3263 


3284 


3304 


3324 


3345 


3365 


3385 


3404 


2 


4 


6 


8 


10 


12 


14 


16 


18 


22 


3424 


3444 


3464 


3483 


3502 


3522 


3541 


3560 


3579 


3598 


2 


4 


6 


8 


10 


12 


14 


15 


17 


23 


3617 


3636 


365s 


3674 


3692 


3711 


3729 


3747 


3766 


3784 


2 


4 


6 


7 


9 


II 


13 


15 


17 


24 


• 3802 


3820 


3838 


3856 


3874 


3892 


3909 


3927 


3 945 


3962 


2 


4 


5 


7 


9 


II 


12 


14 


16 


25 


3979 


3997 


4014 


4031 


4048 


4065 


4082 


4099 


4116 


4133 


2 


3 


5 


7 


9 


10 


12 


14 


15 


26 


4150 


4166 


4183 


4200 


4216 


4232 


4249 


4265 


4281 


4298 


2 


3 


5 


7 


8 


10 


II 


13 


15 


27 


4314 


4330 


4346 


4362 


4378 


4393 


4409 


442s 


4440 


4456 


2 


3 


5 


6 


8 


9 


II 


13 


14 


28 


4472 


4487 


4502 


4518 


4533 


4548 


4564 


4579 


4594 


4609 




3 


5 


6 


8 


9 


1 I 


12 


14 


29 


4624 


4639 


4654 


4669 


4683 


4698 


4713 


4728 


4742 


4757 




3 


4 


6 


7 


9 


10 


12 


13 


30 


4771 


4786 


4800 


4814 


4829 


4843 


4857 


4871 


4886 


4900 




3 


4 


6 


7 


9 


10 


II 


13 


31 


4914 


4928 


4942 


4955 


4969 


4983 


4997 


SOU 


5024 


5038 




3 


4 


6 


7 


8 


10 


II 


12 


32 


5051 


5065 


S079 


5092 


5105 


5119 


S132 


5145 


5159 


5172 




3 


4 


5 


7 


8 


9 


I I 


12 


33 


5185 


5198 


S2II 


5224 


5237 


52SO 


5263 


5276 


5289 


5302 




3 


4 


5 


6 


8 


9 


10 


12 


34 


531S 


5328 


S340 


5353 


5366 


S378 


5391 


5403 


5416 


5428 




3 


4 


5 


6 


8 


9 


10 


I I 


35 


5441 


5453 


546s 


5478 


5490 


5502 


5514 


5527 


5539 


5551 




2 


4 


5 


6 


7 


9 


10 


II 


36 


5563 


5575 


5587 


5599 


5611 


5623 


5635 


5647 


5658 


5670 




2 


4 


5 


6 


7 


8 


10 


11 


37 


5682 


5694 


5705 


5717 


5729 


5740 


5752 


5763 


5775 


5786 




2 


3 


5 


6 


7 


8 


9 


10 


38 


5798 


5809 


5821 


5832 


5843 


585s 


5866 


5877 


5888 


5899 




2 


3 


S 


6 


7 


8 


9 


10 


39 


S9II 


5922 


5933 


5944 


5955 


5966 


5977 


5988 


5999 


6010 




2 


3 


4 


5 


7 


8 


9 


10 


40 


6021 


6031 


6042 


6053 


6064 


6075 


6085 


6096 


6107 


6117 




2 


3 


4 


5 


6 


8 


9 


10 


41 


6128 


6138 


6149 


6160 


6170 


6180 


6191 


6201 


6212 


6222 




2 


3 


4 


5 


6 


7 


8 


9 


42 


6232 


6243 


6253 


6263 


6274 


6284 


6294 


6304 


6314 


6325 




2 


3 


4 


5 


6 


7 


8 


9 


43 


633s 


634s 


635s 


6365 


637s 


6385 


6395 


6405 


6415 


6425 




2 


3 


4 


5 


6 


7 


8 


9 


44 


643s 


6444 


6454 


6464 


6474 


6484 


6493 


6503 


6513 


6522 




2 


3 


4 


5 


6 


7 


8 


9 


45 


6532 


6542 


6551 


6561 


6571 


6580 


6390 


6599 


6609 


6618 




2 


3 


4 


5 


6 


7 


8 


9 


46 


6628 


6637 


6646 


6656 


6665 


6675 


6684 


6693 


6702 


6712 




2 


3 


4 


5 


6 


7 


7 


8 


47 


6721 


6730 


6739 


6749 


6758 


6767 


6776 


6785 


6794 


6803 




2 


3 


4 


S 


5 


6 


7 


8 


48 


6812 


6821 


6830 


6839 


6848 


6857 


6866 


687s 


6884 


6893 




2 


3 


4 


4 


5 


6 


7 


8 


49 


6902 


6911 


6920 


6928 


6937 


6946 


695s 


6964 


6972 


6981 




2 


3 


4 


4 


5 


6 


7 


8 


50 


6990 


6998 


7007 


7016 


7024 


7033 


7042 


7050 


7059 


7067 




2 


3 


3 


4 


5 


6 


7 


8 


51 


7076 


7084 


7093 


7101 


7110 


7118 


7126 


7135 


7143 


7152 




2 


3 


3 


4 


5 


6 


7 


8 


52 


7160 


7168 


7177 


718S 


7193 


7202 


7210 


7218 


7226 


7235 




2 


2 


3 


4 


5 


6 


7 


7 


S3 


7243 


7251 


7259 


7267 


727s 


7284 


7292 


7300 


7308 


7316 




2 


2 


3 


4 


5 


6 


6 


7 


54 


7324 


7332 


7340 


7348 


7356 


7364 


7372 


7380 


7388 


7396 




2 


2 


3 


4 


5 


6 


6 


7 


55 


7404 


7412 


7419 


7427 


7435 


7443 


7451 


7459 


7466 


7474 




2 


2 


3 


4 


5 


5 


6 


7 


56 


7482 


7490 


7497 


750s 


7513 


7520 


7528 


7536 


7S43 


7551 




2 


2 


3 


4 


5 


5 


6 


7 


S7 


7559 


7566 


7574 


7582 


7589 


7597 


7604 


7612 


7619 


7627 




2 


2 


3 


4 


5 


5 


6 


7 


58 


7634 


7642 


7649 


7657 


7664 


7672 


7679 


7686 


7694 


7701 






2 


3 


4 


4 


5 


6 


7 


59 


7709 


7716 


7723 


7731 


7738 


7745 


7752 


7760 


7767 


7774 






2 


3 


4 


4 


5 


6 


7 


60 


7782 


7789 


7796 


7803 


7810 


7818 


782s 


7832 


7839 


7846 






2 


3 


4 


4 


5 


6 


6 


61 


7853 


7860 


7868 


787s 


7882 


7889 


7896 


7903 


7910 


7917 






2 


3 


4 


4 


5 


6 


6 


62 


7924 


7931 


7938 


7945 


7952 


7959 


7966 


7973 


7980 


7987 






2 


3 


3 


4 


5 


6 


6 


63 


7993 


8000 


8O07 


8014 


8021 


8028 


8035 


8041 


8048 


8055 






2 


3 


3 


4 


5 


5 


6 


64 


8062 


8069 


807S 


8082 


8089 


8096 


8102 


8109 


8116 


8122 






2 


3 


3 


4 


S 


5 


6 


65 


8129 


8136 


8142 


8149 


8156 


8162 


8169 


8176 


8182 


8189 






2 


3 


3 


4 


5 


5 


6 



MATHEMATICAL TABLES 



503 



Table 2. — Logarithms — (Continued) 



1 







I 


2 


3 


4 


5 


6 


7 


8 


9 


I 2 3 


4 


5 


6 


7 


8 


9 


66 


8195 


8202 


8209 


8215 


8222 


8228 


823s 


8241 


8248 


82S4 


112 


3 


3 


4 


5 


5 


6 


jL 


67 


8261 


8267 


8274 


8280 


8287 


8293 


8299 


8306 


8312 


8319 


112 


3 


3 


4 


5 


5 


6 


m^ 


68 


832s 


8331 


8338 


8344 


8351 


8357 


8363 


8370 


8376 


8382 


112 


3 


3 


4 


4 


5 


6 


^B 


69 


8388 


8395 


8401 


8407 


8414 


8420 


8426 


8432 


8439 


844s 


112 


2 


3 


4 


4 


S 


6 


H 


70 


8451 


8457 


8463 


8470 


8476 


8482 


8488 


8494 


8500 


8506 


112 


2 


3 


4 


4 


5 


6 


H 


71 


8513 


8519 


8525 


8531 


8537 


8543 


8S49 


8S5S 


8s6i 


8567 


112 


2 


3 


4 


4 


5 


5 


^F 


72 


8573 


8579 


858s 


8591 


8597 


8603 


8609 


861S 


8621 


8627 


112 


2 


3 


4 


4 


5 


5 


y 


73 


8633 


8639 


8645 


86si 


8657 


8663 


8669 


8675 


8681 


8686 


112 


2 


3 


4 


4 


5 


5 




74 


8692. 


8698 


8704 


8710 


8716 


8722 


8727 


8733 


8739 


8745 


112 


2 


3 


4 


4 


5 


5 




75 


8751 


8756 


8762 


8768 


8774 


8779 


8785 


8791 


8797 


8802 


112 


2 


3 


3 


4 


5 


5 


■ 


76 


8808 


8814 


8820 


8825 


8831 


8837 


8842 


8848 


8854 


8859 


112 


2 


3 


3 


4 


5 


5 


■ 


77 


8865 


8871 


8876 


8882 


8887 


8893 


8899 


8904 


8910 


8915 


112 


2 


3 


3 


4 


4 


5 


■ 


78 


8921 


8927 


8932 


8938 


8943 


8949 


8954 


8960 


8965 


8971 


112 


2 


3 


3 


4 


4 


5 


■ 


79 


8976 


8982 


8987 


8993 


8998 


9004 


9009 


9015 


9020 


9025 


112 


2 


3 


3 


4 


4 


5 




80 


9031 


9036 


9042 


9047 


9053 


9058 


9063 


9069 


9074 


9079 


112 


2 


3 


3 


4 


4 


5 


*s 


81 


908s 


9090 


9096 


9101 


9106 


9112 


9117 


9122 


9128 


9133 


112 


2 


3 


3 


4 


4 


5 


1 


82 


9138 


9143 


9149 


9154 


9159 


9165 


9170 


9175 


9180 


9186 


I I 2 


2 


3 


3 


4 


4 


5 


83 


9I9I 


9196 


9201 


9206 


9212 


9217 


9222 


9227 


9232 


9238 


112 


2 


3 


3 


4 


4 


5 




84 


9243 


9248 


9253 


9258 


9263 


9269 


9274 


9279 


9284 


9289 


I I 2 


2 


3 


3 


4 


4 


S 




85 


9294 


9299 


9304 


9309 


9315 


9320 


9325 


9330 


9335 


9340 


112 


2 


3 


3 


4 


4 


5 




86 


9345 


9350 


9355 


9360 


9365 


9370 


9375 


9380 


9385 


9390 


112 


2 


3 


3 


4 


4 


5 




87 


9395 


9400 


9405 


9410 


9415 


9420 


942s 


9430 


9435 


9440 


Oil 


2 


2 


3 


3 


4 


4 




88 


9445 


9450 


9455 


9460 


9465 


9469 


9474 


9479 


9484 


9489 


Oil 


2 


2 


3 


3 


4 


4 




89 


9494 


9499 


9504 


9509 


9513 


9Si8 


9523 


9528 


9533 


9538 


Oil 


2 


2 


3 


3 


4 


4 




90 


9542 


9547 


9552 


9557 


9562 


9566 


9571 


9576 


9581 


9586 


Oil 


2 


2 


3 


3 


4 


4 




91 


9590 


9595 


9600 


960s 


9609 


9614 


9619 


9624 


9628 


9633 


Oil 


2 


2 


3 


3 


4 


4 




92 


9638 


9643 


9647 


9652 


9657 


9661 


9666 


9671 


967s 


9680 


Oil 


2 


2 


3 


3 


4 


4 




93 


9685 


9689 


9694 


9699 


9703 


9708 


9713 


7717 


9722 


9727 


Oil 


2 


2 


3 


3 


4 


4 




94 


9731 


9736 


9741 


9745 


9750 


9754 


9759 


9763 


9768 


9773 


Oil 


2 


2 


3 


3 


4 


4 




95 


9777 


9782 


9786 


9791 


9795 


9800 


9805 


9809 


9814 


9818 


Oil 


2 


2 


3 


3 


4 


4 




96 


9823 


9827 


9832 


9836 


9841 


9845 


9850 


9854 


9859 


9863 


Oil 


2 


2 


3 


3 


4 


4 




97 


9868 


9872 


9877 


9881 


9886 


9890 


9894 


9899 


9903 


9908 


Oil 


2 


2 


3 


3 


4 


4 




98 


9912 


9917 


9921 


9926 


9930 


9934 


9939 


9943 


9948 


9952 


I I 


2 


2 


3 


3 


4 


4 




99 


9956 


9961 


9965 


9969 


9974 


9978 


9983 


9987 


9991 


9996 


Oil 


2 


2 


3 


3 


3 


4 



Table 3. — Antilogarithms 

The supplementary table at the right is used in the same manner as with the previous table. Thus to find the natural number corresponding to the 
logarithm 4749, opposite 47 and under 4 read 2979 and in the same line under 9 read 6 which added to 2979 gives 2985, the natural number required. 








I 


2 


3 


4 


5 


6 


7 


8 


9 


I 


2 


3 . 


4 




6 


7 


8 


9 


.00 


1000 


1002 


1005 


1007 


1009 


1012 


1014 


1016 


1019 


1021 










I 




I 


2 


2 


2 


.01 


1023 


1026 


1028 


1030 


1033 


1035 


1038 


1040 


1042 


1045 










I 




I 


2 


2 


2 


.02 


1047 


1050 


1052 


1054 


1057 


1059 


1062 


1064 


1067 


1069 










I 




I 


2 


2 


2 


.03 


1072 


1074 


1076 


1079 


1081 


1084 


1086 


1089 


logi 


1094 










I 




I 


2 


2 


2 


.04 


1096 


1099 


1102 


1 104 


I107 


II09 


1112 


II14 


1117 


II19 





I 




I 




2 


2 


2 


2 


•OS 


1122 


1125 


1127 


1130 


I132 


1135 


1138 


1 140 


1143 


I146 





I 




1 




2 


2 


2 


2 


.06' 


1148 


1151 


1153 


1156 


1159 


I161 


1164 


I167 


1169 


I172 





I 




I 




2 


2 


2 


2 


.07 


1175 


1178 


1180 


1183 


I186 


I189 


1191 


I194 


1197 


I199 





I 




I 




2 


2 


2 


2 


.08 


1202 


1205 


1208 


1211 


1213 


1216 


1219 


1222 


1225 


1227 





I 




I 




2 


2 


2 


3 


.09 


1230 


1233 


1236 


1239 


1242 


124s 


1247 


1250 


1253 


1256 





I 




I 


I 


2 


2 


2 


3 


. 10 


1259 


1262 


126s 


1268 


1271 


1274 


1276 


1279 


1282 


1285 





I 




I 


I 


2 


2 


2 


3 


. II 


1288 


1291 


1294 


1297 


1300 


1303 


1306 


1309 


I3I2 


1315 





I 




1 


2 


2 


2 


2 


3 


. 12 


1318 


1321 


1324 


1327 


1330 


1334 


1337 


1340 


1343 


1346 









1 


2 


2 


2 


2 


3 


.13 


1349 


1352 


1355 


1358 


1361 


1365 


1368 


1371 


1374 


1377 









I 


2 


2 


2 


3 


3 


■ 14 


1380 


1384 


1387 


1390 


1393 


1396 


1400 


1403 


1406 


1409 









I 


2 


2 


2 


3 


3 


.15 


1413 


1416 


1419 


1422 


1426 


1429 


1432 


1435 


1439 


1442 









I 


2 


2 


2 


3 


3 


.16 


144s 


1449 


1452 


1455 


1459 


1462 


1466 


1469 


1472 


1476 









I 


2 


2 


2 


3 


3 


.17 


1479 


1483 


i486 


1489 


1493 


1496 


1500 


1503 


1507 


1510 









I 


2 


2 


2 


3 


3 


.18 


1514 


1517 


1521 


1524 


1528 


1531 


1535 


1538 


1542 


1545 









I 


2 


2 


2 


3 


3 


• 19 


1549 


1552 


1556 


1560 


1563 


1567 


1570 


1574 


1578 


1581 









I 


2 


2 


3 


3 


3 


.20 


1585 


1589 


1592 


1596 


1600 


1603 


1607 


1611 


I6I4 


1618 









I 


2 


2 


3 


3 


3 


.21 


1622 


1626 


1629 


1633 


1637 


1641 


1644 


1648 


1652 


1656 









2 


2 


2 


3 


3 


3 


.22 


1660 


1663 


1667 


1671 


1675 


1679 


1683 


1687 


1690 


1694 









2 


2 


2 


3 


3 


3 


.23 


1698 


1702 


1706 


1710 


1714 


1718 


1722 


1726 


1730 


1734 









2 


2 


2 


3 


3 


4 


.24 


1738 


1742 


1746 


1750 


1754 


1758 


1762 


1766 


1770 


1774 









3 


2 


2 


3 


3 


4 


• 25 


1778 


1782 


1786 


1791 


1795 


1799 


1803 


1807 


I8II 


1816 









2 


2 


2 


3 


3 


4 


.26 


1820 


1824 


1828 


1832 


.1837 


1841 


1845 


1849 


1854 


1858 









2 


2 


3 


3 


3 


4 


.27 


1826 


1866 


1871 


1875 


1879 


1884 


1888 


1892 


1897 


1901 





- 1 




2 


2 


3 


3 


3 


4 


.28 


190S 


1910 


1914 


1919 


1923 


1928 


1932 


1936 


I94I 


1945 









2 


2 


3 


3 


4 


4 


• 29 


1950 


1954 


1959 


1963 


1968 


1972 


1977 


1982 


1986 


1991 









2 


2 


3 


3 


4 


4 



504 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 















Table 3.— Antilogaeithms- 


-{Continued) 


















1 


I 


2 


3 


4 


5 


6 


7 


8 


9 


I 2 


3 


4 


5 


6 


7 


8 


9 


.30 


199s 


2000 


2004 


2009 


2014 


2018 


2023 


2028 


2032 


2037 


I 


I 


2 


2 


3 


3 


4 


4 




31 


2042 


2046 


2051 


2056 


2061 


2065 


2070 


2075 


2080 


2084 


I 


I 


2 


2 


3 


3 


4 


4 




32 


2089 


2094 


2099 


2104 


2109 


2113 


2118 


2123 


2128 


2133 


I 


I 


2 


2 


3 


3 


4 


4 




33 


2138 


2143 


2148 


2153 


2158 


2163 


2168 


2173 


2178 


2:83 


I 


I 


2 


2 


3 


3 


4 


4 




34 


2188 


2193 


2198 


2203 


2208 


2213 


2218 


2223 


2228 


2234 


I I 


2 


2 


3 


3 


4 


4 


5 




3S 


2239 


2244 


2249 


2254 


2259 


226s 


2270 


2275 


2280 


2286 




2 


2 


3 


3 


4 


4 


'"s 




36 


2291 


2296 


2301 


2307 


2312 


2317 


2323 


2328 


2333 


2339 




2 


2 


3 


3 


4 


4 


5 




37 


2344 


2350 


2355 


2360 


2366 


2371 


2377 


2382 


2388 


2393 




2 


2 


3 


3 


4 


4 


5 




38 


2399 


2404 


2410 


2415 


2421 


2427 


2432 


2438 


2443 


2449 




2 


2 


3 


3 


4 


4 


5 




39 


2455 


2460 


2466 


2472 


2477 


2483 


2489 


2495 


2500 


2506 




2 


2 


3 


3 


4 


5 


5 




40 


2512 


2Si8 


2523 


2529 


2535 


2541 


2S47 


2553 


2559 


2564 




2 


2 


3 


4 


4 


5 


5 




41 


2570 


2576 


2582 


2588 


2594 


2600 


2606 


2612 


2618 


2624 




2 


2 


3 


4 


4 


5 


5 




42 


2630 


2636 


2642 


2649 


2655 


2661 


2667 


2673 


2679 


2685 




2 


2 


3 


4 


4 


5 


6 




43 


2692 


2698 


2704 


2710 


2716 


2723 


2729 


2735 


2742 


2748 




2 


3 


3 


4 


4 


5 


6 




44 


2754 


2761 


2767 


2773 


2780 


2786 


2793 


2799 


2805 


2812 




2 


3 


3 


4 


4 


5 


6 




45 


2818 


282s 


2831 


2838 


2844 


2851 


2858 


2864 


2871 


2877 




2 


3 


3 


4 


5 


5 


6 




46 


2884 


2891 


2897 


2904 


2911 


2917 


2924 


2931 


2938 


2944 




2 


3 


3 


4 


5 


5 


6 




47 


2951 


• 2958 


296s 


2972 


2979 


2985 


2992 


2999 


3006 


3013 




2 


3 


3 


4 


5 


5 


6 




48 


3020 


3027 


3034 


3041 


3048 


3055 


3062 


3069 


3076 


3083 




2 


3 


4 


4 


5 


6 


6 




49 


3090 


3097 


31OS 


3112 


3119 


3126 


3133 


3141 


3148 


3155 




2 


3 


4 


4 


5 


6 


6 




50 


3162 


3170 


3177 


3184 


3192 


3199 


3206 


3214 


3221 


3228 




2 


3 


4 


4 


5 


6 


7 




51 


3236 


3243 


3251 


3258 


3266 


3273 


3281 


3289 


3296 


3304 


I 2 


2 


3 


4 


5 


5 


6 


7 




52 


3311 


3319 


3327 


3334 


3342 


3350 


3357 


336s 


3373 


3381 


I 2 


2 


3 


4 


5 


S 


6 


7 




53 


3388 


3396 


3404 


3412 


3420 


3428 


3436 


3443 


3451 


3459 


I 2 


2 


3 


4 


5 


6 


6 


7 




54 


3467 


3475 


3483 


3491 


3499 


3508 


3516 


3524 


3532 


3540 


I 2 


2 


3 


4 


5 


6 


6 


7 




55 


3548 


3556 


356s 


3573 


3581 


3589 


3597 


3606 


3614 


3622 


I 2 


2 


3 


4 


5 


6 


7 


7 




56 


3631 


3639 


3648 


3656 


3664 


3673 


3681 


3690 


3698 


3707 


I 2 


3 


3 


4 


5 


6 


7 


8 




57 


3715 


3724 


3733 


3741 


3750 


3758 


3767 


3776 


3784 


3793 


I 2 


3 


3 


4 


5 


6 


7 


8 




58 


3802 


3811 


3819 


3828 


3837 


3846 


385s 


3864 


3873 


3882 


I 2 


3 


4 


4 


5 


6 


7 


8 




59 


3890 


3899 


3908 


3917 


3926 


3936 


394S 


3954 


3963 


3972 


I 2 


3 


4 


5 


5 


6 


7 


8 




60 


3981 


3990 


3999 


4009 


4018 


4027 


4036 


4046 


4055 


4064 


I 2 


3 


4 


5 


6 


6 


7 


8 




61 


4074 


4083 


4093 


4102 


4111 


4121 


4130 


4140 


4150 


4159 


I 2 


3 


4 


5 


6 


7 


8 


9 




62 


4169 


4178 


4188 


4198 


4207 


4217 


4227 


4236 


4246 


4256 


I 2 


3 


4 


5 


6 


7 


8 


9 




63 


4266 


4276 


428s 


4295 


4305 


4315 


432 s 


4335 


4345 


4355 


I 2 


3 


4 


5 


6 


7 


8 


9 




64 


4365 


4375 


438s 


4395 


4406 


4416 


4426 


4436 


4446 


4457 


I 2 


3 


4 


5 


6 


7 


8 


9 




65 


4467 


4477 


4487 


4498 


4508 


4519 


4529 


4539 


4550 


4560 


I 2 


3 


4 


5 


6 


7 


8 


9 




66 


4571 


4581 


4592 


4603 


4613 


4624 


4634 


4645 


4656 


4667 


I 2 


3 


4 


5 


6 


7 


9 


10 




67 


4677 


4688 


4699 


4710 


4721 


4732 


4742 


4753 


4764 


4775 


1 2 


3 


4 


5 


7 


• 8 


9 


10 




68 


4786 


4797 


4808 


4819 


4831 


4842 


4853 


4864 


487s 


4887 


I 2 


2 


4 


6 


7 


8 


9 


10 




69 


4898 


4909 


4920 


4932 


4943 


495S 


4966 


4977 


4989 


5000 


I 2 


3 


5 


6 


7 


8 


9 


10 




70 


5012 


5023 


503s 


5047 


S058 


5070 


5082 


SO93 


Sios 


SI17 


I 2 


4 


5 


6 


7 


8 


9 


II 




71 


5129 


5140 


S152 


5164 


5176 


5188 


5200 


5212 


5224 


5236 


I 2 


4 


5 


6 


7 


8 


10 


II 




72 


5248 


5260 


5272 


5284 


5297 


5309 


5321 


5333 


5346 


5358 


I 2 


4 


5 


6 


7 


9 


10 


II 




73 


5370 


5383 


5395 


5408 


5420 


5433 


5445 


5458 


5470 


5483 


I 3 


4 


5 


6 


8 


9 


10 


II 




74 


5495 


5508 


SS2I 


5534 


5546 


5559 


SS72 


558s 


5598 


5610 


I 3 


4 


5 


6 


8 


9 


10 


12 




75 


5623 


5636 


5649 


5662 


5675 


5689 


5702 


5715 


5728 


5741 


I 3 


4 


5 


7 


8 


9 


10 


12 




76 


5754 


5768 


5781 


5794 


5808 


S821 


5834 


5848 


5861 


5875 


I 3 


4 


5 


7 


8 


9 


II 


12 




77 


5888 


5902 


5916 


5929 


5943 


5957 


S970 


5984 


5998 


6012 


I 3 


4 


5 


7 


8 


10 


II 


12 




78 


6026 


6039 


60S3 


6067 


6081 


6095 


6109 


6124 


6138 


6152 


I 3 


4 


6 


7 


8 


10 


II 


13 




79 


6166 


6180 


6194 


6209 


6223 


6237 


6252 


6266 


6281 


629s 


I 3 


4 


6 


7 


9 


10 


II 


13 




80 


6310 


6324 


6339 


6353 


6368 


6383 


6397 


6412 


6427 


6442 


I 3 


4 


6 


7 


9 


10 


12 


13 




81 


6457 


6471 


6486 


6501 


6516 


6531 


6546 


6561 


6577 


6592 


2 3 


5 


6 


8 


9 


II 


12 


14 




82 


6607 


6622 


6637 


6653 


6668 


6683 


6699 


6714 


6730 


674s 


2 3 


5 


6 


8 


9 


II 


12 


14 




83 


6761 


6776 


6792 


6808 


6823 


6839 


6855 


6871 


6887 


6902 


2 3 


5 


6 


8 


9 


II 


13 


14 




84 


6918 


6934 


6950 


6966 


6982 


6998 


701S 


7031 


7047 


7063 


2 3 


5 


6 


8 


10 


II 


13 


15 




85 


7079 


7096 


7112 


7129 


7145 


7161 


7178 


7194 


7211 


7228 


2 3 


5 


7 


8 


10 


12 


13 


15 




86 


7244 


7261 


7278 


7295 


7311 


7328 


7345 


7362 


7379 


7396 


2 3 


5 


7 


8 


10 


12 


13 


IS 




87 


7413 


7430 


7447 


7464 


7482 


7499 


7516 


7534 


7551 


7568 


2 3 


5 


7 


9 


10 


12 


14 


16 




88 


7586 


7603 


7621 


7638 


7656 


7674 


7691 


7709 


7727 


7745 


2 4 


5 


7 


9 


II 


12 


14 


16 




89 


7762 


7780 


7798 


7816 


7834 


7852 


7870 


7889 


7907 


7925 


2 4 


5 


7 


9 


II 


13 


14 


16 




90 


7943 


7962 


7980 


7998 


8017 


803s 


80S4 


8072 


8091 


8110 


2 4 


6 


7 


9 


II 


13 


IS 


17 




91 


8128 


8147 


8166 


818S 


8204 


8222 


8241 


8260 


8279 


8299 


2 4 


6 


8 


9 


1 1 


13 


15 


17 




92 


8318 


8337 


8356 


8375 


8395 


8414 


8433 


8453 


8472 


8492 


2 4 


6 


8 


10 


12 


14 


15 


17 




93 


8511 


8531 


8551 


8570 


8590 


8610 


8630 


8650 


8670 


8690 


2 4 


6 


8 


10 


12 


14 


16 


18 




94 


8710 


8730 


8750 


8770 


8790 


8810 


8831 


8851 


8872 


8892 


2 4 


6 


8 


10 


12 


14 


16 


18 




.95 


8913 


8933 


8954 


8974 


8995 


9016 


9036 


9057 


9078 


9099 


2 4 


6 


8 


10 


12 


15 


17 


19 




.96 


9120 


9141 


9162 


9183 


9204 


9226 


9247 


9268 


9290 


9311 


2 4 


6 


8 


II 


13 


15 


17 


19 




.97 


9333 


9354 


9376 


9397 


9419 


9441 


9462 


9484 


9506 


9528 


2 4 


7 


9 


II 


13 


15 


17 


20 




.98 


9550 


9572 


9594 


9616 


9638 


9661 


9683 


9705 


9727 


9750 


2 4 


7 


9 


II 


13 


16 


18 


20 




.99 


9772 


9795 


9817 


9840 


9863 


9886 


9908 


9931 


9954 


9977 


2 5 


7 


9 


II 


14 


16 


18 


20 



MATHEMATICAL TABLES 



505 



, 






t^ 


^ 


f^ 


^ 


O 


Oi 


o 


t* fO o 


>r> 


m 


^ 


lO 


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t^ 


lO 


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vO 


CO 


Ov 


00 


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01 


vn 


o 


o 


o 


0^ \0 


« 


-o 


0\ 


„ 


01 


w 


a o 


N 


1^ 


H Tt 


m 


o 












no 


vn 




o 






o^ 


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o 


^ 


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■n 


o 


ro 


VO 


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01 


lo 


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o 


01 


lO 


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o 


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Oi 


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w 


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M 




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ro 










" 


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H 


H 


H 


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01 


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01 


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N 


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CI 


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J3 




N 


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K. 


,H 


M 


H 


o 




o 


IH 


w 


m 


^ 


in VO 


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H 


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0\ 


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Ol 


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« 


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Tf 


m 






M 


" 


M 


M 


^ 


M 


H 


H H 


M 


M 


N 


« 


« 


M 


N 


« 


N 


« 


N 


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ro 


m 


« 


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to 


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to 


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in 


in 


m 


m 



506 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Relations of the Trigonometric Functions 






^ Cotangent 




1 




ly^ \ \ \ 


y 




* Cosine > 









sin « = - = 

c cosec 

h 1 

cos q: = - = 

c sec 

o 1 
tan a=T= — i 
cot 

& 1 
cot a = - = r — 

a tan 



Trigonometric Functions and Formulas 

1 tan cos 

sec ~cot 
cot 



= cos X tan 



=vr^ 



cos-" 



sec a = -r = 



cos 

c 1 
cosec q; = - = ^— 
a sin 



= sin X cot =« 
= sin X sec = 
= cos X cosec = 
= tan X cosec 



sm 



cosec tan ^ 



= cot X sec = 



sec 

cosec 

cosec 

sec 

cosec 

cot 

sec 

tan 



sin 
cos' 



• -s/sec^— I. 



cos J 

=^ = Vcosec2— 1. 

tan , 

=3- = Vtan^+l. 

cot / 

=-— = Vcot2+l. 
cos ' 



Table 5. — Natural Trigonometric Functions 











1° 1 


2° 1 


3 









4 





5 





6° 


7° 




/ 


Tan. 


Co-tan. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


> 


/ 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


' 





.00000 


Infinite. 


.01746 


57.2900 


.03492 


28.6363 


.05241 


19.0811 


60 





.06993 


I4^3007 


.08749 


11-4301 


.10510 


9-31436 


.12278 


8.1443s 


60 


I 


.00029 


3437-75° 


•01775 


56-3506 


.03521 


28.3994 


•05270 


18.9755 


59 


I 


.07022 


I4^24ii 


.08778 


11.3919 


.10540 


9^48781 


.12308 


8.12481 


S9 


2 


.00058 


1718.870 


.01804 


55-4415 


■03550 


28.1664 


•05299 


18.871I 


58 


2 


.07051 


14^1821 


.08807 


"■3540 


.10569 


9^46i4i 


.12338 


8.10536 


S8 


3 


.00087 


1145.920 


.01833 


54-5613 


•03579 


27^9372 


•05328 


18.7678 


57 


3 


.07080 


I4^I23S 


.08837 


H^3i63 


.10599 


9^43SiS 


.12367 


8.08600 


57 


4 


.00116 


859-436 


.01862 


53.7086 


.03609 


27^7117 


•05357 


18.6656 


56 


4 


.07110 


14^0655 


.08866 


11^2789 


.10628 


9.40904 


.12397 


8.06674 


S6 


5 


.00145 


687.549 


.01891 


52.8821 


.03638 


27^4899 


•05387 


18.5645 


55 


5 


•07139 


14.0079 


.08895 


11.2417 


.10657 


9^38307 


.12426 


8.04756 


55 


6 


.00175 


572^957 


.01920 


52.0807 


-03667 


27.2715 


•05416 


18.4645 


54 


6 


.07168 


13^9507 


.08925 


11^2048 


.10687 


9-35724 


.12456 


8.02848 


54 


7 


.00204 


491.106 


.01949 


51-3032 


.03696 


27.0566 


•05445 


18.3655 


53 


7 


.07197 


13.8940 


.08954 


II. 1681 


.10716 


9-33154 


.12485 


8.00948 


53 


8 


■00233 


429-718 


.01978 


50-5485 


•03725 


26.8450 


•05474 


18.2677 


52 


8 


.07227 


13-8378 


.08983 


11.1316 


.10746 


9-30S99 


•1251S 


7.99058 


52 


9 


.00262 


381.971 


.02007 


49-8157 


•03754 


26.6367 


•05503 


i8^i7o8 


51 


9 


.07256 


13-7821 


.09013 


11.0954 


•1077s 


9.28058 


.12544 


7.97176 


SI 


10 


.00291 


343-774 


.02036 


49.1039 


•03783 


26.4316 


•05533 


18.0750 


50 


10 


■07285 


13-7267 


.09042 


11^0594 


.10805 


9^25530 


.12574 


7.95302 


50 


11 


.00320 


312.521 


.02066 


48.4121 


.03812 


26.2296 


•05562 


17.9802 


49 


II 


■07314 


13.6719 


.09071 


11^0237 


.10834 


9^230i6 


.12603 


7.93438 


49 


12 


.00349 


286.478 


.02095 


47-7395 


.03842 


26.0307 


•05591 


17.8863 


48 


12 


■07344 


13.6174 


.09101 


io^9882 


.10863 


9^20516 


.12633 


7.91582 


48 


13 


.00378 


264.441 


.02124 


47-0853 


■03871 


25^8348 


.05620 


17.7934 


47 


13 


■07373 


13-5634 


.09130 


10.9529 


.10893 


9.18028 


.12662 


7.89734 


47 


14 


.00407 


245-552 


.02153 


46.4489 


■03900 


25.6418 


.05649 


17.7015 


46 


14 


.07402 


13.5098 


.09159 


10.9178 


.10922 


9-15554 


.12692 


7.87895 


46 


13 


.00436 


229.182 


.02182 


45.8294 


•03929 


25^4517 


.05678 


17.6106 


45 1 


15 


■07431 


13.4566 


.09189 


10.8829 


•10952 


9-13093 


.12722 


7.86064 


45 


16 


.00465 


214.858 


.02211 


45.2261 


.03958 


25^2644 


■05708 


17-5205 


44 


16 


.07461 


13.4039 


.09218 


10.8483 


.10981 


9.10646 


■12751 


7.84242 


44 


17 


.00495 


202.219 


.02240 


44-6386 


.03987 


25.0798 


■05737 


17-4314 


43 


\ ^' 


.07490 


13.3515 


.09247 


10.8139 


.11011 


9.082 1 1 


.12781 


7.82428 


43 


18 


.00524 


190.984 


.02269 


44.0661 


.04016 


24.8978 


.05766 


17-3432 


42 


i 18 


■07519 


13.2996 


.09277 


10.7797 


.11040 


9.05789 


.12810 


7.80622 


42 


19 


•00553 


180.932 


.02298 


43.5081 


.04046 


24.7185 


•05795 


17.2558 


41 1 


1 19 


.07548 


13.2480 


.09306 


10.7457 


.11070 


9.03379 


.12840 


7.78825 


41 


20 


.00582 


171-885 


.02328 


42.9641 


.04075 


24-5418 


.05824 


17-1693 


40 


20 


■07578 


13-1969 


•09335 


10.7119 


. 11099 


9.00983 


.12869 


7.77035 


40 


21 


.00611 


163.700 


-02357 


42-4335 


.04104 


24-3675 


■05854 


17-0837 


39 


21 


.07607 


13.1461 


•09365 


10.6783 


.11128 


8.98598 


.12899 


7^75254 


39 


22 


.00640 


156.259 


,02386 


41-9158 


•04133 


24-1957 


.05883 


16.9990 


38 


22 


•07636 


13.0958 


•09394 


10.6450 


.11158 


8.96227 


.12929 


7^73480 


38 


23 


.00669 


149.465 


.02415 


41.4106 


.04162 


24.0263 


.05912 


16.9150 


37 


\ 23 


.07665 


13-0458 


.09423 


10.6118 


.11187 


8.93867 


.12958 


7^71715 


37 


24 


.00608 


143-237 


.02444 


40.9174 


.04191 


23-8593 


.05941 


16.8319 


36 


24 


■07695 


12.9962 


•09453 


10.5789 


.11217 


8.91520 


.12988 


7^69957 


36 


25 


.00727 


137-507 


•02473 


40-4358 


.04220 


23-6945 


•05970 


16.7496 


35 


25 


.07724 


12.9469 


.09482 


10.5462 


.11246 


8.89185 


.13017 


7.68208 


iS 


26 


.00756 


132.219 


.02502 


39-9655 


.04250 


23-5321 


.05999 


16.6681 


34 


26 


•07753 


12.8981 


.09511 


10.5136 


.11276 


8.86862 


■13047 


7.66466 


34 


27 


.00785 


127-321 


•02531 


39-5059 


.04279 


23-3718 


.06029 


16.5874 


33 


27 


.07782 


12.8496 


•09541 


10.4813 


.11305 


8.84551 


.13076 


7.64732 


33 


28 


.00814 


122.774 


.02560 


39.0568 


.04308 


23-2137 


.06058 


16.5075 


32 


28 


.07812 


12.8014 


-09570 


10.4491 


•11335 


8.82252 


.13106 


7.63005 


32 


29 


.00844 


118.540 


•02589 


38.6177 


•04337 


23-0577 


.060S7 


16.4283 


31 


29 


.07841 


12-7536 


.09600 


10.4172 


• 11364 


8.79964 


.13136 


7.61287 


31 


30 


.00873 


114-589 


.02619 


38-1885 


.04366 


22.9038 


.06116 


16-3499 


30 i 


30 


.07870 


12.7062 


.09629 


10.3854 


•II394 


8.77689 


•13165 


7.59575 


30 


31 


.00902 


110.892 


.02648 


37-7686 


•04395 


22.7519 


.06145 


16^2722 


29 I 


31 


.07899 


12.6591 


.096 58 


10.3538 


.11423 


8-75425 


•13195 


7.57872 


29 


32 


.00931 


107.426 


.02677 


37-3579 


.04424 


22.6020 


.06175 


16.1952 


28, 


32 


.07929 


12.6124 


.09688 


10.3224 


.11452 


8.73172 


•13224 


7.56176 


28 


33 


.00960 


104.171 


.02706 


36.9560 


•04454 


22.4541 


.06204 


16.1190 


27 


33, 


.07958 


12.5660 


.09717 


10.2913 


.11482 


8.70931 


.13254 


7.54487 


27 


34 


.00989 


101.I07 


•02735 


365627 


■04483 


22.3081 


■06233 


16.043s 


26 


34 


■07987 


12.5199 


.09746 


10.2602 


.11511 


8.68701 


•13284 


7.52806 


26 


33 


.01018 


98.2179 


.02764 


36.1776 


.04512 


22.1640 


.06262 


15.9687 


25 


35 


.08017 


12.4742 


.09776 


10.2294 


.11541 


8.66482 


.13313 


7.51132 


2S 


36 


.01047 


95-4895 


-02793 


35-8006 


.04541 


22.0217 


.06291 


15.8945 


24 


36 


.08046 


12.4288 


.09805 


10.1988 


.11570 


8.64275 


.13343 


7-49465 


24 


37 


.01076 


92.9085 


.02822 


35-4313 


.04570 


21.8813 


.06321 


15.8211 


23 


37 


.08075 


12.3838 


.09834 


10.1683 


.11600 


8.62078 


.13372 


7.47806 


23 


38 


.01105 


90-4633 


.02851 


35-0695 


.04599 


21.7426 


.06350 


15.7483 


22 


38 


.08104 


12.3390 


.09864 


10.1381 


.11629 


8.59893 


.13402 


7-46154 


22 


39 


•0113s 


88.1436 


.02881 


34-7151 


.04628 


21.6056 


•06379 


15.6762 


21 


39 


.08134 


12.2946 


■09893 


10.1080 


-11659 


8.57718 


.13432 


7-44.509 


21 


40 


.01164 


85-9398 


.02910 


34-3678 


.04658 


21.4704 


.06408 


15.6048 


20 


40 


■08163 


12-2505 


■09923 


10.0780 


.11688 


8.55555 


.13461 


7-42871 


20 


41 


.01193 


83-8435 


.02939 


34-0273 


.04687 


21-3369 


.06437 


15.5340 


19 


41 


.08192 


12^2067 


.09952 


10.0483 


.11718 


8.53402 


.13491 


7.41240 


19 


42 


.01222 


81.8470 


.02968 


33.6935 


.04716 


21.2049 


.06467 


15.4638 


18 


42 


.08221 


12.1632 


.09981 


10.0187 


.11747 


8-51259 


■13521 


7-39616 


18 


43 


.01251 


79-9434 


.02997 


33.3662 


•04745 


21.0747 


.06496 


15.3943 


17 


43 


.08251 


12.1201 


.16011 


9.98931 


.11777 


8.49128 


.13550 


7-37999 


17 


44 


.01280 


78-1263 


.03026 


330452 


.04774 


20.9460 


•06525 


15-3254 


16 


44 


.08280 


12.0772 


.10040 


9.96007 


.11806 


8.47007 


.13580 


7-36389 


16 


45 


■01309 


76.3900 


■03055 


32-7303 


.04803 


20.8188 


•06554 


15-2571 


15 


45 


.08309 


12.0346 


.10069 


9.93101 


.11836 


8.44896 


■13609 


7.34786 


IS 


46 


•01338 


74.7292 


.03084 


32-4213 


.04832 


20.6932 


.06584 


1S-1893 


14 


46 


•08339 


11^9923 


.10099 


9.90211 


.11865 


8.42795 


.13639 


7.33190 


14 


47 


■01367 


73-1390 


.03114 


32.1181 


.04862 


20.5691 


.06613 


15.1222 


13 


47 


.08368 


11-9504 


.10128 


9.87338 


.11895 


8.40705 


.13669 


7.31600 


13 


48 


■01396 


71.6151 


-03143 


31.8205 


.04891 


20.4465 


.06642 


15-0557 


12 


48 


■08397 


11.9087 


.10158 


9.84482 


.11924 


8.3862s 


.13698 


7.30018 


12 


49 


.01425 


70-1533 


-03172 


31.5284 


.04920 


20.3253 


.06671 


14.9898 


II 


49 


.08427 


11-8673 


.10187 


9.81641 


.11954 


8.36555 


.13728 


7.28442 


11 


50 


•01455 


68.7501 


.03201 


31.2416 


.04949 


20.2056 


.06700 


14.9244 


10 


50 


.08456 


11.8262 


.10216 


9.78817 


.11983 


8.34496 


.13758 


7-26873 


10 


51 


.01484 


67.4019 


-03230 


30.9599 


.04978 


20.0872 


■06730 


14-8596 


9 


51 


.08485 


11-7853 


.10246 


9.76009 


.12013 


8.32446 


.13787 


7-25310 


9 


52 


•01513 


66.1055 


-03259 


30.6833 


.05007 


19.9702 


■06759 


14-7954 


8 


52 


■08514 


11.7448 


.10275 


9.73217 


.12042 


8.30406 


.13817 


7.23754 


8 


53 


•01542 


64.8580 


.03288 


30.4116 


•05037 


19.8546 


■06788 


14-7317 


7 


S3 


.08544 


11.7045 


•10305 


9.70441 


.12072 


8.28376 


.13846 


7.22204 


7 


54 


.01571 


63-6567 


-033 '7 


30.1446 


.05066 


19.7403 


.06817 


14.6685 


6 


54 


■08573 


11.6645 


•10334 


9.67680 


.12101 


8.26355 


■13876 


7.20661 


6 


55 


.01600 


62.4992 


-03346 


29.8823 


.05095 


19.6273 


.06847 


14.6059 


5 


55 


.08602 


11.6248 


■10363 


9.64935 


■ 12131 


8.24345 


.13906 


7-19125 


5 


56 


.01629 


61.3829 


.03376 


29.6245 


.05124 


19.5156 


■06876 


14-5438 


4 


S6 


.08632 


11-5853 


-10393 


9.62205 


.12160 


8-22344 


•13935 


7-17594 


4 


57 


■01658 


60.3058 


-03405 


29.3711 


■05153 


19.4051 


.06905 


14-4823 


3 


57 


.08661 


11.5461 


.10422 


9.59490 


.12190 


8.20352 


•1396s 


7.16071 


3 


S8 


.01687 


59-2659 


.03434 


29.1220 


■05182 


19.2959 


.06934 


14.4212 


2 


58 


.08690 


11.5072 


.10452 


9.56791 


.12219 


8.18370 


•13995 


7-I4S53 


2 


59 


.01716 


58.2612 


.03463 


28.8771 


■05212 


19.1879 


.06963 


14.3607 


I 


59 


.08720 


11.4685 


.10481 


9.54106 


.12249 


8.16398 


.14024 


7-13042 


I 


60 


.01746 


57-2900 


.03492 


28.6363 


■05241 


I9.0811 


•06993 


14-3007 





60 


.08749 


II. 430 1 


.10510 


9.51436 


.12278 


8.14435 


■14054 


7-11537 





/ 


Co-tan. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


t 


Co-TAN. 


Tan. 


Co-tan. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


T 




i 


59" 


88° 


87° 


s 


!6° 


! 




8 


5° 




84° 


83° 1 


82° 





MATHEMATICAL TABLES 



507 



Signs of the Trigonometric Functions in the Four Quadrants 



/ Sine + 


Sine + \ 


/ Cosine — 


Cosine + \ 


/ Tangent — 


Tangent + \ 


/ Cotangent — 


Cotangent + \ 


/ Secant - 


Secant + \ 


Cosecant + 


Cosecant + 


Sine — 


Sine — 


\ Cosine — 


Cosine + / 


\ Tangent + 


Tangent — / 


\ Cotangent + 


Cotangent — / 


\ Secant — 


Secant — / 


\ Cosecant — 


Cosecant + / 



sin^ a+cos^ a= i. 

sec^ Q;=i+tan^ a. 

cosec^ Q:=i+cot^ a. 

sin (a+|8) = sin a cos /3+cos a sin ^. 

sin (a—/?) =sin a cos /?— cos a sin /3. 

cos (a+/3) =cos ex cos /3— sin a sin /3. 

cos (a—/?) =cos a cos /3+sin a sin /3. 

tan (a+j3) = 
tan (a—(. 



1 — tan a tan ^ 
tan a — tan /3 
1+tan a tan /3 
sin 2(2 = 2 sin a cos a. 
cos 20! = 2 cos^a — I. 

2 tan a 
^^^ '"=l-tan^« 



Table 5. — Natueai. Trigonometric Functions — (Continued) 





8 





9 


3 


10° 


11° 






12° 


13° 


14° 


15° 




* 


Tan. 


Co-tan. 


Tan. 


Co-TAN. 


Tan. 


Co-tan. 


Tan. 


Co-TAN. 


/ 


/ 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN 


/ 





.14054 


7.II537 


.15838 


6.3137s 


-17633 


5.67128 


■19438 


5^144SS 


60 





.21256 


4-70463 


.23087 


4-33148 


■24933 


4.01078 


■26795 


3.73205 


60 


I 


.14084 


7.10038 


.15868 


6.30189 


.17663 


5.6616s 


.19468 


5^i36s8 


59 


I 


.21286 


4.69791 


.23117 


4-32573 


.24964 


4.00582 


.26S26 


3.72771 


59 


2 


.14113 


7.08546 


.15898 


6.29007 


-17693 


5-65205 


.19498 


5.12862 


S8 


a 


.21316 


4.6912I 


.23148 


4-32001 


•24995 


4.OC086 


.26857 


3-72338 


58 


3 


■14143 


7.07059 


.15928 


6.27829 


•17723 


5-64248 


•19529 


5.12069 


H 


3 


.21347 


4-68452 


•23179 


4-31430 


.25026 


3-99592 


.26888 


3.71907 


57 


4 


■14173 


7-05579 


.15958 


6.26655 


■17753 


5-63295 


•19559 


5. 11279 


56 


4 


■21377 


4-67786 


•23209 


4-30860 


•25056 


3-99099 


.26920 


3^71476 


ss 


S 


.14202 


7-04105 


.15988 


6.25486 


■17783 


5-62344 


•19589 


5.10490 


55 


5 


.21408 


4.67121 


.23240 


4-30291 


•25087 


3-98607 


.26951 


3.71046 


55 


6 


.14232 


7-02637 


.16017 


6.24321 


■17813 


5-61397 


.19619 


5.09704 


54 


6 


.21438 


4-66458 


■23271 


4-29724 


.25118 


3-98' 17 


.26982 


3.70616 


54 


7 


.142B2 


7-01174 


.16047 


6.23160 


■17843 


5-60452 


.19649 


5.08921 


53 


7 


.21469 


4-65797 


■23301 


4-29159 


•25149 


3-97627 


■27013 


3.70188 


53 


8 


.14291 


6.90718 


.16077 


6.22003 


■17873 


5-59511 


.19680 


5.08139 


52 


8 


.21499 


4-65138 


■23332 


4.28595 


.25180 


3-97139 


■27044 


3.69761 


52 


9 


, -14321 


6.98268 


.16107 


6.20831 


-17903 


5-58573 


.19710 


5.07360 


SI 


9 


.21529 


4-64480 


■23363 


4.28032 


.25211 


3.96651 


.27076 


3-69335 


51 


10 


■14351 


6-96823 


.16137 


6.19703 


•17933 


S-57638 


•19740 


5.06584 


50 


10 


.21560 


4.63825 


■23393 


4-27471 


.25242 


3-96165 


.27107 


3.68909 


50 


II 


.14381 


6-95385 


.16167 


6.18559 


•17963 


5.56706 


•19770 


S-05809 


49 


II 


.21590 


4-63171 


■23424 


4-26911 


•25273 


3-95680 


.27138 


3-68485 


49 


12 


.14410 


6-93952 


.16196 


6.17419 


-17993 


5-55777 


.19801 


S-0S037 


48 


12 


.21621 


4.62518 


■2345s 


4-26352 


•25304 


3-95196 


.27169 


3.68061 


48 


13 


.14440 


6.92525 


.16226 


6.16283 


.18023 


5^54851 


.19831 


5-04267 


47 


13 


.21651 


4.61868 


■2348s 


4-25795 


•25335 


3-94713 


.27201 


3-67638 


47 


14 


.14470 


6.91104 


.16256 


6-15151 


• 18053 


5^53927 


.19861 


5.03499 


46 


14 


.21682 


4-61219 


■23516 


4-25239 


.25366 


3-94232 


.27232 


3-67217 


46 


15 


•14499 


6-89688 


.16286 


6-14023 


.18083 


5^53007 


.19891 


5-02734 


4S 


IS 


.21712 


4-60572 


■23547 


4-24685 


•25397 


3-93751 


.27263 


3-66796 


45 


16 


.14529 


6-88278 


.16316 


6-12899 


.18113 


5.52090 


.19921 


5-01971 


44 


16 


•21743 


4-59927 


•23578 


4.24132 


•25428 


3-93271 


.27294 


3-66376 


44 


17 


•I45S9 


6-86874 


.16346 


6-11779 


.18143 


S-S1176. 


•19952 


S-01210 


43 


17 


•21773 


4-59283 


.23608 


4-23580 


•25459 


3-92793 


.27326 


3-65957 


43 


18 


.14588 


6.8547s 


.16376 


6.10664 


■18173 


5 -50264 


.19982 


5 -0045 1 


42 


18 


.21804 


4-58641 


.23639 


4.23030 


.25490 


3-92316 


■27357 


3-65538 


42 


19 


.14618 


6-840.S2 


.16405 


6-09552 


.18203 


5-49356 


.20012 


4-99693 


41 


19 


.21834 


4.58001 


.23670 


4.22481 


•25521 


3-91839 


■27388 


3^5121 


41 


20 


.14648 


6-82694 


•1643s 


6.08444 


■18233 


S -4845 1 


.20042 


4-98940 


40 


20 


.21864 


4-57363 


.23700 


4-21933 


•25552 


3-91364 


.27419 


3-64705 


40 


21 


.14678 


6-81312 


.16465 


6.073.40 


.18263 


5-47548 


.20073 


4-98188 


39 


21 


.21895 


4.56726 


■23731 


4-21387 


•25583 


3.90890 


•27451 


3^64289 


39 


22 


•14707 


6-79936 


.16495 


6.06240 


• 18293 


5.46648 


.20103 


4-97438 


38 


22 


.21925 


4.56091 


■ 23762 


4-20842 


.25614 


3.90417 


.27482 


3^63874 


38 


2J 


•14737 


6-78564 


•16525 


6.05143 


•18323 


5-4S751 


■20133 


4.96690 


37 


23 


.21956 


4-554s8 


■23793 


4.20298 


•25645 


3-89945 


•27513 


3.63461 


37 


24 


.14767 


6-77199 


•1655s 


6.04051 


•18353 


S-44857 


.20164 


4-95945 


36 


24 


.21986 


4.54S26 


.23823 


4.19756 


•25676 


3-89474 


•27545 


3-63048 


36 


25 


.14796 


6-75838 


.16585 


6.02962 


• 18383 


5-43966 


.20194 


4-95201 


35 


25 


.22017 


4.54196 


•23854 


4-19215 


.25707 


3.89004 


•27576 


3-62636 


33 


26 


.14826 


6.74483 


.16615 


6.01878 


.18414 


5-43077 


.20224 


4.94460 


34 


26 


.22047 


4-S3568 


■23885 


4-18675 


•25738 


3.88536 


•27607 


3.62224 


34 


27 


.14856 


6-73133 


.16645 


6.00797 


.18444 


5-42192 


.20254 


4.93721 


33 


27 


.22078 


4-52941 


.23916 


4.18137 


.25769 


3.88068 


•27638 


3.61814 


33 


23 


.14886 


6-717S9 


.16674 


5.99720 


• 18474 


S-41309 


.202S5 


4.92984 


32 


28 


.22108 


4-52316 


■23946 


4-17600 


.25800 


3.87601 


•27670 


3-61405 


32 


29 


•1491S 


6.70450 


.16704 


5-98646 


• 18504 


5.40429 


■20315 


4.92249 


31 


29 


■22139 


4-51693 


•23977 


4.17064 


.25831 


3.87136 


.27701 


3.60996 


31 


33 


•14945 


6.69116 


•16734 


S-97576 


•18534 


S-39S52 


■2034s 


4.91516 


30 


30 


.22169 


4.51071 


.24008 


4-16530 


.25862 


3.86671 


•27732 


3-60588 


30 


31 


■14975 


6.67787 


.16764 


5.96510 


• 18564 


5.38677 


■20376 


4.90785 


29 


31 


.22200 


4-S0451 


.24039 


4-15997 


•25893 


3.86208 


•27764 


3.60181 


29 


32 


•15005 


6-66463 


.16794 


5-95448 


•18594 


5-37805 


.20406 


4.90056 


28 


32 


.22231 


4.49832 


.24069 


4-15465 


•25924 


3-85745 


■27795 


3-59775 


28 


33 


•15034 


6.65144 


.16824 


594390 


• 18624 


5.36936 


.20436 


4-89330 


27 


33 


.22261 


4-49215 


.24100 


4-14934 


•25955 


3.85284 


.27826 


3-SO370 


27 


34 


.15064 


6.63831 


.16854 


5-93335 


.18654 


5-36070 


.20466 


4-88605 


26 


34 


.22292 


4-48600 


.24131 


4-14405 


.25986 


3.84824 


.27838 


3-58966 


26 


35 


.15094 


6-62523 


.16884 


5-92283 


.18684 


S-35206 


.20497 


4-87882 


25 


35 


.22322 


4-47986 


.24162 


4-13877 


.26017 


3-84364 


.27S89 


3-58562 


25 


30 


.15124 


6.61219 


.16914 


5.91235 


.18714 


S-34345 


.20527 


4-87162 


24 ' 


36 


■22353 


4.47374 


•24193 


4-13350 


.26048 


3-83906 


.27920 


3.5S160 


24 


37 


•15153 


6.59921 


.16944 


5.90191 


•18745 


5-33487 


■20557 


4-86444 


23! 


37 


•223S3 


4.46764 


.24223 


4.12825 


.26079 


383449 


■27952 


3-57758 


23 


3^ 


•15183 


6-58627 


-16974 


5-89151 


•18775 


5-32631 


-20588 


4-85727 


22 


38 


.22414 


4-46155 


■24254 


4.12301 


.26110 


3-82992 


•27983 


3-57357 


22 


39 


•15213 


6-57339 


- 17004 


5-88114 


.18805 


5-31778 


.20618 


4-85013 


21 


39 


.22444 


4.45548 


•2428s 


4.11778 


.26141 


3-82537 


.28015 


3-56957 


21 


40 


•15243 


6.5605s 


•17033 


5.87080 


.18835 


5-30928 


.20648 


4.84300 


20 


40 


.22475 


4.44942 


•24316 


4.11256 


.26172 


3-82083 


.28046 


3-56557 


20 


41 


.15272 


6.54777 


.17063 


5-86051 


.18865 


5.30080 


.20679 


4.83590 


'9 


41 


.22505 


4-44338 


■24347 


4-10736 


.26203 


3.81630 


.28077 


3-S6159 


19 


42 


.15302 


6.53503 


•17093 


5-85024 


.18895 


S-29235 


.20709 


4.82S82 


18 


42 


.22536 


4-43735 


•24377 


4.10216 


.26235 


3-81177 


.28109 


3-55761 


18 


43 


•15332 


6-52234 


•17123 


5.84001 


.18925 


5-28393 


.20739 


4.8217s 


17 


43 


.22567 


4-43134 


.24408 


4.09699 


.26266 


3-80726 


.28140 


3-55364 


17 


44 


•15362 


6-50970 


•17153 


5.82982 


• 18955 


S-27553 


.20770 


4.81471 


16 


44 


.22597 


4-42S34 


■24439 


4.09182 


.26297 


3.80276 


.28172 


3-54968 


16 


45 


•15391 


6-49710 


•17183 


5.81966 


.189S6 


5-26715 


.20800 


4.80769 


IS 


45 


.22628 


4.41936 


-24470 


4.08666 


.26328 


3-79827 


.28203 


3-54573 


15 


46 


.15421 


6-48456 


-17213 


5.80953 


.19016 


5-25880 


.20830 


4.80068 


14 


46 


.22658 


4.41340 


■24501 


4.08152 


•26359 


3-79378 


.28234 


3-54179 


14 


"Z 


•15451 


6-47206 


■17243 


5-79944 


.19046 


5.25048 


.20861 


4-79370 


13 


47 


.22689 


4-40745 


■24532 


4.07639 


•26390 


3-78931 


.28266 


3-53785 


13 


48 


•15481 


6-45961 


•17273 


5.78938 


.19076 


5.24218 


.20891 


4-78673 


12 


48 


.22719 


4.40152 


-24562 


4.07127 


.26421 


3-78485 


.28297 


3-53393 


12 


49 


•15511 


6.44720 


•17303 


5-77936 


.19106 


5^23391 


.20921 


4.77978 


II 


49 


.22750 


4.39560 


•24593 


4.06616 


.26452 


3-78040 


.28329 


3-53001 


II 


5° 


•15540 


6-43484 


•17333 


5-76937 


.19136 


5-22566 


.20952 


4.77286 


10 


50 


.22781 


4.38969 


-24624 


4.06107 


.26483 


3-77595 


.28360 


3.52609 


10 


51 


•15570 


6-42253 


•17363 


5-75941 


.19166 


5-21744 


.20982 


4-76595 


9 


SI 


.22811 


4.38381 


-24655 


4.05599 


.26515 


3-77152 


.28391 


3.52219 


9 


S2 


.15600 


6-41026 


•17393 


5-74949 


.19197 


5.20925 


.21013 


4.75906 


8 


52 


.22842 


4.37793 


-24686 


4-05092 


.26546 


3-76709 


.28423 


3-S1829 


8 


53 


•15630 


6.39804 


•17423 


5.73960 


.19227 


5.20107 


■21043 


4-75219 


7 


53 


.22872 


4-37207 


■24717 


4.04586 


•26577 


3.76268 


■28454 


3-51441 


7 


54 


.15660 


6.38587 


•17453 


5-72974 


•19257 


S-19293 


■21073 


4-74534 


6 


54 


-22903 


4.36623 


■24747 


4-04081 


.26608 


3-75828 


.28486 


3-S1053 


6 


S3 


.15689 


6.37374 


•17483 


5.71992 


.19^87 


5.18480 


.21104 


473851 


s 


55 


•22934 


4.36040 


■24778 


4-03578 


.26639 


3-75388 


.28517 


3-50666 


S 


56 


• 15719 


6.3616s 


.17513 


5^71013 


•19317 


5-17671 


.21134 


4-73170 


"♦l 


56 


.22964 


4-35459 


.24809 


4-03075 


.26670 


3-749SO 


•28549 


3-S0279 


4 


57 


•15749 


6.34961 


•17543 


5.70037 


■19347 


5-16863 


.21164 


4-72490 


3 


57 


■2299s 


4.34879 


.24840 


4-02574 


.26701 


3-74512. 


.28580 


3-49894 


3 


58 


•15779 


6.33761 


• 17573 


5.69064 


■ 19378 


5.16058 


.21195 


4-71813 


2 


S8 


.23026 


4.3.300 


.24871 


4.02074 


■26733 


3-74075 


.28612 


349509 


2 


59 


.15809 


6-32566 


• 17603 


5.68094 


.19408 


S-15256 


.21225 


4-71137 


I 


59 


■23056 


4-33723 


.24902 


4.01576 


.26764 


3-73640 


.28643 


3-4912S 


I 


60 


.15838 


6.31375 


•17633 


5.67128 


.19438 


S-I44S5 


.21256 


4-70463 





60 


.23087 


4-33148 


-24933 


4.01078 


.26795 


3-73205 


•28675 


3.48741 





/ 


Co-tan. 


Tan. 


CO-TAN. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


/ 


f 


Co-TAN. 


Tan. 


Co-TAN- 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


/ 




81° ' 


80° 1 


7 


9° 


7 


S° 






77° 1 


7( 


3° 


75° 


74° 





508 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 













Table 5.— 


Natural Trigonometric Functions- 


-{Continued) 














16° 


17° 


18° 


19° 






20° 


21° 


22° 1 


23° 




/ 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


' 


/ 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


/ 


o 


.28675 


3-48741 


.30573 


3-27085 


.32492 


3.07768 


•34433 


2.90421 


60 





■36397 


2.74748 


.38386 


2.60509 


.40403 


2.47509 


.42447 


2-35585 


60 


I 


.28706 


3-48359 


.30605 


3-26745 


•32524 


3.07464 


•3446s 


2.90147 


59 


I 


.36430 


2.74499 


.38420 


2.60283 


.40436 


2.47302 


.42482 


2-35395 


59 


2 


.28738 


3^47977 


.30637 


3.26406 


•32556 


3.07160 


.34498 


2.89873 


58 


2 


.36463 


2.74251 


•38453 


2^60057 


.40470 


2.47095 


.42516 


2-35205 


58 


3 


.28769 


3-47596 


.30669 


3.26067 


•32588 


3-06857 


•34S30 


2.89600 


57 


3 


.36496 


2.74004 


•38487 


2^59831 


.40504 


2.46888 


•42551 


2.3501S 


57 


4 


.28800 


3.47216 


.30700 


3-25729 


.32621 


3-06554 


•34563 


2.89327 


56 


4 


.36529 


2.73756 


.38520 


2^59606 


■40538 


2.46682 


•42585 


2.3482 s 


56 


S 


.28832 


3-46837 


•30732 


3-25392 


.32653 


3.06252 


•34596 


^■fi°|5 


55 


5 


.36562 


2.73509 


•38553 


2.59381 


•40572 


2.46476 


.42619 


2.34636 


55 


6 


.28864 


3^46458 


.30764 


3-25055 


.32685 


3-05950 


.34628 


2.88783 


54 


6 


■36595 


2.73263 


•38587 


2.59156 


.40606 


2.46270 


■42654 


2.34447 


54 


1 


.28895 


3^46080 


.30796 


3-24719 


■32717 


3-05649 


•34661 


2.88511 


53 


7 


.36628 


2.73017 


.38620 


2.58932 


.40640 


2.46065 


.42688 


2.34258 


S3 


8 


.28927 


3-45703 


.30828 


3-24383 


■32749 


3-05349 


•34693 


2.88240 


52 


8 


.36661 


2.72771 


■38654 


2.58708 


.40674 


2.45860 


.42722 


2.34069 


52 


9 


.28958 


3-45327 


.30860 


3.24049 


•32782 


3-05049 


•34726 


2.87970 


51 


9 


■36694 


2.72526 


■38687 


2.58484 


.40707 


2.4565s 


•42757 


2.33881 


51 


lo 


.28990 


3.44951 


.30891 


3.23714 


•32814 


3-04749 


•34758 


2.87700 


50 


10 


■36727 


2.72281 


.38721 


2.58261 


.40741 


2.45451 


.42791 


2.33693 


50 


11 


.29021 


3^44576 


.30923 


3^2338i 


.32846 


3-04450 


•34791 


2.87430 


49 


II 


.36760 


2.72036 


.38754 


2.58038 


.40775 


2.45246 


.42826 


2-33505 


49 


12 


.29053 


3.44202 


•30955 


3-23048 


.32878 


3-04152 


•34824 


2.87161 


48 


12 


■36793 


2.71792 


•38787 


2.57815 


.40809 


2.45043 


.42860 


2-33317 


48 


13 


.29084 


3^43829 


.30987 


3-22715 


.32911 


3-03854 


•34856 


2.86892 


47 


13 


.36826 


2.71548 


.38821 


2.57593 


.40843 


2.44839 


.42894 


2^33i30 


47 


14 


.29116 


3^43456 


.31019 


3-22384 


•32943 


3-03556 


•34889 


2.86624 


46 


14 


■36859 


2.7130s 


•38854 


2^57371 


.40877 


2.44636 


■42929 


2-32943 


46 


IS 


.29147 


3^43084 


•310S1 


3-22053 


•32975 


3.03260 


.34922 


2.86356 


45 


15 


.36892 


2.71062 


.38888 


2^57150 


.40911 


2.44433 


.42963 


2.32756 


45 


l6 


.29179 


3^427i3 


•31083 


3.21722 


•33007 


3-02963 


•34954 


2.86089 


44 


16 


•36925 


2.70819 


.38921 


2.56928 


•40945 


2.44230 


.42998 


2.32570 


44 


17 


.29210 


3^42343 


•3111S 


3-21392 


■33040 


3.02667 


•34987 


2.85822 


43 


17 


•36958 


2.70577 


.38955 


2.56707 


.40979 


2.44027 


•43032 


2.32383 


43 


l8 


.29242 


3-41973 


•31147 


3.21063 


•33072 


3-02372 


•35019 


2.85555 


42 


18 


•36991 


2.70335 


.38988 


2.56487 


■41013 


2.43825 


•43067 


2.32197 


42 


19 


.29274 


3.41604 


.31178 


3-20734 


•33104 


3-02077 


•35052 


2.85289 


41 


19 


•37024 


2.70094 


.39022 


2.56266 


.41047 


2.43623 


.43101 


2.32012 


41 


20 


•2930s 


3-41236 


.31210 


3.20406 


•33136 


3-01783 


•3508s 


2.85023 


40 


20 


•37057 


2.69853 


•3905s 


2.56046 


.41081 


2.43422 


•43136 


2.31826 


40 


21 


•29337 


3^40869 


.31242 


3-20079 


•33160 


3-01489 


•35117 


2.84758 


39 


21 


.37090 


2.69612 


.39089 


2.55827 


.41115 


2.43220 


•43170 


2.31641 


39 


22 


.29368 


3^40502 


•31274 


3^19752 


■33201 


3.01 196 


•35150 


2.84494 


38 


22 


•37124 


2.69371 


■39122 


2.55608 


.41149 


2.43019 


•4320s 


2.31456 


38 


23 


.29400 


3-40136 


•31306 


3.19426 


•33233 


3.00903 


•35183 


2.84229 


37 


23 


•37157 


2.69131 


■39156 


2.5S389 


.41183 


2.42819 


•43239 


2.31271 


37 


24 


•29432 


3-39771 


•31338 


3.19100 


•33266 


3.00611 


•35216 


2.8396s 


36 


24 


•37190 


2.68892 


•39190 


2.55170 


.41217 


2.42618 


•43274 


2.31086 


36 


25 


.29463 


3-39406 


•31370 


3.18775 


•33298 


3-00319 


■35248 


2.83702 


35 


25 


■37223 


2.68653 


•39223 


= ■54952 


.41251 


2.42418 


•43308 


2.30902 


35 


26 


•29495 


3-39042 


.31402 


3.I845I 


■33330 


3.00028 


•35281 


2.83439 


34 


26 


•37256 


2.68414 


■39257 


2^54734 


.41285 


2.42218 


•43343 


2.30718 


34 


27 


.29526 


3-38679 


•31434 


3-18127 


•33363 


2.99738 


•35314 


2.83176 


33 


27 


.37289 


2.:8i75 


•39290 


2.54516 


■41319 


2.42019 


•43378 


2.30534 


33 


28 


.29558 


3-38317 


■31466 


3-17804 


•33395 


2.99447 


•35346 


2.82914 


32 


28 


.37322 


2.67937 


■39324 


2.54299 


•41353 


2.41819 


.43412 


2-30351 


32 


29 


.29590 


3-37955 


.31498 


3-17481 


•33427 


2-99158 


•35379 


2.82653 


31 


29 


•37355 


2^677oo 


■393S7 


2.54082 


•41387 


2.41620 


•43447 


2.30167 


31 


30 


.29621 


3-37594 


•31530 


3-17159 


•33460 


2.98868 


•35412 


2.82391 


30 


30 


.37388 


2.67462 


-39391 


2.53865 


.41421 


2.41421 


•43481 


2.29984 


30 


31 


.29653 


3-37234 


.31562 


3.16838 


•33492 


2.98580 


•35445 


2.82130 


29 


31 


•37422 


2.67225 


•39425 


2.53648 


■41455 


2.41223 


•43516 


2.29801 


29 


32 


.29685 


3-36875 


•31594 


3^i6si7 


•33524 


2.98292 


•35477 


2.81870 


28 


32 


■37455 


2.66989 


•39458 


2.53432 


.41490 


2.41025 


•43550 


2.29619 


28 


33 


.29716 


3-36516 


.31626 


3.16197 


•33557 


2.98004 


•35510 


2.81610 


27 


33 


.37488 


2^66752 


•39492 


2^53217 


41524 


2.40827 


•43585 


2.29437 


27 


34 


.29748 


3-36158 


.31658 


3-15877 


•33589 


2.97717 


•35543 


2.81350 


26 


34 


■37521 


2.66516 


•39526 


2.53001 


•41558 


2.40629 


.43620 


2.29254 


26 


35 


.29780 


335800 


.31690 


3^15558 


.33621 


2.97430 


■35576 


2.8109I 


25 


35 


■37554 


2.66281 


•39559 


2.52786 


•41592 


2.40432 


■43654 


2.29073 


25 


36 


.29811 


3-35443 


.31722 


3^15240 


•33654 


2.97144 


.35608 


2.80833 


24 


36 


•37588 


2.66046 


•39593 


2.52571 


.41626 


2.4023s 


.43689 


2.28891 


24 


37 


.29843 


3-35087 


■31754 


3.14922 


.33686 


2.96858 


•35641 


2.80574 


23 


37 


.37621 


2.65811 


-39626 


2.52357 


.41660 


2.40038 


•43724 


2.28710 


23 


38 


•29875 


3-34732 


.31786 


3.14605 


•33718 


2.96S73 


■35674 


2.80316 


22 


38 


■37654 


2.65576 


.39660 


2.52142 


.41694 


2.39841 


•43758 


2.28528 


22 


39 


.29906 


3-34377 


.31818 


3.14288 


•33751 


2.96288 


•35707 


2^80059 


21 


39 


•37687 


2.65342 


•39694 


2.51929 


.41728 


2.39645 


•43793 


2^28348 


21 


40 


.29938 


3-34023 


.31850 


3-13972 


•33783 


2.96004 


•35740 


2.79802 


20 


40 


•37720 


2.65109 


•39727 


2^Si7i5 


•41763 


2.39449 


•43828 


2.28167 


20 


41 


.29970 


3-33670 


.31882 


3-13656 


.33816 


2.95721 


•35772 


2.79545 


19 


41 


•37754 


2.64875 


•39761 


2.51502 


•41797 


2.392S3 


.43862 


2.27987 


19 


42 


.30001 


3-33317 


■31914 


3-13341 


•33848 


2.95437 


•35805 


2.79289 


18 


42 


•37787 


2.64642 


•39795 


2.51289 


.41831 


2.39058 


•43897 


2.27806 


18 


43 


•30033 


3-32965 


.31946 


3.13027 


•33881 


2.9515s 


•35838 


2.79033 


17 


43 


.37820 


2.64410 


.39829 


2.51076 


.41865 


2.38862 


•43932 


2.27626 


17 


44 


■3006s 


3-32614 


-31978 


3.12713 


•33913 


2.94872 


•35871 


2.78778 


16 


44 


•37853 


2.64177 


.39862 


2.50864 


.41899 


2.38668 


.43966 


2.27447 


16 


45 


•30097 


3-32264 


.32010 


3.12400 


•33945 


2^94590 


•35904 


2-78523 


15 


45 


.37887 


2-63945 


.39896 


2.50652 


•41933 


2.38473 


.44001 


2.27267 


IS 


46 


.30128 


3-31914 


.32042 


3.12087 


•33978 


2.94309 


■35937 


2.78269 


14 


46 


•37920 


2-63714 


•39930 


2.50440 


.41968 


2.38279 


.44036 


2.27088 


14 


47 


.30160 


3-31565 


•32074 


3-II775 


.34010 


2.94028 


■35969 


2.78014 


13 


47 


■37953 


2.63483 


■39963 


2.50229 


.42002 


2.38084 


.44071 


2.26909 


13 


48 


■30192 


3.31216 


-32106 


3.11464 


•34043 


2^93748 


.36002 


2.77761 


12 


48 


.37986 


2.63252 


•39997 


2.50018 


.42036 


2.37891 


•44105 


2.26730 


12 


49 


.30224 


3 ■30868 


•32139 


3^i"53 


■34075 


2^93468 


■3603s 


2.77507 


11 


49 


.38020 


2.63021 


.40031 


2^49807 


.42070 


2.37697 


.44140 


2.26552 


11 


50 


•30255 


3-30521 


.32171 


3^10842 


•34108 


2.93189 


.36068 


2.77254 


10 


50 


•38053 


2.62791 


.40065 


2^49597 


.42105 


2.37504 


•44175 


2.26374 


10 


SI 


■ 30287 


3-30174 


.32203 


3-10532 


•34140 


2.92910 


.36101 


2.77002 


9 


51 


.38086 


2.62561 


.40098 


2.49386 


•42139 


2.37311 


.44210 


2.26196 


9 


52 


•30319 


3.29829 


•32235 


3.10223 


•34173 


2.92632 


■36134 


2.76750 


8 


52 


.38120 


2.62332 


•40132 


2.49177 


•42173 


2.371 18 


■44244 


2.26018 


8 


53 


■30351 


3-29483 


•32267 


3-09914 


■3420s 


2.92354 


.36167 


2.76498 


7 


53 


•38153 


2.62103 


.40166 


2.48967 


.42207 


2.3692s 


.44279 


2.2S840 


7 


54 


•30382 


3-29139 


.32299 


3.09606 


•34238 


2.92076 


.36199 


2.76247 


6 


54 


.38186 


2.61874 


.40200 


2.48758 


.42242 


2.36733 


•44314 


2.25663 


6 


55 


.30414 


3-28795 


•32331 


3.09298 


•34270 


2.91799 


.36232 


2.75996 


5 


55 


.38220 


2.61646 


•40234 


2.48549 


.42276 


2.36541 


•44349 


2.25486 


5 


56 


■30446 


3.28452 


•32363 


3.08991 


•34303 


2.91523 


.36265 


2.75746 


4 


56 


•38253 


2.61418 


.40267 


2.48340 


■42310 


2^36349 


•44384 


2.2S309 


4 


57 


.30478 


3^28109 


•32396 


3.08685 


•34335 


2.91246 


.36298 


2.75496 


3 


57 


.38286 


2.61190 


.40301 


2.48132 


•42345 


2.36158 


.44418 


2.2SI32 


3 


S8 


•30509 


3.27767 


.32428 


3-08379 


.34368 


2.90971 


•36331 


2.75246 


2 


58 


.38320 


2.60963 


•40335 


2.47924 


•42379 


2.35967 


•44453 


2.24956 


2 


59 


.30541 


3.27426 


.32460 


3.08073 


.34400 


2.90696 


•36364 


2.74997 


1 


59 


•38353 


2.60736 


•40369 


2.47716 


•42413 


2.35776 


.44488 


2.24780 


I 


60 


.30573 


3.27085 


.32492 


3-07768 


•34433 


2.90421 


•36397 


2.74748 1 j 


60 


■38386 


2.60509 


.40403 


2.47509 


.42447 


2.35585 


■44523 


2.24604 





1 


Co-TAN. 


Tan. 


Co-TAN. 1 Tan. 


Co-TAN 


Tan. 


Co-TAN .1 Tan. ' 


~r 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


~r 




73° 


72° 


71° 


70° 




69° 1 


68° 1 


6 


7° 


6 


3° 





MATHEMATICAL TABLES 



509 



Table 5. — Natural Trigonometric Functions — (Continued) 





24° 1 


25° 1 


26° 1 


27° 






28° 


29° 




30° 


31° 




1 


Tan. 


Co-tan. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 







Tan. 


Co-TAN^ 


Tan. C 


O-TAN. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 







•44523 


2.24604 


.46631 


2.14451 


.48773 


2.05030 


.50953 


1.96261 


60 


-53171 


1.88073 


.55431 1 


80405 


•57735 


1.73205 


.60086 


1.66428 


60 


I 


•44558 


2.24428 


.46666 


2.14288 


.48809 


2.04879 


.50989 


1. 96 1 20 


59 


I 


-53208 


1.87941 


.55469 I 


80281 


•57774 


i^73o89 


.60126 


1. 66318 


59 


3. 


•44593 


2.24252 


.46702 


2.1412s 


.48845 


2.04728 


.51026 


1-95979 


58 


2 


•S3246 


1.87809 


-55507 I 


80158 


•57813 


1-72973 


.60165 


1.66209 


58 


3 


.44627 


2.24077 


.46737 


2.13963 


.48881 


2.04577 


.51063 


1.95838 


57 


3 


•53283 


1.87677 


-55545 I 


80034 


•57851 


1-72857 


.60205 


1 .66099 


57 


4 


.44662 


2.23902 


.46772 


2.13801 


.48917 


2.04426 


-51099 


1.95698 


56 


4 


•53320 


1.87546 


-55583 I 


799" 


.57890 


1-72741 


.6024s 


1.65990 


56 


S 


.44697 


2.23727 


.46808 


2.13639 


•48953 


2.04276 


•51136 


1.95557 


55 


5 


.53358 


1-87415 


•55621 I 


79788 


•57929 


1-72625 


.60204 


1.65881 


55 


6 


•44732 


2.23553 


•46843 


2.13477 


^8989 


2.04125 


•51173 


1-95417 


54 


6 


•53395 


1.87283 


-55659 1 


7966s 


•57968 


1-72509 


.60324 


1.65772 


54 


7 


.44767 


2.23378 


.46879 


2.13316 


.49026 


2.03975 


•5 1209 


1.95277 


SI 


7 


•53432 


1.87152 


•55697 I 


79542 


.58007 


1-72303 


.60364 


1.65663 


53 


8 


.44802 


2.23204 


.46914 


2.13154 


.49062 


2.03825 


•51246 


1.95137 


52 


8 


•53470 


I.87021 


•55736 I 


79419 


.58046 


1-72278 


.60403 


1.65534 


52 


9 


•44837 


2.23030 


.46950 


2.12993 


.49098 


2.03675 


-51283 


1.94997 


51 


9 


•53507 


1. 8689 1 


•55774 I 


79296 


■58085 


1-72163 


•60443 


1^65445 


SI 


10 


.44872 


2.22857 


.46985 


2.12832 


.49134 


2.03526 


-51319 


1.94858 


50 


10 


•5354S 


1.86760 


•55812 I 


79174 


.58124 


1-72047 


.60483 


i^6S337 


SO 


II 


.44907 


2.22683 


.47021 


2.I2671 


•49170 


2.03376 


•31356 


1. 947 18 


49 


i " 


-53582 


1.86630 


•55850 I 


79051 


.58162 


1-71932 


.60522 


I^6S228 


49 


12 


.44942 


2.22510 


.47056 


2.12511 


.49206 


2.03227 


•51393 


1.94579 


48 


12 


-53620 


1.86499 


.55888 I 


78929 


.58201 


1-71817 


.60562 


1.65120 


48 


13 


•44977 


2.22337 


.47092 


2.12350 


.49242 


2.03078 


•51430 


1.94440 


47 


13 


.53657 


1.86369 


.55926 I 


78807 


.58240 


1. 71702 


.60602 


1.65011 


47 


14 


.45012 


2.22164 


.47128 


2.12190 


.49278 


2.02929 


•51467 


1.94301 


46 


14 


•53694 


1.86239 


•55964 I 


78685 


.58279 


1-71588 


.60642 


1.64903 


46 


IS 


.45047 


2.21992 


•47163 


2.12030 


•49315 


2.02780 


•51503 


1.94162 


4S j 


IS 


•53732 


1.86109 


.56003 I 


78563 


.58318 


1-71473 


.60681 


1.64795 


45 


16 


.45082 


2.21819 


.47199 


2.11871 


•49351 


2.02631 


•51540 


1.94023 


44 


16 


.53769 


1.85979 


.56041 I 


78441 


-58357 


1-71358 


.60721 


1.64687 


44 


17 


•45117 


2.21647 


.47234 


2.I1711 


•49387 


2.02483 


•51577 


1.9388s 


43 


17 


-53807 


1.85850 


.56079 I 


78319 


-58396 


1-71244 


.60761 


1.64579 


43 


18 


•45152 


2.21475 


.47270 


2.11552 


.49423 


2.0233s 


.51614 


1.93746 


42 


18 


.53844 


1.85720 


.56117 I 


78198 


-5843s 


1-71129 


.60801 


1. 644 7 1 


42 


19 


•45187 


2.21304 


.47305 


2.11392 


•49459 


2.02187 


•51651 


1.93608 


41 


19 


-53882 


1.85591 


.56156 I 


78077 


-58474 


I-7101S 


.60841 


1.64363 


41 


20 


•45222 


2.21132 


.47341 


2.11233 


•49495 


2.02039 


.51688 


1.93470 


40 


20 


.53920 


1.85462 


.56194 I 


77955 


-58513 


1-70901 


.60881 


1.64256 


40 


21 


•45257 


2.20961 


.47377 


2.II075 


•49532 


2.01891 


.51724 


i^93332 


39 


21 


•53957 


1.85333 


.56232 I 


77834 


-58552 


1.70787 


.60921 


1.64148 


39 


22 


•45292 


2.20790 


.47412 


2.10916 


•49568 


2.01743 


.51761 


i^93i9S 


38 


22 


•53995 


1.85204 


.56270 1 


77713 


-58591 


1.70673 


.60960 


1. 6404 1 


38 


23 


•45327 


2.20619 


.47448 


2.10758 


.49604 


2.01596 


.51798 


i^930S7 


37 


23 


.54032 


1-85075 


•56309 I 


77592 


-58631 


1.70560 


.61000 


1.63934 


37 


24 


•45362 


2.20449 


•47483 


2.10600 


.49640 


2.01449 


.51835 


1.92920 


36 


24 


■54070 


I •84946 


.56347 I 


77471 


-58670 


1.70446 


.61040 


1.63826 


36 


25 


•45397 


2.20278 


•47519 


2.10442 


.49677 


2.01302 


.51872 


1.92782 


35 


25 


■54107 


I •8481 8 


•5638s I 


77351 


-58709 


1.70332 


.61080 


1.63719 


35 


26 


•45432 


2.20108 


•47555 


2.10284 


■49713 


2.0115s 


.51909 


1.9264s 


34 


26 


•54145 


1.84689 


.56424 I 


77230 


.58748 


1.70219 


.61120 


1.63612 


34 


27 


•45467 


2.19938 


•47S90 


2.10126 


■49749 


2.01008 


•51946 


1.92508 


33 


27 


■54183 


I. 84561 


.56462 I 


77110 


•58787 


1. 70106 


.61160 


1.63505 


33 


28 


•45502 


2.19769 


.47626 


2.09969 


.49786 


2.00862 


•51983 


1.92371 


32 


28 


■54220 


I •84433 


.56500 I 


76990 


•58826 


1.69992 


.61200 


1.63398 


32 


29 


•45537 


2.19599 


.47662 


2.09811 


.49822 


2.00715 


.52020 


1.9223s 


31 


29 


■54258 


1.84305 


•56539 I 


76869 


•5886s 


1.69879 


.61240 


1.63292 


31 


30 


•45573 


2.19430 


.47698 


2.09654 


.49858 


2.00569 


•52057 


1.92098 


30 


30 


.54296 


1.84177 


•56577 I 


76749 


•58904 


1.69766 


.61280 


1. 63 1 85 


30 


31 


.45608 


2.19261 


•47733 


2.09498 


.49894 


2.00423 


•52094 


1.91962 


29 


31 


■54333 


1.84049 


.56616 I 


76630 


•58944 


1.69653 


.61320 


1-63079 


29 


32 


•45643 


2.19092 


•47769 


2.09341 


■49931 


2.00277 


•52131 


1.91826 


28 


32 


■54371 


1.83922 


.56654 I 


76510 


•58983 


1.69541 


.61360 


1.62972 


2S 


33 


.45678 


2.18923 


.47805 


2.09184 


•49967 


2.00131 


•52168 


1.91690 


27 


33 


■54409 


I •83794 


.56693 I 


76390 


•59022 


1.69428 


.61400 


1.62866 


27 


34 


•45713 


2.18755 


.47840 


2.09028 


.50004 


1.99986 


•5220s 


1.91554 


26 


34 


■54446 


1.83667 


.56731 1 


76271 


•59061 


1.69316 


.61440 


1.62760 


26 


35 


■45748 


2.18587 


•47876 


2.08872 


.50040 


1.99841 


•52242 


1.91418 


25 


35 


.54484 


1.83540 


.56769 I 


76151 


•5910I 


1.69203 


.61480 


1.62654 


25 


36 


■45784 


2.1S419 


.47912 


2.08716 


.50076 


1.9969s 


•52279 


1.91282 


24 


36 


■54522 


1.83413 


.56808 I 


76032 


.59140 


1. 6909 1 


.61520 


1.62548 


24 


37 


.45819 


2.18251 


.47948 


2.08560 


•50113 


1.99550 


•52316 


1.91147 


23 


37 


•54560 


1.83286 


.56846 I 


75913 


•S9179 


1.68979 


.61561 


1.62442 


23 


38 


■45854 


2.18084 


.47984 


2.08405 


.50149 


1.99406 


•52353 


1.91012 


22 


38 


•54597 


1.83159 


.56885 I 


75794 


.59218 


1.68866 


.61601 


1.62336 


22 


39 


■45889 


2.17916 


.48019 


2.08250 


.5018s 


I.99261 


•52390 


1.90876 


21 


39 


•54635 


i^83033 


•56923 I 


75675 


•59258 


1.68754 


.61641 


1.62230 


21 


40 


■45924 


2.17749 


•4805s 


2.08094 


.50222 


1.99116 


•52427 


1.90741 


20 


40 


•54673 


1.82906 


.56962 I 


75556 


•59297 


1.68643 


.61681 


1.62125 


20 


41 


.45960 


2.17582 


.48091 


2.07939 


.50258 


1.98972 


•52464 


1.90607 


19 


41 


■547" 


1.82780 


.57000 1 


75437 


•59336 


1.68531 


.61721 


1. 62019 


19 


42 


•45995 


2.17416 


.48127 


2.0778s 


•5029s 


1.98828 


•52501 


1.90472 


18 


42 


■54748 


1.82654 


•57039 I 


75319 


•59376 


1.68419 


.61761 


1.61914 


18 


43 


.46030 


2.17249 


.48163 


2.07630 


•50331 


1.98684 


•52538 


1.90337 


17 


43 


■54786 


1.82528 


.57078 I 


75200 


•5941s 


1.68308 


.61801 


1.61808 


17 


44 


.4606s 


2.17083 


.48198 


2.07476 


.50368 


1.98540 


.52575 


1.90203 


16 


44 


•54824 


1.82402 


.57116 1 


75082 


-59454 


1.68196 


.61842 


1.61703 


i6 


45 


.46101 


2.16917 


.48234 


2.07321 


.50404 


1.98396 


-52613 


1.90069 


15 


45 


•54862 


1.82276 


•5715s I 


74964 


-59494 


1.68085 


.61882 


1.61598 


15 


46 


.46136 


2.167S1 


.48270 


2.07167 


.50441 


1.98253 


-52650 


1.89935 


14 


46 


•54900 


1.82150 


•57193 I 


74846 


-59533 


1.67974 


.61922 


1.61493 


14 


47 


.46171 


2.16585 


.48306 


2.07014 


•50477 


1.98110 


-52687 


1. 8980 1 


13 


47 


•54938 


1.82025 


•57232 I 


74728 


•59573 


1.67863 


.61962 


1.61388 


13 


48 


.46206 


2.16420 


.48342 


2.06860 


•50514 


1.97966 


-52724 


1.89667 


12 


48 


•54975 


1.81899 


.57271 I 


74610 


-59612 


1.67752 


.62003 


1.61283 


12 


49 


.46242 


2.16255 


•48378 


2.06706 


•50550 


1.97823 


-52761 


1-89533 


11 


49 


•55013 


1.81774 


.57309 I 


74492 


-59651 


1.67641 


.62043 


I.6n79 


11 


50 


•46277 


2.16090 


•48414 


2.06553 


•50587 


1.97680 


-52798 


1-89400 


10 


50 


■55051 


1^81649 


•57348 I 


74375 


-59691 


1.67530 


.62083 


1.61074 


10 


51 


•46312 


2.15925 


.48450 


2.06400 


•50623 


1.97538 


-52836 


1.89266 


9 


51 


■55089 


i^8iS24 


•57386 I 


74257 


-S9730 


1.67419 


.62124 


1.60970 


9 


52 


.46348 


8.15760 


.48486 


2.06247 


.50660 


1-97395 


.52873 


1-89133 


8 


52 


■55127 


I^8i399 


•57425 I 


74140 


-59770 


1.67309 


.62164 


1.60865 


8 


53 


•46383 


2-15596 


.48521 


2.06094 


.50696 


1-97253 


-52910 


1.89000 


7 


53 


■55165 


1.81274 


•57464 I 


74022 


.59809 


1.67198 


.62204 


1.60761 


7 


54 


.46418 


2.15432 


.48557 


2.05942 


•50733 


1. 971 11 


-52947 


1.88867 


6 


54 


.55203 


1.81150 


.57503 I 


73905 


-59849 


1.67088 


.62245 


1.60657 


6 


55 


•46454 


2.15268 


.48593 


2.05790 


-50769 


i^96969 


•52984 


1.88734 


5 


55 


.55241 


1.81025 


.57541 I 


73788 


.59888 


1.66978 


.62285 


1.60553 


5 


56 


.46489 


2.15x04 


.48629 


2.05637 


.50806 


i^96827 


.53022 


1.88602 


4 


56 


.55279 


1.80901 


.57580 I 


73671 


.59928 


1.66867 


.6232s 


1.60449 


4 


57 


.46525 


2.14940 


.48665 


2.0548s 


•50843 


1.9668s 


.53059 


1.88469 


3 


57 


.55317 


1.80777 


.57619 I 


73555 


•59967 


1.66757 


.62366 


1.60345 


3 


58 


.46560 


2.14777 


.48701 


2 .05333 


•50879 


1.96544 


.53096 


1-88337 


2 


S8 


.55355 


1.80653 


.57657 I 


73438 


.60007 


1.66647 


.62406 


1. 60241 


2 


59 


•4659s 


2.14614 


.48737 


2.05182 


.50916 


1.96402 


.53134 


1. 8820s 


I 


59 


•55393 


1.80529 


.57696 1 


73321 


.60046 


1.66538 


.62446 


1.60137 


I 


60 


•46631 


2.14451 


.48773 


2.05030 


•50953 


1. 96261 


■53171 


1.88073 





60 


•5S43I 


1. 8040 5 


•57735 I 


7320s 


.60086 


1.66428 


.62487 


1.60033 





/ 


CO-TAN. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


/ 




Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 


Co-TAN. 


Tan. 






6 


5° 


6 


4° 


63° I 


6 


2° 






6 


1° 


60° 




5 


9° 


5 


8° 





510 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 













Table 5. — Natural Trigonometric Functions — 


(Continued) 














32° 1 


33° 1 


34° f 


35° 






36° II 


37° 1 


38° 


39° 




f 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


/ 1 


/ 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


f 


o 


.62487 


1.60033 


.64941 


1.53986 


.67451 


1.48256 


.70021 


1.42815 


60 





•72654 


1.37638 


-75355 


1.32704 


.78129 


1.27994 


-80978 


1.23490 


60 


I 


■62527 


1.59930 


.64982 


1.53888 


.67493 


1.48163 


.70064 


1.42726 


59 


I 


•72699 


1^37554 


-75401 


1.32624 


.78175 


1.27917 


.81027 


1.23416 


59 


2 


.62568 


1.59826 


•65023 


i^5379i 


.67536 


1.48070 


.70107 


1.42638 


58 


2 


•72743 


1^37470 


•75447 


I •32544 


.78222 


1.27841 


-81075 


1.23343 


58 


3 


.62608 


1.59723 


.65065 


i^S3693 


.67578 


1-47977 


.70151 


1^42550 


57 


3 


.72788 


I •37386 


•75492 


1.32464 


.78269 


1.27764 


.81123 


1.23270 


57 


4 


.62649 


1.59620 


.65106 


1^53595 


.67620 


1^47885 


.70194 


1.42462 


S6 


4 


.72832 


1-37302 


•75538 


1^32384 


.78316 


1.27688 


.81171 


1.23196 


S6 


S 


.62689 


i^595i7 


•65148 


1-53497 


.67663 


1^47792 


.70238 


1.42374 


55 


5 


•72877 


1.37218 


■75584 


1^32304 


.78363 


1.27611 


.81220 


1.23123 


SS 


6 


.62730 


1. 59414 


•65189 


1.53400 


•67705 


1.47699 


.70281 


1.42286 


54 


6 


.72921 


1-37134 


•75629 


1^32224 


.78410 


1.27535 


.81268 


1.23050 


54 


7 


.62770 


I-593II 


•65231 


I-53302 


•67748 


1.47607 


•70325 


1. 42 198 


53 


7 


.72966 


1-37050 


•75675 


1.32144 


•78457 


1.27458 


.81316 


1.22977 


■Si 


8 


.62811 


1.59208 


■65272 


1^53205 


.67790 


I^475i4 


.70368 


1.42110 


52 


8 


.73010 


1-36967 


•75721 


1.32064 


•78504 


1.27382 


.81364 


1.22904 


52 


9 


.62852 


1-59105 


•6S314 


i^53io7 


.67832 


1.47422 


.70412 


1.42022 


51 





•7305s 


1-36883 


•75767 


1.31984 


•78551 


1.27306 


■81413 


1.22831 


SI 


lo 


.62892 


1.59002 


.65355 


1.53010 


•6787s 


1^47330 


•70455 


1.41934 


50 


ro 


•73100 


1 .36800 


.75812 


1-31904 


.78598 


1.27230 


•81461 


1.22758 


50 


II 


•62933 


1.58900 


•65397 


1^52913 


•67917 


1.47238 


.70499 


1.41847 


49 


II 


•73144 


I.36716 


.75858 


1-31825 


.78645 


1.27153 


.81510 


1.22685 


49 


12 


■62973 


1.58797 


.65438 


1.52816 


.67960 


1.47146 


.70542 


1^41759 


48 


12 


•73189 


1-36633 


■75904 


I -31 745 


.78692 


1.27077 


.81558 


1.22612 


48 


13 


.63014 


I -5869s 


.65480 


1. 52719 


.68002 


1-47053 


.70586 


1^41672 


47 


13 


•73234 


1-36549 


■75950 


1. 3 1 666 


•78739 


1. 2 700 1 


.81606 


1.22539 


47 


14 


•6305s 


1.58593 


.65521 


1.52622 


.68045 


1^46962 


.70629 


1^4 1 584 


46 


14 


•73278 


1.36466 


■75996 


1.31586 


.78786 


1.2692s 


.81655 


1.22467 


46 


IS 


•63095 


1.58490 


.65563 


1-52525 


.68088 


1 .46870 


•70673 


1.41497 


45 


15 


•73323 


1-36383 


.76042 


1-31507 


•78834 


1.26849 


•81703 


1.22394 


45 


l6 


■63136 


1.58388 


.65604 


1.52429 


.68130 


1.46778 


.70717 


1.41409 


44 


16 


.73368 


1-36300 


.76088 


1.31427 


.78881 


1.26774 


.81752 


1.22321 


44 


17 


•63177 


1.58286 


.65646 


1-52332 


•68173 


1.46686 


.70760 


1.41322 


43 


17 


•73413 


I.36217 


•76134 


1-31348 


.78928 


1.26698 


.81800 


1.22249 


43 


i8 


•63217 


1.58184 


.65688 


1-5223S 


.68215 


1^46595 


.70804 


1.41235 


42 


18 


•73457 


I-36133 


.76180 


1.31269 


•78975 


1.26622 


.81849 


1.22176 


42 


19 


.63258 


1.58083 


•65729 


1-52139 


.68258 


1.46503 


.70848 


1.41148 


41 


19 


-73502 


I.36051 


.76226 


1-31190 


.79022 


1.26546 


.81898 


1.22104 


41 


20 


•63299 


1.57981 


•65771 


1-52043 


.68301 


1. 4641 1 


•70891 


1.41061 


40 


20 


•73547 


1-35968 


.76272 


1. 31 1 10 


.79070 


1. 26471 


.81946 


1.22031 


40 


21 


•63340 


1^57879 


•65813 


1-51946 


•68343 


1.46320 


•70935 


1.40974 


39 


21 


.73592 


1-35885 


.76318 


1.31031 


.79117 


1-26395 


.81995 


1.21959 


39 


22 


.63380 


i^S7778 


•65854 


1-51850 


.68386 


1.46229 


•70979 


1.40887 


38 


22 


•73637 


1-35802 


■76364 


1.30952 


.79164 


1.26319 


.82044 


1.21886 


38 


23 


.63421 


1-57676 


.65896 


I-SI7S4 


.68429 


1.46137 


.71023 


1 .40800 


37 


23 


.73681 


1-35719 


.76410 


1.30873 


.79212 


1.26244 


.82092 


1.21814 


37 


24 


.63462 


1-57575 


•65938 


1-51658 


.68471 


1.46046 


.71066 


1.40714 


36 


24 


•73726 


I •35637 


•76456 


1.30795 


•79259 


1.26169 


.82141 


1.21742 


36 


25 


■63503 


1-57474 


.65980 


1.51562 


.68514 


I -43955 


.71110 


1.40627 


Z% 


25 


•73771 


1^35554 


.76502 


1-30716 


•79306 


1.26093 


.82190 


1.21670 


35 


26 


■63544 


1-57372 


.66021 


1. 51466 


•68557 


1.45864 


•71154 


1.40540 


34 


26 


.73816 


1-35472 


.76548 


1-30637 


•79354 


1.26018 


.82238 


1.21598 


34 


27 


.63584 


1.57271 


.66063 


1-51370 


.68600 


1-45773 


.71198 


1.40454 


33 


27 


.73861 


1-35389 


•76594 


1-30SS8 


•79401 


1-25943 


.82287 


1.21526 


33 


28 


.63625 


1.57170 


.66105 


I-5I27S 


.68642 


1.45682 


.71242 


1.40367 


32 


28 


•73906 


1-35307 


.76640 


1.30480 


•79449 


1-25867 


.82336 


1.21454 


32 


29 


.63666 


1-57069 


.66147 


1.51179 


.68685 


1^45592 


.71285 


1. 4028 1 


31 


29 


-73951 


1.35224 


.76686 


1. 30401 


•79496 


1-25792 


•82385 


1. 21382 


31 


30 


•63707 


1.56969 


.66189 


1. 5 1084 


.68728 


1^45501 


■71329 


1. 40 1 95 


30 


30 


.73996 


1-35142 


•76733 


1.30323 


•79544 


1-25717 


.82434 


1.21310 


30 


31 


•63748 


1.56868 


.66230 


1.50988 


.68771 


1. 45410 


•71373 


1. 40109 


29 


31 


.74041 


1.35060 


•76770 


1.30244 


•79501 


1-25642 


.82483 


1.21238 


29 


32 


.63789 


1.56767 


.66272 


1.50893 


.68814 


1-45320 


.71417 


1.40022 


28 


32 


.74086 


1.34978 


.76825 


1.30166 


■706,39 


1.25567 


.82531 


1.21166 


28 


33 


.63830 


1.56667 


•66314 


1.50797 


.68857 


1.45229 


.71461 


1 •39036 


27 


33 


•74131 


1.34896 


.76871 


1.30087 


.79686 


1.25492 


.82580 


1.21094 


27 


34 


.63871 


1.56566 


.66356 


1.50702 


.68900 


1-45139 


•7150S 


1.39850 


26 


34 


.74176 


1. 348 1 4 


.76918 


1.30009 


•70734 


1.25417 


.82629 


1. 21023 


26 


3S 


.63912 


1.56466 


.66398 


1.50607 


.68942 


1.45049 


•71549 


1-39764 


25 


35 


.7422! 


1-34732 


■ 76964 


1.29931 


.79781 


1.25343 


.82678 


1.20951 


25 


36 


•630S3 


1.56366 


.66440 


1.50512 


.68985 


1.44958 


•71503 


1-39679 


24 


36 


•74267 


1-34650 


.77010 


1.29853 


.79829 


1.25268 


.82727 


1.20879 


24 


37 


•63094 


1.56265 


.66482 


1.50417 


.69028 


1.44868 


•71637 


I-39S93 


23 


37 


•74312 


1-34568 


■77057 


1.29775 


.79877 


1.25193 


.82776 


1.20808 


23 


38 


•64035 


1. 56165 


.66524 


1.50322 


.69071 


1.44778 


.71681 


1-39507 


22 


38 


•74357 


1-34487 


•77103 


1.29696 


.79924 


1.25118 


.82825 


•■1.20736 


22 


39 


.64076 


1^56065 


.66566 


1.50228 


.69114 


1.446S8 


.71725 


1.39421 


21 


39 


.74402 


1-34405 


•77149 


1.29618 


.79972 


1.25044 


.82874 


1.20665 


21 


40 


.64117 


1.55966 


.66608 


1.50133 


•69157 


1.44598 


.71769 


1-30336 


20 


40 


•74447 


1-34323 


.77196 


1.29541 


.80020 


1.24969 


.82923 


1.20593 


20 


41 


•64158 


1.55866 


.66650 


1.50038 


.69200 


1.44508 


.71813 


1-30250 


19 


41 


.74492 


1.34242 


•77242 


1.29463 


.80067 


1.24895 


.82972 


1.20S22 


19 


42 


.64199 


1.55766 


■66692 


1.49944 


•69243 


1. 44418 


•71857 


1-30165 


18 


42 


•74538 


1. 34 1 60 


•77289 


1.29385 


.80115 


1.24820 


.83022 


1.20451 


18 


43 


.64240 


1.55666 


•66734 


1.49849 


.69286 


1.44329 


.71901 


1.39079 


17 


43 


•74583 


1.34079 


•77335 


1.29307 


.80163 


1.24746 


.83071 


1.20379 


17 


44 


.64281 


1.55567 


.66776 


I.497SS 


-69329 


1.44239 


.71946 


1.38994 


16 


44 


.74628 


1-33998 


•77382 


1.29229 


.80211 


1.24672 


.83120 


1.20308 


16 


45 


•64322 


1-55467 


.66818 


1.49661 


•69372 


1.44149 


.71990 


1.38909 


IS 


45 


.74674 


1-33016 


•77428 


1.29152 


.80258 


1.24597 


•83169 


1.20237 


15 


46 


•64363 


1.55368 


.66860 


1.49566 


.69416 


1 .44060 


•72034 


1.38824 


14 


46 


•74719 


1-33835 


•77475 


1.29074 


.80306 


1.24523 


.83218 


1. 20 1 66 


14 


47 


.64404 


1.55269 


.66902 


1.49472 


•69459 


1.43970 


.72078 


1-38738 


13 


47 


•74764 


1-33754 


•77521 


1.28997 


-80354 


1.24449 


.83268 


1.20095 


13 


48 


.64446 


1.55170 


.66944 


1.49378 


.69502 


1. 43881 


.72122 


1-38653 


12 


48 


.74810 


1-33673 


•77568 


1.28919 


.80402 


1.2437s 


•83317 


1.20024 


12 


49 


.64487 


1.55071 


.669S6 


1.49284 


•69545 


i^43V92 


.72166 


1-38568 


II 


49 


.7485s 


1-33592 


•7761S 


1.28842 


.804 w 


1.24301 


.83366 


I.I99S3 


II 


50 


.64528 


1.54972 


.67028 


1.49190 


.69588 


i^43703 


.72211 


1.38484 


10 


50 


.74900 


1-33511 


.77661 


1.28764 


.80498 


1.24227 


•83415 


1.19882 


10 


51 


•64569 


1.54873 


.67071 


1.49097 


-69631 


1-43614 


•72255 


1-38399 


9 


SI 


• 74946 


1-33430 


■77708 


1.28687 


.80546 


1-24153 


.83465 


1.19811 


9 


52 


.64610 


I.S4774 


.67113 


1.49003 


-6967s 


I.43S-S 


•72299 


1-38314 


8 


52 


.74991 


1-33349 


•77754 


1.28610 


.80594 


1.24079 


•83514 


1. 19740 


8 


S3 


.64652 


1-54675 


•6715s 


1.48909 


.69718 


1^43436 


•72344 


1.38229 


7 


Si 


•75037 


1-33268 


.77801 


1.28533 


.80642 


1.24005 


•83564 


I.I 9669 


7 


54 


.64693 


1-54576 


.67197 


1.48816 


.69761 


1-43347 


.72388 


I -38145 


6 


54 


.75082 


I-33187 


.77848 


1.28456 


.80690 


1.23931 


•83613 


1. 19599 


6 


55 


•64734 


1-54478 


•67239 


1.48722 


.69804 


1-43258 


•72432 


1-38060 


5 


55 


-75128 


1-33107 


.77895 


1.28379 


.80738 


1.23858 


.83662 


1. 19528 


5 


S6 


•6477s 


1-54379 


.67282 


1.48629 


.69847 


1-43169 


•72477 


1-37976 


4 


S6 


•75173 


1.33026 


•77941 


1.28302 


.80786 


1-23784 


•83712 


1.19457 


4 


57 


.64817 


1.54281 


•67324 


1.48536 


.69891 


1.43080 


.72521 


1-37891 


3 


57 


-75219 


1.32946 


■77988 


1.28225 


.808 M 


1.23710 


•83761 


1. 19387 


3 


S8 


.64858 


1-54183 


•67366 


1.48442 


.60934 


1.42992 


.72565 


1-37807 


2 


58 


•75264 


1.32865 


.78035 


1.28148 


.S0882 


1-23637 


.8^811 


I.19316 


2 


59 


.64899 


1.54085 


.67409 


1.48349 


-60077 


1-42903 


.72610 


1.37722 


I 


59 


■75310 


I -3278s 


.78082 


1.28071 


.80930 


1-23563 


.83860 


1. 19246 


I 


60 


.64941 


1.53986 


•67451 


1.48256 


-70021 


1.42815 


•72654 


1-37638 




/ 


60 


•75355 


1-32704 


.78129 


1.27994 


.80978 


1-23490 


.83910 


I.19175 





f 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


~ 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan. 


Co-tan. 


Tan 


/ 




- 5 


70 


5 


6° 


5 


5° 


5 


40 






5 


3° 


52° 1 


5 


1° 


5 


0° 





MATHEMATICAL TABLES 



511 



Table 5. — Natural Trigonometric Functions — (Continued) 



40° 41° 42° 
Tan. Co-tan. Tan. Co-tan. Tan. Co-tan, 



.83910 
.83960 



.84059 
.84108 
.84158 
.84208 
.84258 
.84307 
.84357 
.84407 

•84457 
■84507 
•84556 
.84606 
.84656 
.84706 
.84756 



.84856 
.84906 

■84956 
.85006 
■85057 
.85107 
■85157 
.85207 
■85257 
■85307 
■85358 
.85408 
■85458 
•85509 
•85559 
■85609 
■85660 
■85710 
■85761 
■85811 
•85862 
.85912 

•85963 
•86014 
.86064 
.86115 
.86166 
.86216 
.86267 
.86318 



.86419 
.86470 
.86521 
•86572 
.86623 
.86674 
•86725 
.86776 
.86827 



.86929 



'•I9I75 
1.19105 
I.I9035 
I ■18964 
i^i8894 
1. 18824 
I^i8754 
I^i8684 
1.18614 
I. 18544 
1. 18474 
1. 18404 
1-18334 
1. 18264 
1.18194 
1.18125 
1.18055 
1. 1 7986 
1.17916 
1. 1 7846 
1. 17777 
I" 1 7708 
1.17638 
I^I7569 
1. 17500 
1.17430 
1.17361 
1. 17292 
1. 17223 
1-17154 
i^i7o8s 

1.17016 
1.16947 
1.16878 
1.16809 
1.16741 
1.16672 
1.16603 
1-16535 
1.16466 
1.16398 
1.16329 
1. 16261 
1.16192 
1.16124 
1.16056 
1.15987 
1.15919 
i.i58';i 
1-15783 
1-15715 
1.15647 
1-15579 
1-15511 
I-I5443 
I-IS375 
1-15308 
1.15240 
1. 15172 
1.15104 
1-15037 



Co-TAN. Tan. Co-tan. Tan. 
49° 48° 



.87031 
,87082 
87133 
,87184 
87236 
,87287 
.87338 
,87389 
,87441 

87492 
.87543 
.87595 
.87646 
,87698 
,87749 
,87801 
87852 
87904 
.87955 
,88007 
,88059 
110 
162 
214 
,88265 
88317 



86929 



88421 
.88473 
.88524 
88576 
.88628 



.88732 



,88992 

89045 
,89097 
,89149 
201 
253 
,89306 
89358 
,89410 
,89463 
89515 
.89567 
89620 
89672 
,89725 
,89777 
,80830 



89935 



1.15037 
1.14969 
1.14902 
1.14834 
1.14767 
1.14699 
1. 14632 
1-14565 
1^14498 
1.14430 
I ■14363 

I^i4296 
I^I422g 
1.14162 
1.14095 
1.14028 
I^i396i 
1-13894 
1.13828 
1-13761 
1.13694 
1.13627 
1-13561 
1.13494 
1.13428 
1. 13361 
1-13295 
1-13228 
1.13162 
1. 1 3096 
1.13029 
1.12963 
1. 12897 
1.12831 
1.12765 
1. 12699 
1.12633 
1-12567 
1.12501 
1-12435 
1-12369 

1. 12303 
1.12238 
1.12172 
1. 12106 
1.12041 
1-11975 
1.11909 
1.11844 
1.11778 
1-11713 
1.11648 
I. 11582 
1.11517 
1. 11452 
1. 11387 
1.11321 
1.11256 
1.11191 
1.11126 
l.rlo6l 



.90040 
.90093 
.90146 
.90199 
.90251 
.90304 
■90357 
.90410 
.90463 
.90516 
■90569 
■90621 
.90674 
.90727 
.90781 
.90834 



.90940 
■90993 
.91046 
.91099 

■91153 
.91206 
-91259 
-91313 
.91366 
-91419 
-91473 
.91526 
-91580 
■91633 
.91687 
■91740 
■91794 
.91847 
.91901 
■919SS 
■92008 
.92062 
.92116 
.92170 
.92224 
.92277 
-92331 
-92385 
■92439 
-92493 
-92547 
-92601 
-9265s 
.92709 

.92763 
.92817 
.92872 
.92926 
.92980 
-93034 



-93143 
-93197 
-93252 



1.11061 
1.10996 
1. 10931 
1.10867 
1.10802 
I.I0737 
1.10672 
1.10607 
I-10543 
1.10478 
1.10414 
1.10349 
1.1028s 
1. 10220 
1.10x56 
1.10091 
1.10027 
1.09963 
1.09899 
1.09834 
1.09770 
1.09706 
1.09642 
1.09578 
1.09514 
1 -09450 
1.09386 
1.09322 
1.09258 
1.09195 
1.09131 
1.09067 
1.09003 
1.08940 
1.08876 
1.08813 
1.08749 
1.08686 
1.08622 
1.08559 
1 .08496 

1.08432 
1.08369 
1.08306 
1.08243 
1.08179 
1.08116 
1.08053 
1.07990 
1.07927 
1.07864 
1.07801 
1-07738 
1.07676 
1.07613 
1.075,^0 
1.07487 
1-07425 
1.07362 
1.07299 
1.07237 



Co-TAN. Tan. Co-tan. Tan. 
47° 46° 



43° 

Tan. Co-tan. 



93252 
.93306 
.93360 
.93415 
.93469 
93524 
93578 
93633 
,93688 
93742 
93797 
93852 
.93906 
,93961 
.94016 
•94071 
.94125 
.94180 
.94235 
.94290 
.94345 
,94400 
94455 
94510 
94565 
,94620 
,94676 
94731 
94786 
94841 
,94896 

,94952 
,95007 
,95062 
.95118 
95173 
95229 
95284 
9S340 
95395 
.95451 
.95506 
.95562 
95618 
95673 
95729 
95785 
95841 
.95897 
95952 
96008 
,96064 
,96120 
,96176 
.96232 
.96288 
.96344 
.96400 
.96457 
96513 
96569 



1-07237 
1^07174 
1.07112 
1.07049 
1-06987 
1.06925 
1.06862 
1.06800 
1.06738 
1.06676 
1.06613 
1-06551 
1.06489 
1.06427 
1.0636s 
1.06303 
1. 06241 
1.06179 
1.06117 
1.06056 
1.05994 

1-05932 
1-05870 
1.05809 
1-05747 
1-05685 
1.05624 
1.05562 
1.05501 
1-05439 
1-05378 

1-05317 
1-05255 
1.05194 
1-05133 
1.05072 
1.05010 
1.04949 



1.04827 
1.04766 
1.0470s 
1 .04644 
1.04583 
1.04522 
1.04461 
1.04401 
1.04340 
1.04279 
1.04218 
1.04158 
1.04097 
1.04036 
1.03976 
I -0391s 
1-03855 
1.03794 
1-03734 
1-03674 
1-03613 
1-03553 





44° 






44° 




' 


Tan. 


Co-TAN. 


/ 


' 


Tan. 


CO-TAN- 


r 





96569 


1-03553 


60 


21 


97756 


1.02295 


39 


I 


96625 


1-03493 


59 


22 


97813 


1.02236 


38 


2 


96681 


1-03433 


58 


23 


97870 


1.02176 


37 


3 


96738 


1-03372 


57 


24 


97927 


I.02117 


36 


4 


96794 


1-03312 


56 


25 


97984 


1.02057 


35 


5 


96850 


1.03252 


55 


26 


98041 


1.01998 


34 


6 


96907 


1.03192 


54 


27 


98098 


1-01939 


3i 


7 


96963 


1-03132 


53 


28 


9815s 


1.01879 


32 


8 


97020 


1.03072 


52 


29 


98213 


1.01820 


31 


9 


97076 


1.03012 


51 


30 


98270 


1.01761 


30 


10 


97133 


1.02952 


50 


31 


98327 


1. 01 702 


29 


II 


97189 


1.02892 


49 


32 


98384 


1.01642 


28 


12 


97246 


1.02832 


48 


33 


98441 


1.01583 


27 


13 


97302 


1.02772 


47 


34 


98499 


1.01524 


26 


14 


97359 


1.02713 


46 


35 


98556 


1.01465 


2 5 


IS 


97416 


1.02653 


45 


36 


98613 


1. 01 406 


24 


16 


97472 


1 .02593 


44 


37 


98671 


1.01347 


23 


17 


97529 


1-02533 


43 


38 


98728 


1.01288 


22 


18 


97586 


1.02474 


42 


39 


98786 


1.01229 


21 


19 


97643 


1.02414 


41 


40 


98843 


I.OI170 


20 


20 


97700 


1-02355 


40 










' f 


o-tan. 


Tan. 


/ 


' Co-TAN . 


Tan. 


' 




45° 






45° 





44° 
Tan. Co-TAN. 



98901 
98958 
,99016 
99073 
99131 
99189 
.99247 
,99304 
99362 
.99420 

.99478 
.99536 

•99594 
.99652 
.99710 
99768 
99826 



.99942 



I.0III2 
I-OIO53 
1-00994 
1.00935 
1.00876 

1. 008 1 8 
1-00759 
1. 0070 1 
1.00642 
.1.00583 
1.0052s 
1.00467 
1.00408 
1.00350 
1.00291 
1.00233 
1.0017s 
1.00116 
1.00058 



Co-TAN. Tan. 
45° 



NATURAL SINES AND COSINES 





0° 






0° 








0° 






/ 


Sine 


Cosine 


' 


' 


Sine ( 


,.osine 


' 


/ 


Sine ( 


Cosine 


/ 





.00000 




60 


21 


.006 1 1 


99998 


39 


41 


■01193 


99993 


19 


1 


.00029 




59 


22 


.00640 


00008 


38 


42 


.01222 


99993 


iS 


2 


.00058 




58 


23 


.00669 


09998 


37 


43 


.01251 


99992 


17 


3 


.00087 




57 


24 


.00698 


99998 


36 


44 


.01280 


90992 


16 


4 


.00116 




56 


25 


.00727 


99997 


?,i 


45 


■01309 


99991 


IS 


S 


.00145 




55 


26 


.00756 


99997 


34 


46 


.01338 


99991 


14 


6 


.00175 




54 


27 


.00785 


99997 


33 


47 


■01367 


99991 


13 


7 


.00204 


I 


53 


28 


.00814 


99997 


32 


48 


■01396 


99990 


12 


8 


.00233 




52 


29 


.00844 


99996 


31 


40 


.01425 


99990 


11 


9 


.00262 




51 


30 


.00873 


99996 


30 


50 


•01454 


99989 


10 


10 


.00291 




50 


31 


.00902 


99996 


29 


51 


.01483 


99089 


9 


11 


.00320 


■99999 


49 


32 


-00031 


99996 


28 


52 


•01513 


99989 


8 


12 


■00349 


■99900 


48 


33 


.00960 


90995 


27 


53 


■01542 


90988 


7 


13 


■003 78 


-99900 


47 


34 


.00989 


99995 


26 


54 


.01571 


99988 


6 


14 


.00407 


-99000 


46 


.35 


.0T018 


99995 


2S 


55 


.01600 


99987 


5 


15 


.00436 


-99909 


45 


36 


.01047 


99995 


24 


56 


.01629 


99987 


4 


lb 


.00465 


-99909 


44 


37 


.01076 


00004 


23 


57 


.01658 


99986 


3 


17 


.00495 


-99999 


43 


38 


.01105 


09994 


22 


58 


.01687 


99986 


2 


18 


.00524 


.99900 


42 


30 


.01134 


99994 


21 


59 


.01716 


99085 


I 


19 


-OOSS3 


.99998 


41 


40 


.01164 


99993 


20 


60 


■01745 


99985 





20 


.00582 


•99998 


40 


















/ 


Cosine 
8 


Sine 
9° 


' 


' 


Cosine 
89= 


Sine 
3 


' 


' 


Cosine 
89= 


Sine 


/ 



512 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 













Table 5.— 


Nattiral Trigonometric Functions — (Continued) 














] 


° 


2° 1 


3° 1 


40 






5° 




6° 




70 , 


8° 




' 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


/ 


/ 


Sine ( 


:;osiNE 


Sine ( 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


' 


o 


•01745 


.99985 


.03490 


•99039 


•05234 


.90863 


.06976 


•90756 


60 





.08716 


99619 


10453 


99452 


12187 


.99255 


.13917 


.99027 


60 


I 


.01774 


.99984 


•03519 


.99938 


.05263 


.99861 


.07005 


•99754 


59 


I 


.08745 


99617 


10482 


00449 


12216 


.99251 


.13946 


.99023 


59 


2 


.01803 


.99984 


.03548 


•99937 


.05292 


.99860 


•07034 


•90752 


58 


2 


•08774 


99614 


10511 


99446 


12245 


.99248 


•1397s 


.99019 


58 


3 


.01832 


.99983 


•03577 


.99936 


.05321 


.99858 


•07063 


•90750 


57 


3 


.08803 


99612 


10540 


99443 


12274 


.99244 


.14004 


•99015 


57 


4 


.01862 


.99983 


•03606 


•99035 


•05350 


•99857 


.07092 


•99748 


S6 


4 


.08831 


99609 


10569 


99440 


12302 


.99240 


.14033 


.99011 


56 


5 


.01891 


.99982 


■03635 


•99934 


•05379 


•9985 s 


.07121 


.99746 


55 


S 


.08860 


99607 


10597 


99437 


12331 


■99237 


.14061 


.99006 


55 


6 


.01920 


.99982 


•03664 


•90033 


•05408 


.99854 


■07150 


.99744 


54 


i 6 


.08889 


99604 


10626 


99434 


12360 


■99233 


.14090 


.99002 


54 


7 


.01949 


.99981 


•03693 


.99932 


•05437 


•99852 


■07179 


.99742 


53 


7 


.08918 


99602 


10655 


99431 


12389 


■99230 


.14119 


.98998 


53 


8 


.01978 


.99980 


•03723 


•99931 


.05466 


.99851 


.07208 


•99740 


52 


8 


.08947 


99599 


10684 


99428 


12418 


.99226 


.14148 


•98994 


52 


9 


.02007 


.99980 


•03752 


•99930 


•05495 


■99849 


•07237 


•09738 


51 


9 


.08976 


99596 


10713 


99424 


12447 


.99222 


.14177 


.98990 


5' 


10 


.02036 


•99979 


.03781 


.99929 


•05524 


.99847 


•07266 


•99736 


50 


10 


.09005 


99594 


10742 


90421 


12476 


.99219 


.14205 


.98086 


SO 


11 


.02065 


.99979 


.03810 


.99927 


•05553 


.99846 


.07295 


•99734 


49 


11 


.09034 


99591 


10771 


99418 


12504 


•99215 


.14234 


.98982 


49 


12 


.02094 


•90078 


•03839 


.99926 


•05582 


•99844 


•07324 


•09731 


48 


12 


.09063 


99588 


10800 


00415 


12533 


.99211 


.14263 


.98978 


48 


13 


.02123 


•90977 


.03868 


•90925 


.05611 


.99842 


•07353 


•99729 


47 


13 


.09092 


99586 


10829 


99412 


12562 


.99208 


.14292 


•98973 


47 


14 


.02152 


■99977 


.03897 


•99924 


.05640 


.99841 


.07382 


.99727 


46 


14 


.09121 


99583 


10858 


99409 


12591 


.99204 


.14320 


.98969 


46 


15 


.02181 


.99976 


•03926 


•99923 


.05669 


•99839 


.07411 


•90725 


45 


15 


.09150 


99580 


10887 


99406 


12620 


.99200 


•14349 


.98965 


45 


i6 


.022H 


.99976 


•0395s 


.99922 


.05698 


•99838 


.07440 


•99723 


44 


16 


•09179 


99578 


10916 


99402 


12649 


•99197 


•14378 


.98961 


44 


17 


.02240 


■90975 


•03984 


•99921 


•05727 


.99836 


.07469 


•99721 


43 


! 17 


.09208 


90575 


1094s 


90399 


12678 


•99193 


•14407 


■98957 


43 


i8 


.02269 


.99974 


•04013 


.99919 


•05756 


•99834 


.07408 


•99719 


42 


I 18 


•09237 


99572 


10973 


99396 


12706 


.99189 


.14436 


■98953 


42 


19 


.02298 


.99974 


•04042 


.99018 


•05785 


•99833 


.07527 


.99716 


41 


19 


.09266 


99570 


11002 


99393 


12735 


.99186 


.14464 


.98948 


41 


20 


.02327 


■90973 


.04071 


.99917 


•05814 


.99831 


.07556 


•99714 


40 


20 


.09295 


99567 


11031 


99390 


12764 


.99182 


•14493 


.98944 


40 


21 


.02356 


.99972 


.04100 


.99916 


.05844 


.99829 


.07585 


•99712 


39 


21 


•09324 


99564 


11060 


99386 


12793 


•99178 


•14522 


.98940 


39 


22 


.02385 


.99972 


.04129 


•99015 


•05873 


.99827 


.07614 


.99710 


38 


22 


•09353 


99562 


1 1089 


90383 


12822 


•99175 


•14551 


.98936 


38 


23 


.02414 


•99971 


.04159 


•90913 


.05902 


.99826 


.07643 


.09708 


37 


23 


•09382 


99559 


11118 


00380 


12851 


.99171 


.14580 


.98931 


37 


24 


.02443 


.99970 


.04188 


•00012 


•05931 


.99824 


.07672 


•99705 


36 


24 


.09411 


99556 


11147 


09377 


12880 


•99167 


.14608 


.98927 


36 


25 


.02472 


.99969 


.04217 


•99911 


.05960 


.99822 


.07701 


•99703 


35 


25 


.09440 


90553 


11176 


99374 


12908 


.99163 


•14637 


.98923 


35 


26 


.02501 


•99969 


.04246 


.99910 


.05989 


.99821 


■07730 


•09701 


34 


26 


.09469 


99551 


11205 


903 70 


12937 


.99160 


.14666 


.98919 


34 


27 


.02530 


.99968 


•04275 


.99909 


.06018 


.99819 


•077S9 


.99699 


33 


27 


.09498 


99548 


11234 


99367 


12966 


•99156 


.14695 


.98914 


33 


28 


.02560 


.99967 


.04304 


.99907 


.06047 


.99817 


.07788 


.99696 


32 


28 


.09527 


9954S 


11263 


99364 


12905 


•99152 


.14723 


.98910 


32 


29 


.02589 


.99966 


•04333 


.99906 


.06076 


.99815 


.07817 


•99694 


31 


29 


•09556 


99542 


11291 


99360 


13024 


.99148 


■14752 


.98906 


31 


30 


.02618 


.99966 


.04362 


.99905 


.06105 


.99813 


.07846 


.99692 


30 


30 


•09385 


99540 


11320 


99357 


13053 


.99144 


.14781 


.98902 


30 


31 


.02647 


.99965 


■04391 


•99904 


.06134 


.99812 


•07875 


.99689 


29 


31 


.09614 


99537 


1 1349 


993S4 


1 308 1 


.99141 


.14810 


.98897 


29 


32 


.02676 


.99964 


.04420 


.99902 


.06163 


.99810 


.07004 


.99687 


28 


32 


.09642 


99534 


11378 


993SI 


13110 


•99137 


.14838 


.98893 


28 


33 


.02705 


.99963 


.04449 


.99901 


.06192 


.99808 


•07933 


.99685 


27 


33 


.09671 


09531 


11407 


09347 


13139 


•99133 


.14867 


.98889 


27 


34 


•02734 


.99963 


.04478 


.99900 


.06221 


.99806 


.07962 


.99683 


26 


34 


.09700 


99528 


1 1436 


99344 


13168 


•99129 


.14896 


.98884 


26 


35 


•02763 


.99962 


.04507 


.99898 


.06250 


.99804 


•07001 


.99680 


25 


35 


.09729 


99526 


11465 


99341 


13197 


•09125 


•14025 


.98880 


2S 


36 


.02792 


.99961 


■04536 


.99897 


.06279 


.90803 


•08020 


.99678 


24 


36 


.09758 


99523 


11494 


09337 


13226 


.99122 


• 14054 


.98876 


24 


37 


.02821 


.99960 


■04565 


.99896 


.06308 


.09801 


.08049 


.99676 


23 


37 


•09787 


99520 


11523 


99334 


13254 


.99118 


•14982 


.98871 


23 


38 


.02850 


•99959 


.04594 


.99894 


.06337 


•99799 


.08078 


•99673 


22 


38 


.09816 


90517 


11552 


99331 


13283 


.99114 


• 15011 


.98867 


23 


39 


.02879 


•990S9 


.04623 


.99893 


.06366 


.99797 


.08107 


•99671 


21 


39 


.09845 


99514 


11580 


99327 


13312 


.90110 


• 15040 


.98863 


21 


40 


.02908 


.99958 


•04653 


.99892 


•0639s 


•99795 


.08136 


.99668 


20 


40 


.09874 


99511 


11609 


99324 


13341 


.99106 


• 15069 


.98858 


20 


41 


.02938 


•99057 


.04682 


.99890 


.06424 


•99793 


.08165 


•99666 


19 


41 


.09903 


99508 


1 1638 


99320 


13370 


.99102 


•15097 


■98854 


19 


42 


.02967 


.99956 


.04711 


.90889 


•06453 


.99792 


.08194 


.99664 


18 


42 


•09932 


99506 


11667 


99317 


13399 


.99098 


• 15126 


.98849 


18 


43 


.02996 


•90055 


.04740 


.99888 


.06482 


.99790 


.08223 


.99661 


17 


43 


.09961 


99503 


11696 


99314 


13427 


.99094 


•15155 


.98845 


17 


44 


•03025 


.99954 


.04769 


.99886 


•06511 


.99788 


.08252 


.99659 


16 


44 


.09990 


99500 


11725 


99310 


13456 


.99091 


• 15184 


.98841 


16 


45 


■03054 


•99953 


.04798 


.90885 


.06540 


.99786 


.08281 


•99657 


15 


45 


.10019 


99497 


11754 


99307 


13485 


.99087 


.15212 


.98836 


IS 


46 


.03083 


•99952 


.04827 


.90883 


.06569 


•99784 


.08310 


.99654 


14 


46 


.10048 


90494 


11783 


99303 


13514 


•99*=83 


.15241 


.98832 


14 


47 


•03112 


.99952 


.04856 


.99882 


.06598 


.99782 


.08339 


.99652 


13 


47 


.10077 


99491 


11812 


99300 


13543 


.99079 


.15270 


.98827 


13 


48 


•03141 


■9905 1 


.04885 


.99881 


.06627 


.99780 


.08368 


.99649 


12 


48 


.10106 


99488 


11840 


99297 


13572 


•99075 


•15299 


.98823 


12 


49 


•03170 


.99950 


.04914 


.99879 


.06656 


.99778 


.08397 


■99647 


11 


49 


•I0135 


90485 


11869 


99293 


13600 


•99071 


•15327 


.98818 


11 


SO 


.03199 


•90949 


.04943 


.99878 


.06685 


.99776 


.08426 


.99644 


10 


SO 


.10164 


99482 


1 1898 


99290 


13629 


.99067 


•15356 


.98814 


10 


SI 


.03228 


.99948 


•04972 


.99876 


.06714 


■99774 


•08455 


.99642 


9 


51 


.10192 


99479 


11927 


99286 


13658 


■99063 


•1538s 


.98809 


9 


S2 


•03257 


•90947 


.05001 


•00875 


•06743 


•09772 


.08484 


.99639 


8 


52 


.10221 


99476 


11956 


09283 


13687 


•99059 


•15414 


.98805 


8 


S3 


.03286 


■99946 


.05030 


•90873 


•06773 


.99770 


.08513 


•99637 


7 


S3 


.10250 


99473 


11085 


99279 


13716 


■99055 


•15442 


.98800 


7 


S4 


.03316 


•90945 


.05059 


.99872 


.06802 


.99768 


.08542 


•99635 


6 


54 


.10279 


99470 


12014 


99276 


13744 


.99051 


•15471 


.98796 


6 


55 


•03345 


•90044 


.05088 


.99870 


.06831 


.99766 


.08571 


.99632 


5 


55 


.10308 


99467 


12043 


99272 


13773 


.99047 


• 15500 


.98791 


5 


S6 


•03374 


•90943 


.05117 


.99869 


.06860 


•99764 


.08600 


.99630 


4 


56 


.10337 


99464 


12071 


99269 


13802 


.99043 


•15529 


.98787 


4 


57 


•03403 


■99942 


.05146 


.99867 


.06889 


.99762 


.08629 


.99627 


3 


57 


.10366 


99461 


12100 


99265 


13831 


.99039 


•15557 


.98782 


3 


S8 


•03432 


.99941 


•0517s 


.99866 


.06918 


.99760 


.08658 


■99625 


2 


58 


•10395 


99458 


12129 


99262 


13860 


.90035 


• 15586 


■98778 


3 


59 


•03461 


.99940 


.05205 


.99864 


.06947 


.99758 


.08687 


.99622 


I 


59 


.10424 


99455 


12158 


99258 


13889 


■90031 


.15615 


•98773 


I 


60 


•03490 


■99939 


•05234 


.99863 


.06976 


•99756 


.08716 


.99619 





60 


.10453 


99452 


12187 


99255 


13917 


.99027 


•15643 


.98769 





"7~ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sink 


/ 


/ 


Cosine 


Sine ( 


!;OSINE 


Sine C 


'OSINE 


Sine 


Cosine 


Sine 


f 




8^ 


J° 


8 


7° 


8t 


3° 


& 


5° 






84° 




83° 




& 


1° 


8] 








MATHEMATICAL TABLES 



513 













Table 5. — Natural Trigonometric Functions — 


{Contin 


ued) 














9° 


10° 




11° 


12° 






13° 1 


14 





15° 1 


16° 




' 


Sine 


Cosine 


Sine ( 


::;osine 


Sine 


Cosine 


Sine 


Cosine 


1 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


f 


o 


•15643 


.98769 


•17365 


98481 


.19081 


■98163 


.20791 


.97815 


60 





•22495 


.97437 


•24192 


.97030 


.25882 


.96593 


.27564 


.96126 


60 


1 


.15672 


.98764 


•17393 


98476 


.19109 


■98157 


.20820 


.97809 


59 


I 


.22523 


.97430 


.24220 


■97023 


•25910 


.96585 


.27592 


.96118 


59 


a 


■15701 


.98760 


.17422 


98471 


.19138 


■98152 


•20848 


.97803 


58 


2 


.22552 


.97424 


.24249 


■97015 


•25938 


.96578 


.27620 


.96110 


58 


3 


•15730 


•98755 


•17451 


98466 


.19167 


.98146 


.20877 


•97797 


57 


3 

4 


.22580 


.97417 


.24277 


■97008 


.25966 


.96570 


.27648 


.96102 


57 


4 


•15758 


•98751 


•17479 


98461 


•1919^ 


.98140 


•20905 


•97791 


56 


.22608 


.97411 


.24305 


■97001 


.25994 


.96562 


.27676 


.96094 


56 


S 


.15787 


.98746 


•17508 


98455 


.19224 


■98135 


•20933 


•97784 


55 


5 


.22637 


.97404 


.24333 


.96994 


.26022 


■96555 


.27704 


.96086 


55 


6 


.15816 


•98741 


•I7S37 


98450 


.19252 


.98129 


.20962 


•97778 


54 


6 


.22665 


•97398 


.24362 


.96987 


.26050 


■96547 


.27731 


.96078 


54 


7 


.15845 


•98737 


•17565 


98445 


.19281 


.98124 


.20990 


•97772 


S3 


I 


.22693 


•97391 


.24390 


.96980 


.26079 


■96540 


.27759 


.96070 


53 


8 


.15873 


•98732 


•17594 


98440 


.19309 


.98118 


.21019 


•97766 


52 


.22722 


•97384 


.24418 


•96973 


.26107 


■96532 


.27787 


.96062 


52 





.15902 


.98728 


.17623 


98435 


.19338 


.98112 


.21047 


.97760 


51 


9 

10 


.22750 


■97378 


.24446 


•96966 


.26135 


.96524 


.27815 


.96054 


51 


lO 


•15931 


•98723 


.17651 


98430 


.19366 


.98107 


.21076 


•97754 


SO 


.22778 


■97371 


.24474 


•96959 


.26163 


■96517 


.27843 


.96046 


SO 


II 


.15959 


.98718 


.17680 


98425 


.19395 


.98101 


.21104 


.97748 


49 


It 


.22807 


■97365 


.24503 


•96952 


.26191 


■96509 


.27871 


.96037 


49 


la 


.15988 


.98714 


.17708 


98420 


.19423 


.98096 


.21132 


.97742 


48 


12 


.22835 


■97358 


.24531 


.96945 


.26219 


.96502 


.27899 


•96029 


48 


13 


.16017 


.98709 


.17737 


98414 


.19452 


.98090 


.21161 


•97735 


47 


13 

14 


.22863 


•97351 


.24559 


•96937 


.26247 


■96494 


.27927 


.96021 


^Z 


14 


.16046 


.98704 


.17766 


98409 


.19481 


.98084 


.21189 


•97729 


46 


.22892 


•97345 


.24587 


•96930 


.26275 


.96486 


.2795s 


.96013 


46 


15 


.16074 


.98700 


.17794 


98404 


.19509 


.98079 


.21218 


•97723 


45 


15 


.22920 


•97338 


.24615 


.96923 


.26303 


.96479 


■27983 


.96005 


45 


i6 


.16103 


•98695 


.17823 


98399 


.19538 


•98073 


.21246 


•97717 


44 


16 


.22948 


.97331 


.24644 


.96916 


.26331 


.96471 


.2801X 


.95997 


44 


17 


.16132 


.98690 


.17852 


98394 


.19566 


.98067 


.21275 


•97711 


43 


17 


•22977 


■97325 


.24672 


.96909 


.26359 


.96463 


.28039 


.95989 


43 


i8 


.16160 


•98689 


.17880 


98389 


.19595 


.98061 


.21303 


.97705 


42 


18 


.23005 


■97318 


.24700 


.96902 


.26387 


.96456 


.28067 


.95981 


42 


19 


.16189 


•98681 


.17909 


98383 


.19623 


.98056 


.21331 


.97698 


41 


IQ 


.23033 


■9731 1 


.24728 


.96894 


.26415 


.96448 


.28095 


.95972 


41 


20 


.16218 


.98676 


.17937 


98378 


.19652 


.98050 


.21360 


.97692 


40 


20 


.23062 


■97304 


.24756 


.96887 


.26443 


.96440 


.28123 


.95964 


40 


21 


.16246 


•98671 


.17966 


98373 


.19680 


•98044 


.21388 


.97686 


39 


21 


.23090 


■97298 


.24784 


.96880 


.26471 


.96433 


.28150 


.95956 


39 


22 


.1627s 


.98667 


.17995 


98368 


.19709 


•98039 


.21477 


.97680 


38 


22 


■23118 


■97291 


.24813 


.96873 


.26500 


.96425 


.28178 


.95948 


38 


23 


.16304 


.98662 


.18023 


98362 


.19737 


.98033 


.21445 


•97673 


37 


23 


.23146 


.97284 


.24841 


.96866 


.26528 


.96417 


.28206 


.95940 


37 


24 


.■16333 


•98657 


.18052 


98357 


.19766 


•98027 


.21474 


.97667 


36 


24 


.23175 


.97278 


.24869 


.96858 


.26556 


.96410 


.28234 


.95931 


36 


2S 


.16361 


•98652 


.18081 


98352 


.19794 


.98021 


.21502 


.97661 


3S 


25 


.23203 


.97271 


.24897 


.96851 


.26584 


.96402 


.28262 


.95923 


3S 


26 


.16390 


.98648 


.18109 


98347 


.19823 


.98016 


.21530 


•97655 


34 


26 


.23231 


.97264 


.24925 


.96844 


.26612 


.96394 


.28290 


•95915 


34 


27 


.16419 


.98643 


.18138 


98341 


.19851 


.98010 


.21559 


•97648 


33 


27 


.23260 


•97257 


.24954 


•96837 


.26640 


.96386 


.28318 


•95907 


33 


28 


.16447 


.98638 


.18166 


98336 


.19880 


.98004 


.21587 


•97642 


32 


28 


.23288 


.97251 


.24982 


.96829 


.26668 


.96379 


.28346 


.95898 


32 


29 


.16476 


•98633 


.18195 


98331 


.19908 


.97987 


.21616 


.97636 


31 


2Q 


.23316 


.97244 


.25010 


•96822 


.26696 


.96371 


.28374 


.95890 


31 


30 


.16505 


.98629 


.18224 


98325 


■19937 


.97992 


.21644 


•97630 


30 


30 


.23345 


.97237 


.25038 


•96815 


.26724 


.96363 


.28402 


.95882 


30 


31 


•16533 


.98624 


.18252 


98320 


.19965 


.97987 


.21672 


.97623 


20 


31 
32 


.23373 


.97230 


.25066 


.96807 


.26752 


.96355 


.28429 


•95874 


'^t 


32 


.16562 


.98619 


.18281 


98315 


.19994 


.97981 


.21701 


.97617 


28 


.23401 


.97223 


.25094 


.96800 


.26780 


.96347 


.28457 


•95865 


28 


33 


.16591 


.98614 


.18309 


98310 


.20022 


■97975 


.21729 


.97611 


27 


34 


.23429 


.97217 


.25122 


.96793 


.26808 


.96340 


.28485 


•95857 


27 


34 


.16620 


.98609 


.18338 


98304 


.20031 


■97969 


.21758 


.97604 


26 


.23458 


.97210 


.25151 


.96786 


.26836 


.96332 


.28513 


•95849 


26 


35 


.16648 


.98604 


.18367 


98299 


.20079 


■97963 


.21786 


•97598 


2S 


35 


.23486 


.97203 


.25179 


.96778 


.26864 


.96324 


.28541 


•95841 


25 


36 


.16677 


.98600 


.18395 


98294 


.20108 


■97958 


.21814 


•97592 


24 


36 
37 
38 


.23514 


.97196 


.25207 


•96771 


.26892 


.96316 


.28569 


•95832 


24 


37 


.16706 


■98595 


.18424 


98288 


.20136 


■97952 


.21843 


■97585 


23 


.23542 


.97189 


.25235 


.96764 


.26920 


.96308 


.28597 


.95824 


23 


38 


.16734 


.98590 


.18452 


98283 


.20165 


•97946 


.21871 


■97579 


23 


.23571 


.97182 


.25263 


.96756 


.26948 


.96301 


.28625 


.95816 


22 


39 


.16763 


•98585 


.18481 


98277 


.20193 


•97940 


.21899 


■97573 


21 


39 


■23599 


.97176 


.25291 


.96749 


.26976 


.96293 


.28652 


.95807 


21 


40 


.16792 


.98580 


.18509 


98272 


.20222 


■97934 


.21928 


■97566 


20 


40 


.23627 


.97169 


.25320 


•96742 


.27004 


.96285 


.28680 


.95799 


20 


41 


.16820 


■98575 


.18538 


98267 


.20250 


■97928 


.21956 


■97560 


19 


41 


.23656 


.97162 


.25348 


•96734 


.27032 


.96277 


.28708 


.95791 


19 


42 


.16849 


■98570 


.18567 


98261 


.20279 


■97922 


.21085 


■97553 


18 


42 


.23684 


.97155 


.25376 


•96727 


•27060 


.96269 


.28736 


■95782 


18 


43 


.16878 


■98565 


• 18595 


98256 


.20307 


•97916 


.22013 


■97547 


17 


43 


•23712 


.97148 


.25404 


•96719 


J 7088 


.96261 


.28764 


■95774 


17 


44 


.16906 


■98561 


.18624 


98250 


.20336 


•97910 


.22041 


•97541 


16 


44 


•23740 


.97141 


•25432 


•96712 


.27116 


.96253 


.28792 


.95766 


16 


45 


.16935 


■98556 


.18652 


98245 


.20364 


•97905 


.22070 


•97534 


IS 


45 


.23769 


.97134 


.25460 


•96705 


.27144 


.96246 


.28820 


.95757 


15 


46 


.16964 


■98551 


.18681 


98240 


.20393 


■97899 


.22098 


■97528 


14 


46 


.23797 


.97127 


.25488 


.96697 


.27172 


.96238 


.28847 


.95749 


14 


47 


.16992 


■98546 


.18710 


98234 


.20421 


•97893 


.22126 


•97521 


13 


47 


.23825 


.97120 


.25516 


.96690 


.27200 


.96230 


.28875 


.95740 


13 


48 


.17021 


.98541 


.18738 


98229 


.20450 


•97887 


.22155 


•97515 


12 


48 


.23853 


.97113 


.25545 


.96682 


.27228 


.96222 


.28903 


.95732 


12 


49 


.17050 


■98536 


.18767 


98223 


.20478 


•97881 


.22183 


•97508 


II 


49 


.23882 


.97106 


.25573 


.96675 


.27256 


.96214 


.28931 


.95724 


II 


SO 


.17078 


■98531 


.18795 


98218 


.20507 


•97875 


.22212 


■97502 


10 


50 


.23910 


.97100 


.25601 


.96667 


.27284 


.96206 


.28959 


.95715 


10 


SI 


.17107 


.98526 


.18824 


98212 


.20535 


•97869 


.22240 


.97496 


9 


51 


.23938 


.97093 


.25629 


.96660 


.27312 


.96198 


.28987 


■95707 


9 


S2 


.17136 


.98521 


.18852 


98207 


.20563 


•97863 


.22268 


■97489 


8 


52 


.23966 


.97086 


.25657 


.96653 


■27340 


.96190 


.29015 


■95698 


8 


53 


.17164 


.98516 


.18881 


98201 


.20592 


•97857 


.22297 


■97483 


7 


53 


.23995 


.97079 


.25685 


•96645 


.27368 


.96182 


.29042 


.95690 


7 


54 


•17193 


.98511 


•18910 


98196 


.20620 


•97851 


.22325 


■97476 


6 


54 


.24023 


.97072 


.25713 


•96638 


.27396 


■96174 


.29070 


.95681 


6 


55 


.17222 


.98506 


.18938 


98190 


.20649 


■97845 


.22353 


■97470 


S 


55 


.24051 


.97065 


.25741 


.96630 


.27424 


.96166 


.29098 


.95673 


S 


56 


.17250 


■98501 


.18967 


98185 


.20677 


■97839 


.22382 


•97463 


4 


56 


.24079 


.97058 


.25769 


.96623 


.27452 


.96158 


.29126 


.95664 


4 


57 


.17279 


.98496 


.18995 


98179 


.20706 


■97833 


.22410 


•97457 


3 


57 


.24108 


.97051 


.25798 


.96615 


.27480 


.96150 


.29154 


■95656 


3 


58 


.17308 


.98491 


.19024 


98174 


■20734 


■97827 


.22438 


•97450 


2 


S8 


.24136 


.97044 


.25826 


.96608 


.27508 


.96142 


.29182 


•95647 


2 


59 


■17336 


.98486 


.19052 


98168 


■20763 


.97821 


-02467 


.97444 


I 


59 


.24164 


•97037 


■25854 


•96600 


.27536 


.96134 


.29209 


•95639 


I 


60 


■17365 


.98481 


.10081 


98163 


.20791 


■97815 


.22495 


■97437 





60 


.24192 


•97030 


.25882 


•96593 


.27564 


.96126 


.29237 


•95630 







Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


~^ 




a 


)° 


79° 




7{ 


i° 


r 


r° 






7 


5° 


7. 


5° 


7 


do 


7 


3° 





33 



514 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 













Table 5. — Natural Trigonometric Functions — 


'Continued) 














17° 1 


18° 


19° 1 


20° 






21 





22° II 


23° 




24«' 




/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


/ 


t 


Sine 


CosnJE 


Sine 


Cosine 


Sine C 


'OSINE 


Sine 


Cosine 


t 


o 


•29237 


■95630 


■30902 


.95106 


■32557 


■94552 


.34202 


.93969 


60 





•35837 


•93358 


•37461 


.92718 


■39073 


92050 


.40674 


■91355 


60 


I 


.29265 


■95622 


■30929 


■95097 


.32584 


.94542 


.34229 


■93959 


S9 


I 


•35864 


•93348 


•37488 


.92707 


■39100 


92039 


.40700 


■91343 


59 


2 


.29293 


•95613 


■30957 


.95088 


.32612 


■94533 


•34257 


•93949 


S8 


3 


•35891 


•93337 


•37515 


.92697 


■39127 


92028 


■40727 


■91331 


58 


3 


■29321 


■95605 


■3098s 


■95079 


•32639 


■94523 


•34284 


•93939 


57 


3 


•35918 


•93327 


•37542 


.92686 


■39153 


92016 


■40753 


■91319 


S7 


4 


.29348 


■95596 


■31012 


.95070 


•32667 


•94514 


•343 1 1 


■93929 


S6 


4 


■35945 


•93316 


•37569 


.92675 


■39180 


92005 


■40780 


■91307 


56 


5 


.29376 


•95588 


■31040 


■95061 


.32694 


•94504 


•34339 


•93919 


55 


S 


.35973 


.93306 


•37595 


.92664 


■39207 


91994 


.40806 


■91295 


ss 


6 


.29404 


•95579 


.31068 


■95052 


■32722 


•94495 


•34366 


■93909 


54 


6 


.36000 


.9329s 


•37622 


.92653 


■39234 


91982 


■40833 


■91283 


54 


7 


.29432 


•95571 


■31095 


■95043 


•32749 


■94485 


•34393 


•93899 


53 


7 


.36027 


■93285 


•37649 


.92642 


■39260 


91971 


■40860 


■91272 


53 


8 


.29460 


■95562 


-31123 


-95033 


•32777 


•94476 


•34421 


■93889 


52 


8 


.36054 


■93274 


.37676 


■92631 


■39287 


91959 


.40886 


.91260 


52 


9 


.29487 


■95554 


■31151 


•95024 


•32804 


•94466 


•34448 


■93879 


51 


9 


.36081 


■93264 


•37703 


.92620 


■39314 


91948 


.40913 


.91248 


51 


lO 


•29515 


•95545 


■31178 


■95015 


•32832 


•94457 


•34475 


-93869 


SO 


10 


.36108 


■93253 


•37730 


.92609 


•39341 


91936 


•40939 


.91236 


50 


II 


•29543 


■95536 


■31206 


.95006 


■32859 


•94447 


•34503 


■93859 


49 


II 


■36135 


.93243 


•37757 


■92598 


■30367 


91925 


.40966 


.91224 


■♦s 


12 


•29571 


■95528 


■31233 


■94997 


■32887 


.94438 


•34530 


.93849 


48 


12 


.36162 


■93232 


•37784 


■92587 


■39394 


91914 


.40992 


.91212 


48 


13 


•29599 


■95519 


.31261 


■94988 


■32914 


■94428 


■34557 


-93839 


47 


13 


■36190 


.93222 


•37811 


■92576 


.39421 


91902 


.41019 


.91200 


47 


14 


.29626 


■9S5II 


.31289 


■94979 


■32942 


.94418 


■34584 


.93829 


46 


14 


.36217 


■93211 


•37838 


■92565 


■39448 


91891 


.41045 


.91188 


46 


15 


.29654 


■95502 


■31316 


■94970 


■32969 


■94409 


■34612 


•93819 


45 


15 


•36244 


■93201 


•3786s 


■92554 


■39474 


91879 


.41072 


.91176 


45 


l6 


.29682 


■95493 


■31344 


■94961 


■32997 


■94399 


-34639 


■93809 


44 


16 


■36271 


■93190 


.37892 


.92543 


■39501 


91868 


.4109S 


.91164 


44 


17 


.29710 


■95485 


■31372 


■94952 


■33024 


.94390 


■34666 


■93799 


43 


17 


.36298 


.93180 


•37919 


.92532 


■39528 


91856 


•41125 


.91152 


43 


i8 


•29737 


■95476 


•31399 


•94943 


■33051 


.94380 


-34694 


■93789 


42 


18 


■36325' 


■93169 


•37946 


.92521 


■39555 


91845 


•41151 


.91 140 


42 


19 


.29765 


■95467 


■31427 


•94933 


■33079 


■94370 


•34721 


•93779 


41 


19 


■36352 


■93159 


•37973 


■92510 


39581 


91833 


.41178 


.91128 


41 


20 


•29793 


•95459 


•31454 


.94924 


■33106 


-94361 


•34748 


.93769 


40 


20 


•36379 


•93148 


•37999 


•92499 


.39608 


91822 


.41204 


.91116 


40 


21 


•29821 


•95450 


■31482 


•9491S 


■33134 


-94351 


•34775 


.93759 


39 


21 


.36406 


.93137 


.38026 


■92488 


•39635 


91810 


•41231 


.91104 


^ 


22 


.29849 


■93441 


■31510 


.94906 


■33161 


■94342 


•34803 


.93748 


38 


22 


.36434 


.93127 


■38053 


•92477 


•39661 


91799 


-41257 


.91092 


38 


23 


.29876 


■95433 


•31537 


.94897 


■33189 


-94332 


-34830 


•93738 


37 


23 


.36461 


.93116 


■38080 


.92466 


.39688 


91787 


.41284 


.91080 


37 


24 


.29904 


•95424 


•3156s 


.94888 


■33216 


-94322 


-34857 


.93728 


36 


24 


.36488 


.93106 


•38107 


■92455 


•3971S 


91775 


•41310 


.91068 


36 


25 


.29932 


•95415 


•31593 


.94878 


■33244 


■94313 


-34884 


.93718 


35 


25 


■36515 


•93095 


•38134 


■92444 


•39741 


91764 


•41337 


.91056 


3S 


26 


.29960 


•95407 


•31620 


.94869 


-33271 


-94303 


-34912 


■93708 


34 


26 


•36542 


■93084 


.38161 


■92432 


•39768 


91752 


.41363 


.91044 


34 


27 


.29987 


•95398 


•31648 


■94860 


■33298 


■94293 


•34939 


■93698 


33 


27 


•36569 


■93074 


.38188 


■92421 


•3979S 


9174I 


.41390 


.91032 


33 


28 


.30015 


•95389 


■3167s 


-94851 


■33326 


.94284 


•34966 


.93688 


32 


28 


•36596 


.93063 


■38215 


.92410 


.39822 


91729 


.41416 


.91020 


32 


29 


•30043 


-95380 


■31703 


.94842 


•33353 


•94274 


•34993 


■93677 


31 


29 


.36623 


■93052 


■38241 


■92399 


.39848 


91718 


•41443 


.91008 


31 


3° 


.30071 


•95372 


•31730 


.94832 


■33381 


.94264 


■35021 


■93667 


30 


30 


■36650 


■93042 


.38268 


.92388 


•39875 


91706 


.41469 


.90996 


30 


31 


•30098 


■95363 


•31758 


•94823 


•33408 ■ 


•94254 


■35048 


■93657 


29 


31 


•36677 


•93031 


■3829s 


■92377 


•39902 


91694 


.41496 


.90984 


29 


32 


•30126 


■95354 


.31786 


.94814 


•33436 


•94245 


■35075 


■93647 


28 


1 32 


.36704 


.93020 


■38322 


■92366 


■39928 


91683 


.41522 


.90972 


28 


33 


•30154 


•95345 


■31813 


.94805 


•33463 


•94235 


•35102 


■93637 


27 


1 33 


■36731 


.93010 


•38349 


.92355 


■39955 


91671 


.41549 


.90960 


27 


34 


•30182 


■95337 


■31841 


•94795 


•33490 


.94225 


•35130 


■93626 


26 


34 


■36758 


.92999 


•38376 


■92343 


■39982 


91660 


•41575 


.90948 


26 


35 


•30209 


•95328 


.31868 


■94786 


•33518 


•94215 


•35157 


.93616 


25 


35 


■36785 


.92988 


•38403 


■92332 


.40008 


91648 


.41602 


■90936 


2S 


36 


■30237 


■95319 


■31896 


■94777 


•33545 


■94206 


•35184 


.93606 


24 


36 


.36812 


•92978 


•38430 


■92321 


■4003s 


91636 


.41628 


.90924 


24 


37 


.30265 


•95310 


■31923 


.94768 


•33573 


.94196 


•35211 


■93596 


23 


37 


■36839 


.92967 


•38456 


■92310 


.40062 


91625 


•41655 


.90911 


23 


38 


.30292 


■95301 


■31951 


■94758 


•33600 


.94186 


•35239 


■9358s 


22 


38 


■36867 


■92956 


.38483 


■92299 


.40088 


91613 


.41681 


.90899 


22 


39 


.30320 


■95293 


■31979 


■94749 


•33627 


•94176 


•35266 


.93575 


21 


39 


■36894 


■92945 


■38510 


.92287 


.40115 


91601 


•41707 


.90887 


21 


40 


•30348 


■95284 


.32006 


■94740 


•33655 


•94167 


•35293 


■9356s 


20 


40 


.36921 


■92935 


■38537 


■92276 


.40141 


91590 


•41734 


■90875 


20 


41 


■30376 


■95275 


•32034 


■94730 


•33682 


•94IS7 


•35320 


■9355S 


'? 


41 


■36948 


.92924 


■38564 


■92265 


.40168 


91578 


.41760 


■90863 


19 


42 


•30403 


■95266 


.32061 


.94721 


•33710 


•94147 


•35347 


■93544 


18 


42 


■3697s 


■92913 


■38591 


■92254 


.40195 


91566 


.41787 


■90851 


18 


43 


•30431 


■95257 


.32089 


.94712 


•33737 


•94137 


•35375 


■93534 


17 


43 


■37002 


.92902 


■38617 


■92243 


.40221 


9155s 


.41813 


.90839 


17 


44 


■30459 


■95248 


.32116 


.94702 


•33764 


•94127 


•35402 


•93524 


16 


44 


■37029 


.92892 


■38644 


■92231 


.40248 


91543 


.41840 


.90826 


16 


45 


■30486 


■95240 


•32144 


■94693 


•33792 


.94118 


•35429 


•93514 


IS 


45 


■37056 


.92881 


■38671 


.92220 


■40275 


91531 


.41866 


.90814 


IS 


46 


■30514 


■95231 


•32171 


.94684 


•33819 


.94108 


•35456 


•93503 


14 


46 


•37083 


.92870 


■38698 


.92209 


■40301 


91519 


.41892 


.90802 


14 


47 


■30542 


■95222 


■32199 


•94674 


•33846 


.94098 


•35484 


•93493 


13 


47 


•37110 


■92859 


•38725 


.92198 


■40328 


91508 


.41919 


■90790 


13 


48 


■30570 


■95213 


■32227 


.94665 


•33874 


.94088 


•35511 


•93483 


12 


48 


•37137 


.92849 


•38752 


.92186 


.40355 


91496 


•41945 


.90778 


12 


49 


■30597 


■95204 


■32254 


•94656 


•33901 


.94078 


•35538 


•93472 


II 


1 49 


•37164 


■92838 


•38778 


.92175 


.40381 


91484 


.41072 


.90766 


II 


5° 


.30625 


•95195 


■32282 


.94646 


•33929 


.94068 


•35565 


.93462 


10 


50 


•37191 


■92827 


•3880s 


.92164 


.40408 


91472 


.41998 


•90753 


10 


SI 


•30653 


■95186 


.32309 


•94637 


•33956 


.94058 


•35592 


■93452 


9 


SI 


•37218 


.92816 


•38832 


.92152 


■40434 


91461 


.42024 


■90741 


9 


52 


■30680 


■95177 


.32337 


.94627 


•33983 


•94049 


•35619 


•93441 


8 


S2 


•37245 


■92805 


-38859 


.92141 


.40461 


91449 


.42051 


.90729 


8 


53 


.30708 


.95168 


•32364 


.94618 


.34011 


.94039 


•35647 


•93431 


7 


S3 


.37272 


.92794 


.38886 


.92130 


.40488 


91437 


.42077 


.90717 


7 


54 


■30736 


■95159 


■32392 


.94609 


•34038 


.94029 


■35674 


•93420 


6 


54 


•37299 


-92784 


.38912 


.92119 


■40514 


91425 


.42104 


.90704 


6 


55 


•30763 


•95150 


■32419 


•94599 


•34065 


.94019 


■35701 


■93410 


S 


55 


-37326 


.92773 


-38939 


.92107 


.40541 


91414 


•42130 


.90692 


S 


56 


■30791 


■95142 


-32447 


•94590 


•34093 


.94009 


■35728 


■93400 


4 


56 


•37353 


.92762 


.38966 


.92096 


■40567 


91402 


■42156 


.90680 


4 


57 


■30819 


■95133 


•32474 


■94580 


•34120 


■93999 


■35755 


■93389 


3 


57 


•37380 


•92751 


•38993 


.92085 


■40594 


91390 


■42183 


.90668 


3 


58 


■30846 


■95124 


■32502 


•94571 


•34147 


■93989 


■35782 


•93379 


2 


58 


-37407 


•92740 


.39020 


■92073 


.4062 1 


91378 


■42209 


•9065s 


2 


59 


.30874 


•95115 


■32529 


■94561 


.34175 


•93979 


■35810 


.93368 


I 


59 


.37434 


•92729 


.39046 


.92062 


.40647 


91366 


■42235 


.90643 


I 


60 


.30902 


■95106 


■32557 


■94552 


■34202 


.93969 


■35837 


•93358 





60 


•37461 


.92718 


.39073 


.92050 


.40674 


91355 


.42262 


.90631 





f 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sdie 


Cosine 


Sine 


t 


/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


' 




7 


2° • 


7 


\° 


7 


D° 1 


6 


9° 






68 





6 


7° 


66° 




6 


5° 





MATHEMATICAL TABLES 



515 













Table s^ — 


Natural Trigonometric Functions- 


-{Continued) 














25° 




26° 1 


27° 1 


28° 






29° I 


30° 




31° 




32° 






1 


Sine ( 


ZOSINE 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


' 


/ 


Sine 


Cosine 


Sine C 


'OSINE 


Sine ( 


Cosine 


Sine ( 


-OSINE 


9 


o 


.42262 


90631 


■43837 


.89879 


•45399 


.89101 


.46947 


.88295 


60 





.48481 


.87462 


.50000 


86603 


•S1504 


85717 


.52992 


84805 


6c 


I 


.42288 


90618 


■43863 


.89867 


■45425 


.89087 


■46973 


.88281 


59 


1 


.48506 


.87448 


.50025 


86588 


•51529 


85702 


.53017 


84789 


59 


2 


•42315 


90606 


■43889 


.89854 


■45451 


.89074 


.46999 


.88267 


58 


2 


•48532 


■87434 


.50050 


86573 


•5I5S4 


85687 


.53041 


84774 


58 


3 


•42341 


90594 


.43916 


.89841 


■45477 


.89061 


■47024 


.88254 


57 


3 


•48557 


.87420 


.50076 


86559 


.51579 


85672 


.53066 


84759 


57 


4 


.42367 


90582 


■43942 


.89828 


■45503 


.89048 


47050 


.88240 


S6 


4 


•48583 


.87406 


•50I0I 


86544 


.51604 


85657 


.53091 


84743 


56 


5 


•42394 


90569 


■43968 


.89816 


■45529 


■8903s 


.47076 


.88226 


S% 


5 


.48608 


■87391 


.50126 


86530 


•51628 


85642 


.53115 


84728 


55 


■ 6 


.42420 


90557 


•43994 


.89803 


•45554 


.89021 


.47101 


.88213 


54 


6 


.48634 


•87377 


•50151 


8651S 


.51653 


85627 


.53140 


84712 


54 


7 


.42446 


90545 


.44020 


•89790 


•45580 


.89008 


.47127 


.88199 


S3 


7 


.48659 


■87363 


•50176 


86501 


.51678 


85612 


.53164 


84697 


53 


8 


•42473 


90532 


.44046 


•89777 


.45606 


.88995 


•47153 


.8818s 


52 


8 


.48684 


•87349 


.50201 


86486 


•51703 


85597 


.53189 


84681 


52 


9 


.42499 


90520 


.44072 


•89764 


•45632 


.88981 


•47178 


.88172 


51 


9 


.48710 


.87335 


.50227 


86471 


.51728 


85582 


.53214 


84666 


51 


lO 


•42525 


90507 


.44098 


.89752 


.45658 


.88968 


47204 


.88158 


SO 


10 


•48735 


■87321 


.50252 


86457 


•SI753 


85567 


.53238 


84650 


SO 


II 


.42552 


9049s 


.44124 


•89739 


•45684 


■88955 


47229 


.88144 


49 ' 


II 


.48761 


■87306 , 


•50277 


86442 


■5 1 778 


85551 


.53263 


8463s 


49 


12 


•42578 


90483 


■44151 


•89726 


•45710 


.88942 


•47255 


.88130 


48 


12 


.48786 


•87292 


•50302 


86427 


.51803 


85536 


.53288 


84619 


48 


13 


.42604 


90470 


■44177 


•89713 


•45736 


.88928 


.47281 


.88117 


47 


13 


.48811 


•87278 


■50327 


86413 


.51828 


85521 


.53312 


84604 


47 


14 


.42631 


90458 


■44203 


.89700 


.45762 


.88915 


.47306 


.88103 


46 


14 


.48837 


•87264 


.50352 


86398 


.51852 


85506 


.53337 


84588 


46 


15 


.42657 


90446 


■44229 


.89687 


•45787 


.88902 


47332 


.88089 


4S 


IS 


.48862 


•87250 


•503 7 7 


86384 


.51877 


85491 


■53361 


84573 


45 


i6 


.42683 


90433 


■44255 


.89674 


•45813 


.88888 


•47358 


■8807s 


44 


16 


.48888 


•8723s 


•S0403 


86369 


.51902 


85476 


■53386 


84557 


44 


17 


.42709 


90421 


.44281 


.89662 


•45839 


.88875 


47383 


.88062 


43 


17 


.48913 


•87221 


•50428 


86354 


.51927 


85461 


■53411 


84542 


43 


i8 


.42736 


90408 


.44307 


.89649 


•45865 


.88862 


•47409 


.88048 


42 


18 


.48938 


.87207 


•50453 


86340 


.51952 


85446 


■53435 


84526 


42 


'9 


.42762 


90396 


■44333 


.89636 


.45891 


.88848 


•47434 


.88034 


41 


19 


.48964 


•87193 


•50478 


86325 


.51977 


85431 


.53460 


84511 


41 


20 


.42788 


90383 


■44359 


.89623 


■45917 


.88835 


47460 


.88020 


40 


20 


.48989 


.87178 


•50503 


86310 


.52002 


85416 


.53484 


8449s 


40 


21 


42815 


90371 


■4438s 


.89610 


.45942 


.88822 


•47486 


.88006 


39 


21 


.49014 


.87164 


.50528 


86295 


.52026 


85401 


.53509 


84480 


30 


22 


42841 


90358 


.44411 


■89597 


.45968 


.88808 


•47511 


■87993 


38 


22 


.49040 


•87150 


•50553 


86281 


.52051 


8538s 


.53534 


84464 


38 


23 


.42867 


90346 


■44437 


■89584 


•45994 


■88795 


■47537 


■87979 


37 1 


23 


.49065 


•87136 


•50578 


86266 


.52076 


85370 


.53558 


84448 


37 


24 


.42894 


90334 


■44464 


■89571 


.46020 


.88782 


■47562 


.87965 


36 


24 


.49090 


.87121 


•50603 


S6251 


.52101 


8S3SS 


.53583 


84433 


36 


25 


.42920 


90321 


.44490 


■89558 


.46046 


.88768 


47588 


.87951 


33 


25 


.49116 


.87107 


•50628 


86237 


.52126 


85340 


.53607 


84417 


35 


26 


.42946 


90309 


.44516 


■89545 


.46072 


■88755 


•47614 


.87937 


34 


26 


49141 


.87093 


•50654 


86222 


.52151 


8532s 


.53632 


84402 


34 


27 


•42972 


90296 


•44542 


■89532 


.46097 


.88741 


•47639 


■87923 


33 


27 


.49166 


•87079 


.50679 


86207 


.52175 


85310 


■53656 


84386 


33 


28 


.42999 


90284 


.44568 


.89519 


■46123 


.88728 


•47665 


.87909 


32 


28 


.49192 


•87064 


•50704 


86192 


.52200 


85294 


■53681 


84370 


32 


29 


•43025 


90271 


•44594 


.89506 


.46149 


.88715 


.47690 


.87896 


31 


29 


.49217 


.87050 


•50729 


86178 


■52225 


85279 


■53705 


84355 


31 


30 


•43051 


90259 


.44620 


,89493 


•4617s 


.88701 


•47716 


.87882 


30 


30 


.49242 


.87036 


•50754 


86163 


.52250 


85264 


■53730 


84339 


30 


31 


•43077 


90246 


.44646 


.89480 


46201 


.88688 


•47741 


.87868 


29 


31 


.49268 


.87021 


-50779 


86148 


■5227s 


85249 


■53754 


84324 


20 


32 


.43104 


90233 


.44672 


.89467 


.46226 


.88674 


•47767 


■87854 


28 


32 


•49293 


.87007 


•50804 


86133 


.52299 


85234 


■53779 


84308 


28 


33 


•43130 


90221 


.44698 


.89454 


.46252 


.88661 


•47793 


.87840 


27 


33 


•49318 


.86993 


.50829 


86119 


■52324 


85218 


■53804 


84292 


27 


34 


•43156 


90208 


•44724 


.89441 


.46278 


.88647 


.47818 


.87826 


26 


34 


•49344 


.86978 


.50854 


86104 


■52349 


85203 


■53828 


84277 


26 


35 


•43182 


90196 


•44750 


.89428 


■46304 


.88634 


■47844 


.87812 


25 


35 


•49369 


.86964 


•50879 


86089 


■52374 


85188 


■53853 


84261 


25 


36 


•43209 


90183 


■44776 


■8941s 


■46330 


.88620 


.47869 


■87798 


24 


36 


•49394 


.86949 


•50904 


86074 


■52399 


85173 


■53877 


84245 


24 


37 


•43235 


90171 


.44802 


.89402 


■46355 


.88607 


■47895 


.87784 


23 


37 


.49419 


■86935 


•50929 


86059 


■52423 


85157 


■53902 


84230 


23 


38 


.43261 


90158 


.44828 


■89389 


■46381 


•88593 


.47920 


■87770 


22 


38 


•4944S 


.86921 


•S0954 


86045 


■52448 


85142 


■53926 


84214 


22 


39 


.43287 


90146 


■44854 


.89376 


.46407 


.88580 


.47946 


■87756 


21 


39 


.49470 


.86906 


•S0979 


86030 


■52473 


85127 


■53951 


84198 


21 


40 


•43313 


90133 


.44880 


■89363 


■46433 


.88566 


■47971 


■87743 


20 


40 


■4949S 


.86892 


.51004 


86015 


.52498 


85112 


■53975 


84182 


20 


41 


•43340 


goi2o 


.44906 


■89350 


■46458 


•88553 


■47997 


.87729 


19 


41 


■49521 


.86878 


.51029 


86000 


.52522 


85096 


.54000 


84167 


19 


42 


■43366 


90108 


.44932 


■89337 


.46484 


•88539 


.48022 


■8771s 


18 


42 


.49546 


.86863 


•S1054 


8598s 


.52547 


85081 


.54024 


84151 


I?. 


43 


•43392 


90095 


.44958 


.89324 


.46510 


.88526 


48048 


.87701 


17 


43 


■49571 


.86849 


•51079 


85970 


.52572 


85066 


■54049 


84135 


17 


44 


•43418 


90082 


■44984 


■893 1 1 


■46536 


.88512 


.48073 


.87687 


16 


44 


■49596 


.86834 


.51104 


85956 


.52597 


85051 


■54073 


84120 


16 


45 


•43445 


90070 


.45010 


.89298 


•46561 


.88499 


.48099 


■87673 


IS 


45 


.49622 


.86820 


.51129 


85941 


.52621 


85035 


■54097 


84104 


15 


46 


•43471 


90057 


•45036 


.89285 


■46587 


.88485 


.48124 


■87659 


14 


46 


.49647 


.86805 


•51154 


85926 


.52646 


85020 


.54122 


84088 


14 


47 


•43497 


9004s 


•45062 


.89272 


■46613 


.88472 


.48150 


.87645 


13 


47 


.49672 


.86791 


•51179 


859II 


52671 


85005 


.54146 


84072 


13 


48 


•43523 


90032 


.45088 


■89259 


■46639 


.88458 


•4817s 


.87631 


12 


48 


.49697 


.86777 


.51204 


85896 


.52696 


84989 


■54171 


84057 


12 


49 


•43549 


90019 


•45114 


.89245 


.46664 


.88445 


•48201 


.87617 


II 


49 


■49723 


.86762 


.SI229 


85881 


.52720 


84974 


.54195 


84041 


II 


SO 


•43575 


90007 


.45140 


.89232 


.46690 


.88431 


.48226 


.87603 


10 


SO 


.49748 


.86748 


•51254 


85866 


.52745 


84959 


.54220 


.84025 


10 


51 


.43602 


89994 


.45166 


.89219 


.46716 


.88417 


.48252 


.87589 


9 


SI 


■49773 


■86733 


.51279 


85851 


.52770 


84943 


■54244 


84009 


9 


52 


.43628 


89981 


45192 


.89206 


.46742 


.88404 


•48277 


■87575 


8 


S2 


■49798 


.86719 


.51304 


85836 


.52794 


84928 


■54269 


83994 


8 


53 


■43654 


89968 


.45218 


.89193 


.46767 


.88390 


•48303 


■87561 


7 


S3 


.49824 


.86704 


•51329 


85821 


.52819 


84913 


.54293 


83978 


7 


54 


.43680 


89956 


■45243 


.89180 


■46793 


■88377 


•48328 


.87546 


6 


54 


.49849 


.86690 


•51354 


85806 


.52844 


84897 


.54317 


83962 


6 


55 


•43706 


89943 


.45269 


.89167 


.46819 


■88363 


•48354 


■87532 


5 


55 


.49874 


.8667s 


■51379 


85792 


.52869 


84882 


.54342 


83946 


S 


56 


•43733 


89930 


■4529s 


•89153 


.46844 


.88349 


•48379 


.87518 


4 


S6 


■49899 


.86661 


•51404 


85777 


.52893 


84866 


.54366 


83930 


4 


57 


•43759 


89918 


■45321 


.89140 


.46870 


.88336 


•4840s 


.87504 


3 


57 


.49924 , 


.86646 


•51429 


85762 


.52918 


84851 


.54391 


83915 


3 


S8 


•43785 


8990s 


■45347 


.89127 


.46896 


.88322 


•48430 


.87490 


2 


58 


■49950 


.86632 


•51454 


85747 


.52943 


84836 


.54415 


83899 


2 


59 


.43811 


89892 


■45373 


.89114 


.46921 


.88308 


.48456 


.87476 


I 


59 


■49975 


.86617 


•51479 


85732 


■52967 


84820 


.54440 


83883 


I 


60 


•43837 


89879 


■45399 


.89101 


.46947 


.88295 


.48481 


.87462 





60 


.50000 


.86603 


•51504 


85717 


.52992 


84805 


.54464 


83867 







Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


"7~ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


~ 




64° 




6 


3° 


6 


2° 


6 


1° 






60° 


59° 




58° 




57° 







516 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 













Table 


5-— 


Natural Trigonometkic Functions- 


-{Continued) 














33° 1 


34° 1 


35° 




36° 








37° 


38° 


■39° 


40° 




/ 


Sink 


Cosine 


Sine 


Cosine 


Sine C 


;osine 


Sine ( 


^osnie 


/ 


f 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


/ 


o 


54464 


.83867 


•55919 


.82904 


•57358 


8I9IS 


•58779 


80902 


60 





.60182 


.79864 


.61566 


.78801 


.62932 


.77715 


.64279 


.76604 


"to 


I 


54488 


.83851 


•55943 


.82887 


•57381 


81899 


.58802 


80885 


59 


I 


.60205 


.79846 


.61589 


.78783 


.6295s 


.77696 


.64301 


.76586 


59 


2 


54513 


.83835 


•55968 


.82871 


•57405 


81882 


.58826 


80867 


58 


a 


.60228 


.79829 


.61612 


.78765 


.62977 


.77678 


.64323 


.76567 


S8 


3 


54537 


.83819 


•55992 


.82855 


.57429 


81865 


.58849 


80850 


57 


3 


.60251 


.79811 


.61635 


.78747 


.63000 


.77660 


.64346 


■76548 


57 


4 


54561 


.83804 


.56016 


.82839 


•57453 


81848 


•58873 


?°§^3 


56 


4 


.60274 


.79793 


.61658 


.78729 


.63022 


•77641 


.64368 


■76530 


S6 


5 


54586 


.83788 


.56040 


.82822 


■57477 


81832 


.58896 


80816 


55 


5 


.60298 


.79776 


.61681 


.78711 


•63045 


•77623 


.64390 


.76511 


SS 


6 


54610 


.83772 


.56064 


.82806 


•57501 


8I8I5 


.58920 


80799 


54 


6 


.60321 


•79758 


.61704 


.78694 


.63068 


.77605 


.64412 


.76492 


54 


7 


54635 


.83756 


.56088 


.82790 


•57524 


81798 


.58943 


80782 


53 


7 


.60344 


•79741 


.61726 


.78676 


.63090 


.77586 


.64435 


•76473 


53 


8 


54659 


.83740 


.56112 


•82773 


•57548 


81782 


■58967 


80765 


52 


8 


.60367 


•79723 


.61749 


.78658 


•63113 


.77568 


.64457 


.76455 


52 


9 


54683 


■83724 


.56136 


•82757 


•57572 


81765 


.58990 


80748 


51 


9 


.60390 


.79706 


.61772 


.78640 


.63135 


.77550 


■64479 


•76436 


SI 


lO 


54708 


.83708 


.56160 


.82741 


•57596 


81748 


.59014 


80730 


50 


10 


.60414 


.79688 


•6179s 


.78622 


.63158 


.77531 


.64501 


.76417 


50 


II 


54732 


83692 


.56184 


.82724 


•57619 


8I73I 


..59037 


80713 


49 


II 


.60437 


.79671 


.61818 


.78604 


.63180 


.77513 


.64524 


.76398 


49 


13 


54756 


.83676 


.56208 


.82708 


•57643 


81714 


.59061 


80696 


48 


12 


.60460 


.79653 


.61841 


.78586 


.63203 


.77494 


.64546 


.76380 


48 


13 


54781 


.83660 


.56232 


.82692 


•57667 


81698 


.59084 


80679 


47 


13 


.60483 


.79635 


.61864 


.78568 


.63225 


■77476 


.64568 


.76361 


47 


14 


54805 


.83645 


•56256 


■8267s 


•57691 


8I68I 


.59108 


80662 


46 


14 


.60506 


.79618 


.61887 


.78550 


.63248 


.77458 


.64590 


.76342 


46 


IS 


54829 


.83629 


.56280 


.82659 


•57715 


81664 


•59131 


80644 


45 


15 


.60529 


.79600 


.61909 


.78532 


.63271 


.77439 


.64612 


•76323 


45 


16 


S4854 


.83613 


•5630s 


.82643 


•57738 


81647 


•59154 


80627 


44 


16 


.60553 


.79583 


■61932 


.78514 


•63293 


.77421 


.64635 


.76304 


44 


17 


54878 


.83597 


•56329 


.82626 


•57762 


81631 


.59178 


80610 


43 


17 


.60576 


.79565 


•6195s 


.78496 


•63316 


.77402 


.64657 


.76286 


43 


18 


54902 


.83581 


•56353 


.82610 


•57786 


8I6I4 


.59201 


80593 


42 


18 


.60599 


.79547 


.61978 


.78478 


•63338 


.77384 


.64679 


.76267 


42 


19 


54927 


.83565 


•56377 


•82593 


•57810 


81597 


•59225 


80576 


41 


19 


.60622 


•79530 


.62001 


.78460 


•63361 


.77366 


.64701 


.76248 


41 


20 


S49SI 


.83549 


.56401 


.82577 


.57833 


81580 


•59248 


80558 


40 


20 


.60645 


•79512 


.62024 


.78442 


•63383 


.77347 


.64723 


.76229 


40 


21 


S497S 


•83533 


•56425 


.82561 


•57857 


81563 


.59272 


80541 


39 


21 


.60668 


•79494 


.62046 


.78424 


.63406 


.77329 


.64746 


.76210 


39 


22 


54999 


.83517 


.56449 


.82544 


•57881 


81546 


•59295 


80524 


38 


22 


.60691 


•79477 


.62069 


.78405 


•63428 


.77310 


.64768 


.76192 


38 


23 


55024 


.83501 


-56473 


.82528 


■57904 


81530 


•59318 


80507 


37 


23 


.60714 


.79459 


.62092 


.78387 


•63451 


.77292 


.64790 


.76173 


37 


24 


55048 


.8348s 


•56497 


.82511 


•57928 


81513 


•59342 


80489 


36 


24 


.60738 


•79441 


.62115 


.78369 


•63473 


.77273 


.64812 


•76154 


36 


2S 


55072 


.83469 


•56521 


.82495 


■57952 


81496 


•59365 


80472 


35 


25 


.60761 


.79424 


.62138 


.78351 


.63496 


.77255 


■64834 


-76135 


35 


26 


55097 


■83453 


•5654s 


.82478 


•57976 


81479 


•59389 


8045 s 


34 


26 


.60784 


.79406 


.62160 


.78333 


.63518 


.77236 


.64856 


.76116 


34 


27 


55121 


.83437 


•56569 


.82462 


•57999 


81462 


•59412 


80438 


33 


27 


.60807 


.79388 


.62183 


.78315 


.63540 


.77218 


.64878 


.76097 


33 


28 


55145 


.83421 


•56593 


.82446 


.58023 


81445 


•59436 


80420 


32 


28 


.60830 


.79371 


.62206 


.78297 


•63563 


.77199 


.64901 


.76078 


3a 


29 


55169 


.83405 


•56617 


.82429 


.58047 


81428 


•59459 


80403 


31 


29 


.60853 


.79353 


.62229 


■78279 


.63585 


.77181 


■64923 


■76059 


31 


30 


55194 


.83389 


•56641 


.82413 


.58070 


8I4I2 


.59482 


80386 


30 


30 


.60876 


•79335 


.62251 


.78261 


.63608 


•77162 


•6494s 


.76041 


30 


31 


SS218 


•83373 


•56665 


.82396 


.58094 


8139s 


.59506 


80368 


29 


31 


.60899 


.79318 


.62274 


.78243 


.63630 


.77144 


•64967 


.76022 


29 


32 


55242 


.83356 


.56689 


.823 


.58118 


81378 


•59529 


80351 


28 


32 


.60922 


.79300 


.62297 


.78225 


.63653 


.77125 


.64989 


.76003 


28 


33 


55266 


.83340 


•56713 


.82363 


.58141 


8I36I 


.59552 


80334 


27 


33 


.60945 


.79282- 


.62320 


.78206 


.63675 


.77107 


.65011 


•75984 


27 


34 


55291 


.83324 


•56736 


■82347 


.5816s 


81344 


.59576 


80316 


26 


34 


.60968 


.79264 


.62342 


.78188 


.63698 


.77088 


•65033 


•75965 


26 


35 


55315 


.83308 


.56760 


•82330 


.58189 


81327 


.59599 


80299 


25 


35 


.60991 


•79247 


.62365 


.78170 


.63720 


.77070 


•6505s 


•75946 


2S 


36 


55339 


.83292 


.56784 


■82314 


.58212 


81310 


.59622 


80282 


24 


36 


.61015 


.79229 


.62388 


.78152 


.63742 


.77051 


•65077 


•759^7 


24 


37 


55363 


.83276 


.56808 


.82297 


.58236 


81293 


.59646 


80264 


23 


37 


.61038 


.79211 


.62411 


.78134 


.63765 


.77033 


.65100 


.75908 


23 


38 


55388 


.83260 


.56832 


.82281 


.58260 


81276 


.59669 


80247 


22 


38 


.61061 


.79193 


.62433 


.78116 


.63787 


.77014 


.65122 


.75889 


22 


39 


55412 


.83244 


.56856 


.82264 


.58283 


81259 


.59693 


80230 


21 


39 


.61084 


.79176 


•62456 


.78098 


.63810 


.76996 


.65144 


.75870 


21 


40 


55436 


■83228 


.56880 


.82248 


.58307 


81242 


•59716 


80212 


20 


40 


.61107 


•79158 


•62479 


.78079 


.63832 


.76977 


.65166 


.75851 


20 


41 


55460 


.83212 


■56904 


.82231 


.58330 


81225 


•59739 


80195 


19 


41 


.61130 


.79140 


.62502 


.78061 


.63854 


•76959 


.65188 


.75832 


'2 


42 


55484 


.83195 


■56928 


.82214 


■58354 


81208 


•59763 


80178 


18 


42 


•61 1 53 


.79122 


.62524 


.78043 


.63877 


.76940 


.65210 


.75813 


i3 


43 


55509 


•83179 


■56952 


.82198 


■58378 


8II9I 


•59786 


80160 


17 


43 


.61176 


.79105 


•62547 


•78025 


.63899 


.76921 


.65232 


.75794 


17 


44 


5SS33 


.83163 


.56976 


.82181 


.58401 


8II74 


.59809 


80143 


16 


44 


.61199 


•79087 


.62570 


.78007 


.63922 


.76903 


■65254 


•7S775 


16 


45 


55557 


•83147 


■S7000 


.82165 


.58425 


8II57 


•59832 


80125 


15 


45 


.61222 


.79069 


.62592 


.77988 


.63944 


.76884 


■65276 


.75756 


IS 


46 


55581 


.83131 


•57024 


.82148 


.58449 


81 140 


•59856 


80108 


14 


46 


.61245 


.79051 


.62615 


.77970 


.63966 


.76866 


.65298 


•75738 


14 


47 


55605 


•83115 


•57047 


.82132 


.58472 


8II23 


•59879 


80091 


13 


47 


.61268 


•79033 


.62638 


•77952 


.63989 


.76847 


.65320 


•7S719 


13 


48 


55630 


.83098 


•57071 


.8211S 


.58496 


81 106 


•59902 


80073 


12 


48 


.61291 


■79016 


.62660 


•77934 


.64011 


.76828 


.65342 


•75700 


12 


49 


55654 


.83082 


•57095 


.82098 


•58519 


81089 


.59926 


80056 


II 


49 


.61314 


.78998 


.62683 


•77916 


.64033 


.76810 


.65364 


.75680 


II 


50 


S5678 


.83066 


.57119 


.82082 


•58543 


81072 


•59949 


80038 


10 


SO 


•61337 


.78980 


.62706 


.77897 


.64056 


•76791 


.65386 


.75661 


10 


SI 


55702 


.83050 


.57143 


.82065 


•58567 


810SS 


.59972 


80021 


9 


' SI 


.61360 


.78962 


.62728 


.77879 


.64078 


.76772 


.65408 


■75642 


9 


S2 


55726 


•83034 


.57167 


.82048 


•58590 


81038 


•59995 


80003 


8 


( 52 


.61383 


.78944 


.62751 


.77861 


.64100 


.76754 


.65430 


.75623 


8 


53 


5S7SO 


.83017 


.57191 


.82032 


•58614 


8I02I 


.60019 


79986 


7 


1 S3 


.61406 


.78926 


.62774 


.77843 


.64123 


.76735 


.65452 


.75604 


7 


S4 


5S77S 


.83001 


.57215 


.82015 


.58637 


81004 


.60042 


79968 


6 


S4 


.61429 


.78908 


.62796 


.77824 


.64145 


■76717 


■65474 


.75585 


6 


S5 


55799 


.82985 


■57238 


.81999 


.58661 


80987 


.60065 


79951 


5 


S5 


.61451 


.78891 


.62819 


.77806 


.64167 


.76698 


.65496 


.75566 


S 


SO 


55823 


.82969 


•57262 


.81982 


.58684 


80970 


.60089 


79934 


4 


S6 


•61474 


•78873 


.62842 


.77788 


.64190 


.76679 


.65518 


.75547 


4 


57 


55847 


.82953 


.57286 


.81965 


.58708 


80953 


.60112 


79916 


3 


57 


.61497 


•78855 


.62864 


.77769 


.64212 


.76661 


.65540 


.75528 


3 


S8 


55871 


.82936 


.57310 


.81949 


.58731 


80936 


.60135 


79899 


2 


58 


.61520 


•78837 


.62887 


•77751 


.64234 


.76642 


.65562 


•75509 


2 


59 


55895 


.82920 


.57334 


.81932 


.58755 


80919 


.60158 


79881 


I 


S9 


.61543 


.78819 


.62909 


■77733 


.64256 


.76623 


.65584 


.75490 


I 


60 


SS919 


.82904 


.S73S8 


.81915 


.58779 


80902 


.60182 


79864 





60 


.61566 


.78801 


.62932 


•77715 


.64279 


.76604 


.65606 


.75471 





■~~c 


:;osiNE 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 




/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 




5( 


)° 


5. 


>° 


64° 




53° 








5^ 


50 


5] 





5C 


►° 


4£ 


>° 





MATHEMATICAL TABLES 



517 













Table 5. — Natural Trigonometric Functions — (Continued) 














41° 


42° 


43° 


440 












r ( 


2° ( 


3° 




/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


f 





Sec. 

I 


Co-sec. 


Sec. 


Co-sec. 


Sec. 


Co-sec. 


Sec. 


Co-sec. 


/ 


o 


.65606 


■7S47I 


■66913 


•74314 


.68200 


•7313s 


.69466 


•71934 


60 


Infinite. 


I^OUOI 


57.299 


1 .0006 


28.654 


1.0014 


19.107 


60 


I 


.65628 


■75452 


■6693 s 


•74295 


.68221 


.73116 


.69487 


•71914 


59 


I 


I 


3437-70 


I.OOOI 


56.359 


1.0006 


28.417 


1.0014 


19.002 


59 


2 


.65650 


•75433 


■66956 


.74276 


.68242 


.73096 


■69508 


.71894 


S8 


2 


I 


1718.90 


1 .0002 


53.450 


1.0006 


28.184 


1.0014 


18.897 


58 


^ 


.65672 


•75414 


.66978 


■74256 


.68264 


•73076 


•69529 


•71873 


57 


3 


I 


1145-90 


1 .0002 


54^570 


1 .0006 


27-955 


1.0014 


18.794 


57 


4 


.65694 


•75395 


.66999 


■74227 


.68285 


•73056 


•69549 


•71853 


56 


4 


1 


859-44 


1 .0002 


53-/18 


1.0006 


27-730 


1.0014 


18.692 


56 


S 


.65716 


•75375 


.67021 


.74217 


.68306 


.73036 


•69570 


•71833 


SS 


S 


I 


687-55 


1.0002 


52.891 


1.0007 


27-508 


1.0014 


18.591 


55 


6 


•65738 


•75356 


.67043 


.74198 


.68327 


•73016 


.69591 


•71813 


54 


6 


I 


572.96 


1 .0002 


52.090 


1.0007 


27.290 


1 .00 1 5 


18.491 


54 


7 


•65759 


•75337 


.67064 


•74178 


.68349 


.72996 


.69612 


.71792 


53 


7 


I 


491. II 


1.0002 


51-313 


1 .0007 


27.075 


1.0015 


18.393 


S3 


8 


.65781 


•75318 


.67086 


•74159 


.68370 


•72976 


■69633 


•71772 


52 


8 


I 


429-72 


1 .0002 


50.558 


1.0007 


26.864 


1.0015 


18.295 


Sa 


9 


.65803 


•75299 


.67107 


•74139 


■68391 


•72957 


•69654 


•71752 


51 


9 


I 


381-97 


1 .0002 


49.826 


1.0007 


26.65s 


i.oois 


18.198 


SI 


lo 


.65825 


.75280 


.67129 


•74120 


.68412 


•72937 


•69675 


.71732 


SO 


10 


I 


343-77 


1 .0002 


49-114 


1.0007 


26.450 


1.001S 


18.103 


SO 


11 


.65847 


.75261 


.67151 


.74100 


■68434 


.72917 


.69696 


•71711 


49 


11 


I 


312-52 


1.0002 


48^422 


1.0007 


26.249 


1.001s 


18.008 


49 


12 


.6586P 


•75241 


.67172 


.74080 


.68455 


.72897 


.69717 


.71691 


48 


12 


I 


286.48 


1.0002 


47^750 


1.0007 


26.050 


1.0016 


17.914 


48 


13 


.65891 


.75222 


.67194 


.74061 


.68476 


•72877 


•69737 


.71671 


47 


13 


I 


264.44 


1.0002 


47^096 


1.0007 


25-854 


1.0016 


17.821 


47 


14 


•65913 


•75203 


•67215 


•74041 


.68497 


•72857 


•69758 


•71650 


46 


14 


I 


245-55 


1.0002 


46.460 


1 .0008 


25.661 


1.0016 


17.730 


46 


IS 


•65935 


•75184 


•67237 


.74022 


.68518 


•72837 


•69779 


•71630 


4S 


IS 


I 


229.18 


1 .0002 


45.840 


1.0008 


25.471 


1.0016 


17-639 


45 


I6 


•65956 


■75165 


■67258 


./4002 


.68539 


.72817 


.69800 


.71610 


44 


16 


r 


214.86 


1 .0002 


45-237 


1.0008 


25.284 


1.0016 


17-549 


44 


17 


.65978 


.75146 


.67280 


•73983 


.68561 


•72797 


.69821 


•71590 


43 


17 


I 


202.22 


1 .0002 


44-650 


1 .0008 


25.100 


1.0016 


17.460 


43 


l8 


.66000 


•75126 


•67301 


•73963 


.68582 


•72777 


.69842 


•71569 


42 


18 


I 


190.99 


1.0002 


44-077 


1.0008 


24.918 


1.O0I7 


17-372 


42 


19 


.66022 


•75107 


■67323 


•73944 


.68603 


•72757 


.69862 


•71549 


41 


19 


I 


180.73 


1.0003 


43^520 


1 .0008 


24-739 


1.0017 


17-285 


41 


20 


.66044 


.75088 


•67344 


•73924 


.68624 


•72737 


.69883 


•71529 


40 


20 


I 


171.89 


1.0003 


42.976 


1.0008 


24.562 


1.0017 


17.198 


40 


21 


.66066 


.75069 


.67366 


•73904 


.68645 


•72717 


.69904 


.71508 


39 


21 


I 


163.70 


1.0003 


42^445 


1.0008 


24.358 


1.0017 


17.113 


39 


32 


.66088 


•75050 


•67387 


•73885 


■68666 


.72697 


•69925 


.71488 


38 


22 


I 


156.26 


1.0003 


41.928 


1.0008 


24.216 


1.0017 


17.028 


38 


23 


.66109 


•75030 


.67409 


■73865 


.68688 


.72677 


.69946 


.71468 


37 


23 


I 


149.47 


1.0003 


41.423 


I 0009 


24.047 


1.00 1 7 


16.944 


37 


24 


.66131 


.75011 


.67430 


•73846 


.68709 


.72657 


.69966 


•71447 


36 


24 


I 


143-24 


1.0003 


40^930 


1.0009 


23.880 


1.0018 


16.861 


36 


25 


.66153 


•74992 


•67452 


•73826 


■68730 


.72637 


.69987 


.71427 


3S 


35 


I 


137-51 


1.0003 


40.448 


1.0009 


23.716 


1.0018 


16.779 


35 


26 


.66175 


•74973 


•67473 


•73806 


.68751 


.72617 


.70008 


.71407 


34 


26 


I 


132^22 


1.0003 


39-978 


1.0009 


23-553 


1.0018 


16.698 


34 


27 


.66197 


•74953 


■67495 


•73787 


.68772 


■72597 


.70029 


.71386 


33 


27 


I 


127^32 


1.0003 


39.518 


1 .0009 


23-393 


i.ooiS 


16.617 


33 


28 


.66218 


•74934 


.67516 


•73767 


■68793 


■72577 


.70049 


.71366 


32 


28 


I 


122.78 


1.0003 


39069 


1.0009 


23-235 


1.0018 


16.538 


32 


29 


.66240 


■74915 


•67538 


•73747 


.68814 


•72557 


.70070 


•71345 


31 


29 


I 


118.54 


1.0003 


3S631 


1.0009 


23-079 


1.0018 


16.459 


31 


30 


.66262 


.74896 


•67559 


•73728 


.68835 


•72537 


.70091 


•7132s 


30 


30 


I 


11459 


1.0003 


38.201 


1.0009 


22.925 


1.0019 


16.380 


30 


31 


.66284 


.74876 


.67580 


•73708 


.68857 


•72517 


.70112 


•71305 


29 


31 


I 


110.90 


1.0003 


37.782 


1. 00 10 


22.774 


1.0019 


16.303 


29 


32 


.66306 


•74857 


.67602 


■73688 


.68878 


•72497 


.70132 


.71284 


28 


32 


I 


107.43 


1.0003 


37.371 


I.OOIO 


22.624 


1.0019 


16.226 


28 


33 


.66327 


•74838 


.67623 


•73669 


.68899 


■72477 


•70153 


.71264 


27 


33 


I 


104.17 


1.0004 


36.969 


1. 0010 


22.476 


1.0019 


16.150 


27 


34 


.66349 


.74818 


.67645 


•73649 


■68920 


•72457 


.70174 


•71243 


26 


34 


I 


lOI.II 


1.0004 


36.576 


I.OOIO 


22.330 


1.0019 


16.07s 


26 


3| 


•66371 


•74799 


.67666 


•73629 


.68941 


•72437 


•70195 


•71223 


25 


35 


I 


95-223 


1.0004 


36-191 


1.0010 


22.186 


1.0019 


16.000 


25 


36 


.66393 


.74780 


.67688 


■73610 


.68962 


.72417 


•70215 


•71203 


24 


36 


I 


93405 


1.0004 


35^8i4 


1.0010 


22.044 


1.0020 


15926 


24 


32 


.66414 


.74760 


.67709 


■73590 


.68983 


•72397 


■70236 


.71182 


23 


37 


I 


92.914 


1.0004 


35^445 


I.OOIO 


21.904 


1.0020 


15-8,53 


23 


38 


.66436 


•74741 


•67730 


•73570 


.69004 


■72377 


■70257 


.71162 


22 


38 


1. 0001 


92^469 


1 .0004 


35-084 


I.OOIO 


21.765 


1 .0020 


15-780 


22 


39 


.66458 


.74722 


.67752 


•73S5I 


.69025 


■72357 


.70277 


.71141 


21 


39 


1. 000 1 


88.149 


1.0004 


34.729 


I.OOll 


21.629 


1.0020 


15-708 


21 


40 


.66480 


•74703 


•67773 


•73531 


.69046 


•72337 


.70298 


.7x121 


20 


40 


1. 000 1 


85.946 


1.0004 


34-382 


I.OOll 


21.494 


1.0020 


15-637 


20 


41 


.66501 


.74683 


•67795 


•73511 


.69067 


•72317 


■70319 


.71100 


19 


41 


I.OOOI 


83.849 


1.0004 


34-042 


I.OOll 


21.360 


1. 002 1 


15-566 


19 


42 


.66523 


.74664 


.67816 


•73491 


.69088 


•72297 


•70339 


■71080 


18 


42 


1 .0001 


81.853 


1 .0004 


33-708 


l.OOII 


21.228 


1. 002 1 


15.496 


18 


43 


•66545 


.74644 


.67837 


•73472 


.69109 


.72277 


.70360 


■71059 


17 


43 


I.OOOI 


79950 


1.0004 


33-381 


I.OOll 


21.098 


1. 002 1 


15-427 


17 


44 


.66566 


•74625 


•67859 


•73452 


•69130 


.72257 


.70381 


.71039 


16 


44 


I.OOOI 


78.133 


1 .0004 


33 -060 


I.OOll 


20.970 


1.0021 


15-358 


16 


45 


.66588 


.74606 


.67880 


•73432 


.69151 


.72236 


.70401 


.71019 


IS 


45 


I.OOOI 


76.396 


1.0005 


32-745 


I.OOll 


20.843 


1. 002 1 


15-290 


IS 


46 


.66610 


•74586 


.67901 


•73413 


.69172 


.72216 


.70422 


.70998 


14 


46 


I.OOOI 


74-736 


1.0005 


32^437 


1.0012 


20.717 


1.0022 


15.222 


14 


47 


.66632 


•74567 


.67923 


•73393 


.69193 


.72196 


•70443 


.70978 


13 


47 


I.OOOI 


73^i46 


1.000s 


32-134 


1.0012 


20.593 


1.0022 


15.155 


13 


48 


.66653 


•74548 


•67944 


•73373 


.69214 


.72176 


•70463 


•70957 


12 


48 


I.OOOI 


71.622 


1 .0005 


31-836 


1.0012 


20.471 


1.0022 


15-089 


12 


49 


■6667s 


•74528 


•67965 


•73353 


■69235 


•72156 


.70484 


•70937 


II 


49 


I.OOOI 


71.160 


1.0005 


31-544 


1.0012 


20.350 


1.0022 


15-023 


II 


50 


.66697 


•74509 


.67987 


•73333 


■69256 


•72136 


•70505 


.70916 


10 


SO 


I.OOOI 


68.757 


1.0005 


31-257 


1.0012 


20.230 


1.0022 


14958 


10 


SI 


.66718 


.74489 


.68008 


•73314 


,69277 


.72116 


•70525 


.70896 


9 


SI 


I^OOOI 


67.409 


i.ooos 


30-976 


1.0012 


20.112 


1.0023 


14,893 


9 


52 


.66740 


•74470 


.68029 


•73294 


.69298 


.72095 


.70546 


.70875 


8 


52 


I.OOOI 


66.113 


1 .0005 


30.699 


1.0012 


19-995 


1.0023 


14,829 


8 


S3 


.66762 


•74451 


.68051 


•73274 


.69319 


.72075 


•70567 


■70855 


' 


S3 


I.OOOI 


64.866 


1.0005 


30.428 


1.0013 


19.880 


1.0023 


14-765 


7 


54 


.66783 


•74431 


.68072 


•73254 


•69340 


•72055 


■70587 


.70834 


6 


S4 


I.OOOI 


63.664 


1 .0005 


30.161 


1.0013 


19.766 


1.0023 


14,702 


6 


55 


.66805 


•74412 


.68093 


•73234 


•69361 


•7203s 


.70608 


.70813 


S 


SS 


I.OOOI 


62.507 


1 .0005 


29.899 


1.0013 


19-653 


1.0023 


14,640 


S 


S6 


.66827 


•74392 


.6811S 


■73215 


.69382 


.72015 


.70628 


•70793 


4 


56 


I.OOOI 


61.391 


1.0006 


29.641 


1.0013 


19-541 


1.0024 


14,578 


4 


H 


.66848 


-74373 


.68136 


•73195 


.69403 


•71995 


.70649 


.70772 


3 


57 


I.OOOI 


61.314 


1 .0006 


29.388 


I.0OI3 


19-431 


1.0024 


14-S17 


3 


S8 


.66870 


•74353 


■68157 


•73175 


.69424 


•71974 


.70670 


•70752 


2 


58 


I.OOOI 


59.274 


1.0006 


29139 


1.0013 


19.322 


1.0024 


14-456 


2 


S9 


.66891 


•74334 


.68179 


•73155 


■69445 


•71954 


.70690 


■70731 


I 


59 


1.0001 


58^270 


1.0006 


28.894 


1.0013 


19.214 


1.0024 


14-395 


I 


60 


•66913 


•74314 


■68200 


•73135 


.69466 


•71934 


.70711 


■70711 





60 


I.OOOI 


57^299 


1.0006 


28.654 


I.00I4 


19.107 


1.0024 


14-335 





f 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 1 


Sine 


/ 


/ 


Co-sec. 


Sec. 


Co-sec- 


Sec. 


Co-sec. 


Sec. 


Co-sec. 


Sec. 






48 


1° 1 


47 





46 





45 









8< 


)° 


88° 1 


8 


r 


8 


3° 





518 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 5. — Natural Trigonometkic Functions — {Continued) 





40 




5° 


6 


° 


70 




j 


8 





9° 


10° 


11° 




1 


Sec. 


Co-SEC. 


Sec. 


Co-SEC. 


Sec. 


Co-SEC 


Sec. 


Co-SEC. 


' 


/ 


Sec. 


Co-SEC. 


Sec. 


Co-SEC. 


Sec. 


CO.SEC. 


Sec. 


Co-SEC. 


' 





1.0024 


14-335 


1 .0038 


11.474 


1.0055 


9.5668 


1.0075 


8.2055 


60 


T 


1.0098 


7-1853 


1.0125 


6.3924 


1-0154 


S-7588 


1.0187 


S.2408 


60 


I 


I.002S 


14.276 


1 .0038 


11.436 


1.0055 


9.5404 


1.0075 


8.I86I 


59 


I 


1.0099 


7.1704 


1.0125 


6.3807 


1-0155 


5-7493 


1.0188 


5.2330 


59 


2 


I.002S 


14.217 


1 .0039 


11.398 


1.0056 


9-5'4i 


1.0076 


8.1668 


S8 


2 


1 .0099 


7-1557 


I.OI25 


6.3690 


1-015S 


5-7398 


1.0188 


5.2252 


58 


3 


1.0025 


14-159 


1.0039 


11.360 


1.0056 


9.4880 


1.0076 


8.1476 


57 


3 


1 .0099 


7.1409 


1. 0126 


6.3574 


1.0156 


S-7304 


1.0189 


5.2174 


57 


4 


1.0025 


14.IOI 


1.0039 


11-323 


1.0056 


9.4620 


1.0076 


8.1285 


S6 


4 


I.OIOO 


7-1263 


I.OI26 


6.3458 


1.0156 


S-72IO 


1.0189 


S-2097 


S6 


S 


1.0025 


14.043 


1.0039 


11.286 


1.0057 


9-4362 


1.0077 


8.1094 


55 


5 


I.OIOO 


7.1117 


I.OI27 


6.3343 


1-0157 


5-7117 


1.0190 


5-2019 


55 


6 


1.0026 


13-986 


1 .0040 


11.249 


1.0057 


9-4105 


1.0077 


8.0905 


54 


6 


1.0 lOI 


7.0972 


I.OI27 


6.3228 


1.0157 


5-7023 


1.0191 


5. 1942 


54 


7 


1.0026 


13-930 


1 .0040 


II.2I3 


1 .005 7 


9-3850 


1.0078 


8.0717 


53 


7 


I.OIOI 


7.0827 


I.OI28 


6.3113 


1.0158 


5-6930 


1.0191 


S-1865 


53 


8 


1.0026 


13-874 


1 .0040 


II. 176 


1.0057 


9-3596 


1.0078 


8.0529 


52 


8 


I.0I02 


7.0683 


1.0128 


6.2999 


1.0158 


5.6838 


1.0192 


5-1788 


52 


9 


1.0026 


13.818 


1 .0040 


II. 140 


1 .0058 


9-3343 


1.0078 


8.0342 


51 


9 


1.0102 


7-0539 


I.OI29 


6.288s 


1.0159 


5-6745 


1.0192 


5-1712 


SI 


10 


1.0026 


13-763 


1. 004 1 


11.104 


1.0058 


9.3092 


1.0079 


8.0156 


50 


10 


1. 0102 


7.0396 


I.OI29 


6.2772 


1.0159 


5-6653 


1.0193 


5.1636 


SO 


II 


1.0027 


13-708 


1.0041 


11.069 


1.0058 


9.2842 


1.0079 


7.9971 


'♦S 


II 


I.OI03 


7.0254 


I.OI30 


6.2659 


1.0160 


5-6561 


1.0193 


S-1560 


49 


12 


1.0027 


13-654 


1. 0041 


1 1 -033 


1.0059 


9-2593 


1.0079 


7.9787 


48 


12 


I.OI03 


7.OH2 


I.OI30 


6.2546 


1.0160 


5.6470 


1.0194 


5-1484 


48 


13 


1.0027 


13.600 


1.0041 


10.988 


1.0059 


9.2346 


1.0080 


7.9604 


47 


13 


I.OI04 


6.9971 


I.0I3I 


6.2434 


1.0161 


5-6379 


1.019s 


5.1409 


47 


14 


1.0027 


13-547 


1 .0042 


10.963 


1.0059 


9.2100 


1 .0080 


7-9421 


46 


14 


I.OI04 


6.9830 


1.0I3I 


6.2322 


1.0162 


5.6288 


I.OI9S 


S-1333 


46 


15 


1.0027 


13-494 


1.0042 


10.929 


1 .0060 


9-i8s5 


1 .0080 


7.9240 


45 


IS 


I.OI04 


6.9690 


I.OI32 


6.2211 


IX)l62 


5-6197 


1.0196 


5.1258 


45 


16 


1 .0028 


13-441 


1 .0042 


10.894 


1 .0060 


9.1612 


1. 008 1 


7-9059 


44 


16 


i.oios 


69550 


1.0132 


6.2100 


1.0 163 


56107 


1. 0196 


5-1183 


44 


17 


1.0028 


13-389 


1.0043 


10.860 


1 .0060 


9-1370 


1.0081 


7-8S79 


43 


17 


1. 0105 


6.941 1 


1-0133 


6.1990 


1.0163 


5.6017 


1.0197 


5.1109 


43 


18 


1.0028 


13-337 


1.0043 


10.826 


1.0061 


9.1129 


1.0082 


7.8700 


42 


18 


1.0106 


6.9273 


1-OI33 


6.1880 


1.0164 


5.5928 


1.0198 


S-1034 


42 


19 


1.0028 


13.286 


1.0043 


10.792 


1. 0061 


9.0890 


1.0082 


7-8522 


41 


19 


1. 0106 


6.9135 


1-0134 


6.1770 


1.0164 


S-5838 


i.oigS 


5.0960 


41 


20 


1.0029 


13-235 


1.0043 


10.758 


1. 006 1 


9.0651 


1.0082 


7-8344 


40 


20 


1.0107 


6.8998 


1.0134 


6.1661 


1.0165 


5-5749 


1.0199 


5.0886 


40 


21 


1.0029 


13.184 


1.0044 


10.725 


1.0062 


9.0414 


1.0083 


7.8168 


39 


21 


1.0107 


6.8861 


1.013s 


6.1552 


1.0165 


5.5660 


1.0199 


5.0812 


39 


22 


1.0029 


13-134 


1.0044 


10.692 


1 0062 


9-0179 


1.0083 


7.7992 


38 


22 


I.OI07 


6.872s 


1-013S 


6.1443 


1. 0166 


S-5572 


1.0200 


5-0739 


38 


23 


1.0029 


13.084 


1 .0044 


10.659 


1.0062 


8.9944 


1 .0084 


7.7817 


37 


23 


1.0108 


6.8589 


1.0136 


6.1335 


1.0166 


5-5484 


1.0201 


5.0666 


37 


24 


1.0029 


13-034 


1 .0044 


10.626 


1.0063 


8.9711 


1.0084 


7.7642 


36 


24 


I.OI08 


6.8454 


1. 0136 


6.1227 


1.0167 


S-SJJ6 


1.0201 


5-0593 


36 


25 


1.0030 


12-985 


1.0045 


10.593 


1.0063 


8.9479 


1.0084 


7.7469 


35 


25 


1.0109 


6.8320 


1.0136 


6.1120 


1.0167 


5-5308 


1.0202 


5-0520 


35 


26 


1.0030 


12-937 


1.0045 


10.561 


1.0063 


8.9248 


1 .0085 


7.7296 


34 


26 


1.0109 


6.818s 


1.0137 


6.1013 


1.0168 


5-5221 


1.0202 


S-0447 


34 


27 


1.0030 


12.888 


1.0045 


10.529 


1.0064 


8.9018 


1.0085 


7.7124 


33 


27 


l.OIIO 


6.8052 


1-0137 


6.0906 


1.0169 


5-5134 


1.0203 


50375 


33 


28 


1.0030 


12.840 


1.0046 


10.497 


1.0064 


8.8790 


1.008s 


7-6953 


32 


28 


I.OIIO 


6.7919 


1.0138 


6.0800 


1. 0169 


5-S047 


1 .0204 


5-0302 


32 


29 


1.0031 


12.793 


1.0046 


10.465 


1.0064 


8.8563 


1.0086 


7-6783 


31 


29 


I.OIII 


6.7787 


1.0138 


6.0694 


1. 01 70 


5-4960 


1 .0204 


5-0230 


31 


30 


1.0031 


12.745 


1.0046 


10.433 


1.0065 


8.8337 


1.0086 


7.6613 


30 


30 


I.OIII 


6.7655 


1.0139 


6.0588 


1.0170 


5 -48 74 


1.020s 


5-0158 


30 


31 


I.OO3I 


12.698 


1 .0046 


10.402 


1.0065 


8.8112 


1.0087 


7.6444 


29 


31 


l.OllI 


6.7523 


1.0139 


6.0483 


1.0171 


5-4788 


1.0205 


5-0087 


29 


32 


I.OO3I 


12.652 


1.0047 


10.371 


1.0065 


8.7888 


1.0087 


7.6276 


28 


32 


I.OII2 


6.7392 


1.0140 


6.0379 


1.0171 


5-4702 


1.0206 


5-0015 


28 


33 


1.0032 


12.606 


1.0047 


10.340 


1.0066 


8.7665 


1.0087 


7.6108 


27 


33 


1.0H2 


6.7262 


1.0140 


6.0274 


1.0172 


S-4617 


1.0207 


4-9944 


27 


34 


1.0032 


12.560 


1 .0047 


10.309 


1.0066 


8.7444 


1.0088 


7-5942 


26 


34 


I.OII3 


6.7132 


1.0141 


6.0170 


1.0172 


5 -4532 


1.0207 


4-9873 


26 


35 


1.0032 


12.514 


1 .0048 


10.278 


1.0066 


8.7223 


1.0088 


7-5776 


25 


35 


I.OII3 


6.7003 


1.0141 


6.0066 


1.0173 


5-4447 


1 .0208 


4.9802 


25 


36 


1.0032 


12.469 


1.0048 


10.248 


1.0067 


8.7004 


1.0089 


7.5611 


24 


36 


I.OII4 


6.6874 


1.0142 


5-9963' 


1.0174 


5-4362 


1.0208 


4-9732 


24 


37 


1.0032 


12.424 


1 .0048 


10.217 


1.0067 


8.6786 


1.0089 


7-5446 


23 


37 


I.OII4 


6.6745 


1.0142 


5.9860 


1.0174 


S-4278 


1.0209 


4.9661 


23 


38 


1.0033 


12.379 


1.0048 


10.187 


1.0067 


8.6569 


1.0089 


7.5282 


22 


38 


I.OII5 


6.6617 


1.0143 


S-9758 


1.0175 


5-4194 


1.0210 


4-9591 


22 


39 


1.0033 


12.335 


1 .0049 


10.157 


1.0068 


8.6353 


1.0090 


7-51 19 


21 


39 


i.oiis 


6.6490 


1.0143 


5-9655 


1.0I7S 


5-4110 


1.0210 


4-9521 


21 


40 


1.0033 


12.291 


1 .0049 


10.127 


1.0068 


8.6138 


I'oogo 


7-4957 


20 


40 


1.0II5 


6.6363 


1.0144 


5-9554 


I.0176 


5.4026 


1.0211 


4.9452 


20 


41 


1.0033 


12.248 


1.0049 


10.098 


1.0068 


8.5924 


1.0090 


7-4795 


19 


41 


I.OII6 


6.6237 


1.0144 


5-9452 


1.0176 


5-3943 


1 .02 1 1 


4-9382 


19 


42 


1.0034 


12.204 


1.0050 


10.068 


1 .0069 


8.5711 


1.0091 


7-4634 


18 


42 


I.0II6 


6.611I 


1.0145 


5-9351 


1.0177 


5 -3860 


1.0212 


4-9313 


18 


43 


1.0034 


I2.l6l 


1.0050 


10.039 


1 .0069 


8.5499 


1. 009 1 


7-4474 


17 


43 


1.0117 


6.5985 


1.0145 


5-9250 


1.0177 


5-3777 


1.0213 


4.9243 


17 


44 


1.0034 


12.118 


1.0050 


10.010 


1 .0069 


8.5289 


1 .0092 


7-4315 


16 


44 


1.0II7 


6.5860 


1.0146 


5-9150 


1.0178 


5-3695 


1.0213 


4-9175 


16 


45 


1.0034 


12.076 


1 .0050 


9.9812 


1.0070 


8.5079 


1.0002 


7.4156 


15 


45 


I.0II8 


6.5736 


1.0146 


5.9049 


1.0179 


5-3612 


1.0214 


4.9106 


IS 


46 


1.0035 


12.034 


1.0051 


9-9525 


1.0070 


8.4871 


1.0092 


7-3998 


14 


46 


I.0II8 


6.5612 


1.0147 


5.8950 


1. 01 79 


S-3530 


1.021S 


4-9037 


14 


47 


1.0035 


11.992 


1.0051 


9.9239 


1 .0070 


8.4663 


1.0093 


7.3840 


13 


47 


I.0II9 


6.5488 


1.0147 


5-8850 


1.0180 


5 3449 


1 .02 1 5 


4.8969 


13 


48 


1.003s 


11.950 


1.0051 


9-8955 


1. 0071 


8.4457 


1.0093 


7.3683 


12 


48 


I.OI19 


6.536s 


1.0148 


5-8751 


1.0180 


5-3367 


1.0216 


4.8901 


12 


49 


1.0035 


11.909 


1.0052 


9.8672 


1.0071 


8.4251 


1.0094 


7-3527 


11 


49 


I.OI19 


6.5243 


1.0148 


5-8652 


i.oiSr 


S-3286 


1. 02 16 


4-8833 


II 


50 


1.0036 


11.868 


1.0052 


9.8391 


1.0071 


8.4046 


1.0094 


7.3372 


10 


SO 


1.0120 


6.5121 


1.0149 


5-8554 


1.0181 


S-320S 


1.0217 


4-8765 


10 


SI 


1.0036 


11.828 


1.0052 


9.8II2 


1.0072 


8.3843 


1.0094 


7.3217 


9 


51 


I.0I20 


6.4999 


1.0150 


5-8456 


1. 0182 


S-3124 


1.0218 


4-8697 


9 


S2 


1.0036 


11.787 


1.0053 


9-7834 


1.0072 


8.3640 


1.0095 


7.3063 


8 


52 


1.0I2I 


6.4878 


1.0150 


5-8358 


1.0182 


5-3044 


1. 02 18 


4.8630 


8 


53 


1 .0036 


11.747 


1.0053 


9.7558 


1.0073 


8.3439 


1.0095 


7.2909 


7 


53 


I.0I2I 


6.4757 


i.oisi 


5.8261 


1.0183 


5-2963 


1.0219 


4-8563 


7 


54 


1.0037 


11.707 


1.00S3 


9-7283 


1.0073 


8.3238 


1.0096 


7-2757 


6 


54 


I.OI22 


6.4637 


1.0151 


5-8163 


1.0184 


S-2883 


1.0220 


4.8496 


6 


55 


1.0037 


11.668 


1.0053 


9.7010 


1.0073 


8.3039 


1.0096 


7.2604 


5 


55 


I.OI22 


6.4517 


1.0152 


5-8067 


1.0184 


5-2803 


1.0220 


4.8429 


5 


S6 


1.0037 


11.628 


1.0054 


9-6739 


1 .0074 


8.2840 


1.0097 


7-2453 


4 


S6 


I.OI23 


6.4398 


1.0152 


S-7970 


1.0185 


5.2724 


1.0221 


4-8362 


4 


57 


1.0037 


11.589 


1.0054 


9.6469 


1 .0074 


8.2642 


1.0097 


7.2302 


3 


57 


1.0123 


6.4279 


1-0153 


S-7874 


1.0185 


5-2645 


1.0221 


4.8296 


3 


S8 


1.0038 


II-5SO 


1.0054 


9.6200 


1.0074 


8.2446 


1.0097 


7.2152 


2 


S8 


1.0124 


6.4160 


1-0153 


5-7778 


1.0186 


5-2566 


1.0222 


4.8229 


2 


59 


1.0038 


H.512 


1.0055 


9-5933 


1.0075 


8.2250 


1 .0098 


7.2002 


I 


59 


I.OI24 


6.4042 


1-0154 


5-7683 


1.0186 


S.2487 


1.0223 


4.8163 


I 


60 


1 .0038 


11-474 


I.O055 


9.5668 


1.0075 


8.2055 


1 .0098 


7.1853 





60 


1.0125 


6.3924 


1.0154 


S-7588 


1.0187 


5-2408 


1.0223 


4.8097 





/ 


Co-SEC. 


Sec. 


Co-SEC. 


Sec. 


Co-SEC. 


Sec. 


Co-SEC. 


Sec 


/ 


Co-SEC. 


Sec. 


Co-SEC. 


Sec. 


Co-SEC. 


Sec 


Co-SEC. 


Sec 


' 




8< 


)° 


8 


40 


8c 


!° 


8^ 


JO 






8 


1° 


8( 


)° 


7c 


P 


7^ 


\° 





MATHEMATICAL TABLES 



519 













Table 5. — 


NTattjeal Trigonometric Functions- 


-{Continued) 














12° 


13° 1 


14° 


15° 






16° 1 


17° 1 


18° 1 


19° 




9 


Sec. 


Co-sec. 


Sec. 


Co-sec. 


Sec. 


Co-sec. 


Sec. 


Co-sec. 


' 


' 


Sec. 


Co-sec. 


Sec. 


Co-SEC. 


Sec. 


Co-SEC. 


Sec. 


Co-SEC. 


/ 





1.0223 


4.8097 


1.0263 


4-4454 


1.0306 


4-1336 


I -03 53 


3-8637 


60 





1 .0403 


3-6279 


1-0457 


3-4203 


1-0515 


3-2361 


1-0576 


3-0715 


60 


I 


1.0224 


4.8032 


1.0264 


4.4398 


1.0307 


4.1287 


1-03S3 


3-8595 


5g 


I 


1.0404 


3-6243 


1-0458 


3 


4170 


1.0516 


3-2332 


I-OS77 


3.0690 


59 


a 


I.022S 


4.7966 


1.0264 


4.4342 


1.0308 


4-1239 


1-0354 


3-8553 


S8 


2 


1.0405 


3.6206 


1.0459 


3 


4138 


1.0517 


3-2303 


1-0578 


3.0664 


58 


3 


1.022s 


4.7901 


1.0265 


4.4287 


1.0308 


4.1191 


1-0355 


3-8512 


^l 


3 


1.0406 


3-6169 


1.0460 


3 


4106 


1.0518 


3-2274 


1-0579 


3-0638 


57 


4 


1.0226 


4.7835 


1.0266 


4-4231 


1.0309 


4-1144 


1.0356 


3.8470 


56 


4 


1.0406 


3-6133 


1. 046 1 


3 


4073 


1.0519 


3-2245 


1.0580 


3.0612 


56 


5 


1.0226 


4.7770 


1.0266 


4-4176 


1.0310 


4.1096 


1.0357 


3-8428 


55 


5 


1.0407 


5.6096 


1. 046 1 


3 


4041 


1.0520 


3.2216 


1. 0581 


3-0586 


%% 


6 


1.0227 


4.7706 


1.0267 


4.4121 


1-0311 


4.1048 


1.0358 


3-8387 


54 


6 


1.0408 


3.6060 


1 .0462 


3 


4009 


1.0521 


3.2188 


1.0582 


3-0561 


54 


7 


1.0228 


4.7641 


1.0268 


4-4065 


1.0311 


4.1001 


1.0358 


3-8346 


S3 


7 


1.0409 


3.6024 


1.0463 


3 


3977 


1.0522 


3-2159 


1.0584 


3-0535 


53 


8 


1.0228 


4.7576 


1.0268 


4.40 1 1 


1.0312 


4-0953 


I.03S9 


3-8304 


52 


8 


1.0410 


3-5987 


1 .0464 


3 


3945 


1-0523 


3-2I3I 


i.os8s 


3-0509 


Sa 


9 


1.0229 


4-7SI2 


1.0269 


4-3956 


'1-0313 


4.0906 


1.0360 


3-8263 


SI 


9 


1.0411 


3-5951 


1.046 s 


3 


3913 


1-0524 


3.2102 


1.0586 


3-0484 


51 


lo 


1.0230 


4.7448 


1.0270 


4-3910 


1-0314 


4.0859 


1.0361 


3.8222 


SO 


10 


1.0412 


3-5915 


1.0466 


3 


3881 


1.0525 


3-2074 


1.0587 


3-0458 


so 


II 


1.0230 


4.7384 


1.0271 


4-3847 


1.0314 


4.0812 


1.0362 


3.8181 


49 


11 


1.0413 


3-5879 


1.0467 


3 


3849 


1.0526 


3-2045 


1.0588 


3-0433 


4g 


12 


1.0231 


4.7320 


1.0271 


4-3792 


1-031S 


4-0765 


1.0362 


3-8140 


48 


12 


1.0413 


3-5843 


1.0468 


3 


3817 


1.0527 


3-2017 


1.0589 


3.0407 


48 


13 


1.0232 


4-7257 


1.0272 


4-3738 


1.0316 


4.0718 


1.0363 


3.8100 


47 


13 


1.0414 


3-5807 


1 .0469 


3 


3785 


1.0528 


3-1989 


1.0590 


3.0382 


47 


14 


1.0232 


4-7193 


1.0273 


4-3684 


1.0317 


4.0672 


1.0364 


3-8059 


46 


14 


1.0415 


3-5772 


1.0470 


3 


3754 


1.0529 


3-1960 


1.0591 


3-03S7 


46 


IS 


1.0233 


4-7130 


1.0273 


4-3630 


1-0317 


4-0625 


1.0365 


3.8018 


4S 


IS 


1.0416 


3-5736 


1.0471 


3 


3722 


1.0530 


3-1932 


1.0592 


3-0331 


45 


i6 


1.0234 


4-7067 


1.0274 


4-3576 


1.0318 


4-0579 


1.0366 


3-7978 


44 


16 


1.0417 


3-5700 


1.0472 


3 


3690 


1-0531 


3-1904 


1-0593 


3.0306 


44 


17 


1.0234 


4.7004 


1.027s 


4-3522 


1.0319 


4-0532 


1.0367 


3-7937 


43 


17 


1.0418 


35665 


1-0473 


3 


3659 


1-0532 


3-1876 


1.0594 


3.0281 


43 


i8 


1.0235 


4.6942 


1.0276 


4,3469 


1.0320 


4.0486 


1.0367 


3-7897 


42 


18 


1.0419 


3-5629 


1.0474 


3 


3627 


1-0533 


3-1848 


1-0595 


3-0256 


42 


19 


1.0235 


4.6879 


1.0276 


4-3415 


1.0320 


4.0440 


1.0368 


3-7857 


41 


19 


1.0420 


3-5594 


1-047S 


3 


3596 


1-0534 


3.1820 


1-0596 


3-0231 


41 


20 


1.0236 


4.6817 


1.0277 


4-3362 


1.0321 


4-0394 


1.0369 


3-7816 


40 


20 


1.0420 


3-5559 


1.0476 


3 


3565 


1-053S 


3-1792 


1.0598 


3.0206 


40 


21 


1.0237 


4-6754 


1.0278 


4-3309 


1.0322 


4-0348 


1.0370 


3-7776 


39 


21 


1.0421 


3-5523 


1-0477 


3 


3534 


1.0536 


3-1764 


1.0S99 


3.0181 


39 


22 


1.0237 


4.6692 


1.0278 


4-3256 


1.0323 


4.0302 


1.0371 


3-7736 


38 


22 


1.0422 


3.5488 


1.0478 


3 


3502 


I -0537 


3-1736 


1.0600 


3-0156 


33 


23 


1.0238 


4.6631 


1.0279 


4-3203 


1.0323 


4.0256 


1.0371 


3-7697 


37 


23 


1.0423 


3-5453 


1.0478 


3 


3471 


1-0538 


3.1708 


1. 0601 


3-0131 


37 


24 


1.0239 


4.6569 


1.0280 


4-3150 


1.0324 


4.0211 


1.0372 


3-7657 


36 


24 


1.0424 


3-5418 


1.0479 


3 


3440 


1-0539 


3.1681 


1.0602 


3.0106 


36 


25 


1.023P 


4-6507 


1.0280 


4-3098 


1-0325 


4-0165 


1-0373 


3-7617 


35 


25 


1.042 s 


3-5383 


1 .0480 


3 


3409 


1-0540 


3-1653 


1.0603 


3.0081 


35 


26 


1.0240 


4.6446 


1.0281 


4-3045 


1.0326 


4.0120 


1-0374 


3-7577 


34 


26 


1.0426 


3-5348 


1. 048 1 


3 


3378 


1.0541 


3-1625 


1.0604 


3-0056 


34 


27 


1.0241 


4-6385 


1.0282 


4-2993 


1.0327 


4.0074 


1.0375 


3-7538 


33 


27 


1.0427 


3-5313 


1.0482 


3 


3347 


1.0542 


3-1598 


1.0605 


3-0031 


33 


28 


1.0241 


4.6324 


1.0283 


4.2941 


1-0327 


4.0029 


1.0376 


3-7498 


32 


28 


1.0428 


3-5279 


1.0483 


3 


3316 


1-0543 


3-1570 


1.0606 


3-0007 


32 


29 


1.0242 


4.6263 


1.0283 


4.2888 


1.0328 


3.9984 


1.0376 


3-7459 


31 


29 


1.0428 


3-5244 


1.0484 


3 


3286 


1-0544 


3-1543 


1.0607 


2.9982 


31 


30 


1.0243 


4.6202 


1.0284 


4.2836 


1.0329 


3-9939 


I-0377 


3.7420 


30 


30 


1.0429 


3-5209 


1.0485 


3 


3255 


1-054S 


3-1515 


1.0608 


2.9957 


30 


31 


1.0243 


4.6142 


1.0285 


4-2785 


1.0330 


3-9894 


1.0378 


3-7380 


29 


31 


1 .0430 


3-5175 


1.0486 


3 


3224 


1.0546 


3-1488 


1 .0609 


2.9933 


29 


32 


1.0244 


4.6081 


1.0285 


4-2733 


1.0330 


3-9850 


1.0379 


3-7341 


28 


32 


1.0431 


3-5140 


1.0487 


3 


3194 


1-0547 


3-1461 


I.0611 


2.9908 


28 


33 


1.024s 


4.6021 


1.0286 


4.2681 


1-0331 


3 -9805 


1.0380 


3-7302 


27 


33 


1.0432 


3-5106 


1.0488 


3 


3163 


1-0548 


3-1433 


1. 0612 


2.9884 


27 


34 


1.0245 


4.5961 


1.0287 


4-2630 


1-0332 


3-9760 


1.0381 


3-7263 


26 


34 


1-0433 


3-5072 


1.0489 


3 


3133 


1-0549 


3-1406 


1.0613 


2.9859 


26 


3S 


1.0246 


4.5901 


1.0288 


4-2579 


1-0333 


3-9716 


1.0382 


3.7224 


25 


35 


1.0434 


3-5037 


1.0490 


3 


3102 


1-0550 


3-1379 


1.0614 


2.9835 


25 


36 


1.0247 


4.5841 


1.0288 


4-2527 


1-0334 


3.9672 


1.0382 


3.7186 


24 


36 


1-0435 


3-5003 


1. 049 1 


3 


3072 


1-0551 


3-i3'52 


1.061S 


2.9810 


24 


^l 


1.0247 


4.5782 


1 .0289 


4-2476 


1-0334 


3-9627 


1.0383 


37147 


23 


37 


1-0436 


3.4969 


1.0492 


3 


3042 


1.0552 


3-1325 


I.0616 


2.9786 


23 


38 


1.0248 


4.5722 


1.0290 


4-2425 


1-0335 


3-9583 


1.0384 


3-7108 


22 


38 


1-0437 


3-4935 


1.0493 


3 


3011 


I-05S3 


3.1298 


1.0617 


2.9762 


22 


39 


1.0249 


4-5663 


1.0291 


4-2375 


1-0336 


3-9539 


1.038s 


3.7070 


21 


39 


1-0438 


3-4901 


1.0494 


3 


2981 


1-0554 


3-1271 


1.0618 


2.9738 


21 


40 


1.0249 


4-5604 


1.0291 


4-2324 


1-0337 


3-9495 


1.0386 


3-7031 


20 


40 


1-0438 


3-4867 


1.0495 


3 


2951 


1-0555 


3-1244 


1.0619 


2.9713 


20 


41 


1.0250 


4-SS4S 


1 .0292 


4-2273 


1-0338 


3-94SI 


1.0387 


3-6993 


19 


41 


1.0439 


3-4833 


1.0496 


3 


2921 


1-0556 


3-I2I7 


1.0620 


2.9689 


19 


42 


1.0251 


4.5486 


1.0293 


4-2223 


1-0338 


3-9408 


1.0387 


3-6955 


18 


42 


1.0440 


3-4799 


1.0497 


3 


2891 


I-OS57 


3-1190 


1.0622 


2.9665 


18 


43 


1.0251 


4-5428 


1.0293 


4-2173 


I -0339 


3-9364 


1.0388 


3-6917 


17 


43 


1.0441 


3-4766 


1.0498 


3 


2861 


1-0558 


3-1163 


1.0623 


2.9641 


17 


44 


1.0252 


4-5369 


1.0294 


4.2122 


1.0340 


3.9320 


1.0389 


3-6878 


16 


44 


1.0442 


3-4732 


1.0499 


3 


2831 


1-OS59 


3-1137 


1.0624 


2.9617 


16 


4S 


I.0253 


4-53" 


1.0295 


4.2072 


1.0341 


3-9277 


1.0390 


3.6840 


IS 


45 


1.0443 


3.4698 


1.0500 


3 


2801 


1.0560 


3-1110 


1.0625 


2-9593 


15 


46 


1.02S3 


4S253 


1.0296 


4.2022 


1.0341 


3-9234 


1.0391 


3.6802 


14 


46 


1.0444 


3-4665 


1.0501 


3 


2772 


1.0561 


3-1083 


1.0626 


2.9569 


14 


47 


1.0254 


4-S195 


1.0296 


4.1972 


1.0342 


3-9199 


1.0392 


3.6765 


13 


47 


1.0445 


3-4632 


1.0502 


3 


2742 


1.0562 


3-1057 


1.0627 


2 -9545 


13 


48 


I.025S 


4-5137 


1.0297 


4-1923 


1-0343 


3-9147 


1-0393 


3.6727 


12 


48 


1 .0446 


3-4598 


1.0503 


3 


2712 


1.0563 


3-1030 


1.0628 


2.9521 


12 


49 


I.02SS 


4-5079 


1.0298 


4-1873 


1-0344 


3.9104 


1-0393 


3.6689 


11 


49 


1.0447 


3-4565 


1.0504 


3 


2683 


1.0565 


3.1004 


1.0629 


2.9497 


11 


SO 


1.0256 


4-5021 


1.0299 


4.1824 


1.0345 


3-9061 


1.0394 


3.6651 


10 


SO 


1.0448 


3-4532 


I.OS05 


3 


2653 


1.0566 


3.0977 


1.0630 


2.9474 


10 


SI 


1.0257 


4.4964 


1.0299 


4-1774 


1-0345 


3.9018 


1-039S 


3-6614 


g 


SI 


1.0448 


3.4498 


1.0506 


3 


2624 


1.0567 


3.0951 


1.0632 


2.9450 


9 


S2 


1.0257 


4.4907 


1 .0300 


4-1725 


1-0346 


3-8976 


1-0396 


3-6576 


8 


52 


1.0449 


3-4465 


1.0507 


3 


2594 


1.0568 


3-0925 


1.0633 


2.9426 


8 


53 


1.0258 


4.4850 


1.0301 


4.1676 


1-0347 


3-8933 


1-0397 


3-6539 


7 


S3 


1.0450 


3-4432 


1.0508 


3 


2565 


1.0569 


3.0898 


1.0634 


2.9402 


7 


54 


1.0259 


4-4793 


1.0302 


4.1627 


1-0348 


3.8990 


1.0398 


3-6502 


6 


54 


1.0451 


3-4399 


1.0509 


3 


2535 


1.0570 


3-0872 


1.063s 


2.9379 


6 


55 


1.0260 


4-4736 


1.0302 


4-1578 


1-0349 


3.8848 


1.0399 


3.6464 


5 


55 


1.0452 


3-4366 


1.0510 


3 


2506 


1-0571 


3.0846 


1.0636 


2.935s 


5 


56 


1.0260 


4.4679 


1.0303 


4.1529 


1-0349 


3-880S 


1.0399 


3.6427 


4 


56 


1.0453 


3-4334 


1.0511 


3 


2477 


1.0572 


3.0820 


1.0637 


2.9332 


4 


^l 


i.o26r 


4.4623 


1.0304 


4.1481 


1-0350 


3-8763 


1.0400 


3.6390 


3 


57 


1.0454 


3-4301 


1.0512 


3 


2448 


1-OS73 


3-0793 


1.0638 


2.9308 


3 


S8 


1.0262 


4-4566 


1.0305 


4-1432 


1-0351 


3-8721 


1.0401 


3-6353 


2 


58 


I -0453 


3-4268 


1-0513 


3 


2419 


I-OS74 


3.0767 


1.0639 


2.9285 


2 


S9 


1.0262 


4.4510 


1.0305 


4-1384 


1.0352 


3-8679 


1.0402 


3-6316 


I 


59 


1-0456 


3-4236 


I.0514 


3 


2390 


1-OS75 


3-0741 


1. 064 1 


2.9261 


I 


60 


1.0263 


4-4454 


1.0306 


4.1336 


1-0353 


3-8637 


1.0403 


3.6279 





60 


1.0457 


3-4203 


1.0SIS 


3 


2361 


1-0576 


3-0715 


1.0642 


2.9238 





' 


Co-sec. 


Sec. 


Co-sec. 


Sec. 


Co-sec. 


Sec. 


CO-SEC. 


Sec. 


/ 


Co-sec. 


Sec. 


CO-SEC. 


Sec. 


Co-SEC. 


Sec. 


Co-SEC. 


Sec. 


/ 




7' 


7° 


7 


3° 


7 


5° 


7 


J" 






7; 


}° 


r. 


JO 


7] 


L° 


7( 


)° 





520 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table s- — Natural Trigonometric Functions — (Continued) 





20° 1 


21° 1 


22° I 


23° I 






24° 


25° 


26° 


27° 




t 


Sec. 


Co-SEC. 


Sec. 


Co-SEC 


Sec. 


Co-SEC. 


Sec. 


Co-SEC. 


/ 


/ 


Sec. 


Co-SEC. 


Sec. 


Co-SEC. 


Sec 


Co-SEC. 


Sec 


Co-SEC. 


/ 


o 


1 .0642 


2.9238 


1.0711 


2.7904 


1.0785 


2.6695 


1.0864 


2.5593 


60 





1.0946 


2.4586 


1.1034 


2.3663 


1.1126 


2.2812 


1.1223 


2.2027 


60 


I 


1.0643 


2.921S 


1.0713 


2.7883 


1.0787 


2.6675 


1.086s 


2.5575 


59 


I 


1.0948 


2.4570 


1.1035 


2.3647 


1.1127 


2.2798 


1.1225 


2.2014 


59 


2 


1.0644 


2.9I91 


1.0714 


2.7862 


1.0788 


2.6656 


1.0866 


2.5558 


58 


2 


1.0949 


2.4554 


1.1037 


2.3632 


1.1129 


2.2784 


1.X226 


2.2002 


58 


3 


1.0645 


2.9168 


1.071S 


2.7841 


1.0789 


2.6637 


1.0868 


2.5540 


57 


3 


1.0951 


2.4538 


1.1038 


2.3618 


1.1x31 


2.2771 


1.1228 


2.1989 


57 


4 


1.0646 


2.914s 


1.0716 


2.7820 


1.0790 


2.66i8 


1.0869 


2.5523 


56 


4 


1.0952 


2.4522 


1.1040 


2.3603 


1.1132 


2.2757 


1.1230 


2.1977 


S6 


5 


1.0647 


2.9122 


1.0717 


2.7799 


1.0792 


2.6599 


1.0870 


2.5506 


55 


S 


1.0953 


2.4506 


1.1041 


2.3588 


1.1134 


2.2744 


1.1231 


2.1964 


55 


6 


1 .0648 


2.9098 


1.0719 


2.7778 


1.0793 


2.6580 


1.0872 


2.5488 


54 


6 


1.095s 


2.4490 


1. 1043 


2.3574 


1.1135 


2.2730 


1.1233 


2.1952 


54 


7 


1.0650 


2.9075 


1.0720 


2.7757 


1.0794 


2.6561 


1.0873 


2.5471 


S3 


7 


1.0956 


2.4474 


1.1044 


2.3559 


I.II37 


2.2717 


1.123s 


2.1939 


53 


8 


1.0651 


2.9052 


1.0721 


2.7736 


1.0795 


2.6542 


1.0874 


2.54S3 


52 


8 


1.0958 


2.4458 


1.1046 


2.3544 


1.1139 


2.2703 


1.1237 


2.1927 


52 


9 


1.0652 


2.9029 


1.0722 


2.7715 


1.0797 


2.6523 


1.0876 


2.S436 


51 


9 


1.0959 


2.4442 


1.1047 


2.3530 


1.1140 


2.2690 


1.1-38 


2.1914 


51 


lo 


1.0653 


2.9006 


1.0723 


2.7694 


1.0798 


2.6504 


1.0877 


2.5419 


50 


10 


1.0961 


2.4426 


1.1049 


2.3515 


1.1142 


2.2676 


1.1240 


2.1902 


50 


II 


1.0654 


2.8983 


1.0725 


2.7674 


1.0799 


2.6485 


1.0878 


2.5402 


49 


II 


1.0962 


2.4411 


1.1050 


2.3501 


11143 


2.2663 


1. 1242 


2.1889 


49 


12 


1.0655 


2.8960 


1.0726 


2.7653 


1.0801 


2.6466 


1.0880 


2.5384 


48 


12 


1.0963 


2.4395 


1.1052 


2.3486 


1.1145 


2.2650 


1.1243 


2.1877 


48 


13 


1.0656 


2.8937 


1.0727 


2.7632 


1.0802 


2.6447 


1. 088 1 


2.5367 


47 


13 


1.0965 


2.4379 


1.1053 


2.3472 


1.1147 


2.2636 


1.1245 


2.1865 


47 


14 


1.0658 


2.8915 


1.0728 


2.7611 


1.0803 


2.6428 


1.0882 


2.5350 


46 


14 


1 .0966 


2.4363 


1.105s 


2-3457 


1.1x48 


2 2623 


1.1247 


2.1852 


46 


IS 


1.0659 


2.8892 


1.0729 


2.7591 


1.0804 


2.64x0 


1.0884 


2.5333 


45 


15 


1.0968 


2.4347 


1.1056 


2.3443 


1.1150 


2.2610 


1.1248 


2.1840 


45 


i6 


1.0660 


2.8869 


1.0731 


2.7570 


1.0806 


2.6391 


1.0885 


2.5316 


44 


16 


1.0969 


2.4332 


1.1058 


2.3428 


1.1151 


2.2596 


1.1250 


2.1828 


44 


17 


1. 066 1 


2.8846 


1.0732 


2.7550 


1.0807 


2.6372 


1.0886 


2.5299 


43 


17 


1.0971 


2.4316 


1.1059 


2.3414 


1.1153 


2.2583 


1.1252 


2.181S 


43 


i8 


1 .0662 


2.8824 


1.0733 


2.7529 


1.0808 


2.6353 


1.0888 


2.5281 


42 


18 


1.0972 


2.4300 


i.io6i 


2.3399 


I.1I55 


2.2570 


1.1253 


2.1803 


42 


19 


1.0663 


2.8801 


I.0734 


2.7509 


1.08x0 


2.6335 


1.0889 


2.5264 


41 


19 


1.0973 


2.4285 


1.1062 


2.338s 


1.1156 


2.2556 


1.1255 


2.1791 


41 


20 


1 .0664 


2.8778 


1.0736 


2.7488 


1.081 1 


2.6316 


1. 0891 


2.5247 


40 


20 


1.097s 


2.4269 


1.1064 


2.3371 


1.1158 


2.2543 


I.12S7 


2.1778 


40 


21 


1 .0666 


2.8756 


1.0737 


2.7468 


1.0812 


2.6297 


1.0892 


2.5230 


39 


21 


1.0976 


2.4254 


1.1065 


2.3356 


1.1159 


2.2530 


1.1258 


2.1766 


39 


22 


1.0667 


2.8733 


1.0738 


2.7447 


1.0813 


2.6279 


1.0893 


2.5213 


38 


22 


1.0978 


2.4238 


1.X067 


2.3342 


1.1161 


2.2517 


1. 1260 


2.1754 


38 


23 


1.0668 


2.871 1 


1.0739 


2.7427 


1.081S 


2.6260 


1.089s 


2.5196 


^1 


23 


1.0979 


2.4222 


1.1068 


2.3328 


1.1x63 


2.2503 


1.1262 


2.1742 


37 


24 


1.0669 


2.8688 


1.0740 


2.7406 


1.0816 


2.6242 


1 .0896 


25179 


36 


24 


1. 098 1 


2.4207 


1.1070 


2.3313 


1.1164 


2.2490 


1.1264 


2.1730 


36 


2S 


1.0670 


2.8666 


1.0742 


2.7386 


1.0817 


2.6223 


1.0897 


2.5163 


35 


25 


1.0982 


2.4x91 


1.1072 


2.3299 


1.1x66 


2.2477 


1.126s 


2.1717 


3S 


26 


1. 067 1 


2,8644 


1.0743 


2.7366 


1.0819 


2.6205 


1 .0899 


2.5146 


34 


26 


1 .0984 


2.4176 


1.1073 


2.328s 


1.1167 


2.2464 


1. 1267 


2.1705 


34 


27 


1.0673 


2.8621 


1.0744 


2.7346 


1 .0820 


2.6186 


1.0900 


2.5129 


33 


27 


1.0985 


2.4160 


1. 107s 


2.3271 


1.1x69 


2.2451 


1.1269 


2.1693 


33 


28 


1.0674 


2.8599 


1.074s 


2.732s 


1. 082 1 


2.6168 


1.0902 


2.5112 


32 


28 


1.0986 


2.4145 


1.1076 


2.3256 


1.1171 


2.2438 


1. 1270 


2.1681 


32 


29 


1.0675 


2.8577 


1.0747 


2.7305 


1.0823 


2.6x50 


1.0903 


2.5095 


31 


29 


1.0988 


2.4130 


1.1078 


2.3242 


1.1172 


2.2425 


1. 1272 


2.1669 


31 


30 


1.0676 


2.8554 


1.0748 


2.728s 


1.0824 


2.6131 


1.0904 


2.5078 


30 


30 


1.0989 


2.4114 


1.1079 


2.3228 


1.1174 


2.24II 


1.1274 


2.1657 


30 


31 


1.0677 


2.8532 


1.0749 


2.726s 


1.0825 


2.6113 


1.0906 


2.5062 


29 


31 


1.0991 


2.4099 


I.1081 


2.3214 


I.1176 


2.2398 


1.127s 


2.164s 


29 


32 


1.0678 


2.8510 


1.0750 


2.724s 


1.0826 


2.6095 


1.0907 


2.5045 


28 


32 


1.0992 


2.4083 


1.1082 


2.3200 


1.1177 


2.2385 


1.1277 


2.1633 


28 


33 


1.0679 


2.8488 


1.0751 


2.722s 


1.0828 


2.6076 


1.0908 


2.5028 


27 


33 


1.0994 


2.4068 


1.1084 


2.3186 


1.1179 


2.2372 


1. 1279 


2.1620 


27 


34 


1. 068 1 


2.8466 


I.0753 


2.720s 


1.0829 


2.6058 


1.0910 


2.5011 


26 


34 


1.0995 


2.4053 


1. 1085 


2.3172 


1.1180 


2.2359 


1.1281 


2.1608 


26 


3S 


1.0682 


2.8444 


1.0754 


2.7185 


1.0830 


2.6040 


1.0911 


2.499s 


25 


35 


1.0997 


2.4037 


1.1087 


2.3158 


1.1182 


2.2346 


1.1282 


2.1596 


2S 


36 


1.0683 


2.8422 


1.07SS 


2.7165 


1.0832 


2.6022 


1.0913 


2.4978 


24 


36 


1.0998 


2.4022 


1.1088 


2.3143 


1.1184 


2.2333 


1. 1284 


2.1584 


24 


37 


1.0684 


2.8400 


1.0756 


2.714s 


1.0833 


2.6003 


1.0914 


2.4961 


23 


37 


1. 1000 


2.4007 


1.1090 


2.3x29 


1.1185 


2.2320 


1.1286 


2.1572 


23 


38 


1.0685 


2.8378 


1.0758 


2.7125 


1.0834 


2.5985 


1.091S 


2.4945 


22 


38 


1. 1001 


2.3992 


1.1092 


2.3I1S 


1.1187 


2.2307 


1.X287 


2.1560 


22 


39 


1.0686 


2.8356 


I.07S9 


2.7105 


1.0836 


2.5967 


1.0917 


2.4928 


21 


39 


I. 1003 


2.3976 


1.1093 


2.3IOI 


1.1189 


2.2294 


1.1289 


2.1548 


21 


40 


1.0688 


2.8334 


1.0760 


2.7085 


1.0837 


2.5949 


1.0918 


2.4912 


20 


40 


1.1004 


2.3961 


1. 109s 


2.3087 


1. 1190 


2.2282 


1.1291 


2.1536 


20 


41 


1.0689 


2.8312 


1. 076 1 


2.706s 


1.0838 


2.5931 


1.0920 


2.489s 


19 


41 


i.ioos 


2.3946 


1.1096 


2.3073 


1.1192 


2.2269 


1.1293 


2.1525 


19 


42 


I .o6qo 


2.8290 


1.0763 


2.7045 


1.0840 


2.5913 


1.0921 


2.4879 


18 


42 


1.1007 


2.3931 


1.1098 


2.3059 


1.1193 


2.2256 


1.1294 


2.1513 


18 


43 


1. 069 1 


2.8269 


1.0764 


2.7026 


1.0841 


2.589s 


1.0922 


2.4862 


17 


43 


1. 1008 


2.3916 


1.1099 


2.3046 


1. 1195 


2.2243 


1.1296 


2.1501 


17 


44 


1.0692 


2.8247 


1.076s 


2.7006 


1 .0842 


2.5877 


1.0924 


2.4846 


16 


44 


1. 1010 


2.3901 


I.IIOI 


2.3032 


1.1197 


2.2230 


1.1298 


2.1489 


16 


45 


1.0694 


2.8225 


1.0766 


2.6986 


1.0844 


2.5859 


1.0925 


2.4829 


IS 


45 


I.IOII 


2.3886 


1.1102 


2.3018 


1.1198 


2.2217 


1.1299 


2.X477 


15 


46 


■ 1.0695 


2.8204 


1.0768 


2.6967 


1.0845 


2.5841 


1.0927 


2.4813 


14 


46 


1.1013 


2.3871 


I.X104 


2.3004 


1.1200 


2.2204 


1.1301 


2.1465 


14 


47 


1 .0696 


2.8182 


1.0769 


2.6947 


1.0846 


2.5823 


1.0928 


2.4797 


13 


47 


1. 1014 


2.3856 


1.1106 


2.2990 


1.1202 


2.2192 


1.1303 


2.1453 


13 


48 


1.0697 


2.8160 


1.0770 


2.6927 


1.0847 


2.5805 


1.0929 


2.4780 


12 


48 


1.1016 


2.3841 


1.1107 


2.2976 


1.X203 


2.2179 


1.1305 


2.1441 


12 


49 


1.0698 


2.8139 


1.0771 


2.6908 


1.0849 


2.S787 


1. 093 1 


2.4764 


II 


49 


1. 1017 


2.3826 


1.H09 


2.2962 


1.1205 


2.2166 


1.X306 


2.X430 


II 


SO 


1.0699 


2.8117 


1.0773 


2.6888 


1.0850 


2.5770 


1.0932 


2.4748 


10 


50 


1.1019 


2.3811 


1. 1110 


2.2949 


I.1207 


2.2153 


1.1308 


2.1418 


10 


51 


1. 0701 


2.8096 


1.0774 


2.6869 


1.0851 


2.5752 


1.0934 


2.4731 


9 


51 


1. 1020 


2.3796 


1.1112 


2.293s 


1.1208 


2.2I4I 


1.1310 


2.1406 


9 


52 


1.0702 


2.8074 


I.077S 


2.6849 


1.0853 


2.5734 


1.0935 


2.471s 


8 


52 


1.1022 


2.3781 


1.1113 


2.2921 


1. 1210 


2.2128 


1. 1312 


2.1394 


8 


S3 


1.0703 


2.8053 


1.0776 


2.6830 


1.0854 


2.5716 


1.0936 


2.4699 


7 


S3 


I. 1023 


2.3766 


I. Ills 


2.2907 


1.1212 


2.2115 


1.1313 


2.X382 


7 


S4 


1.0704 


2.8032 


1.0778 


2.6810 


I.08SS 


2.5699 


1.0938 


2.4683 


6 


54 


1. 1025 


2.3751 


1. 1116 


2.2894 


1.1213 


2.2103 


1.1315 


2.1371 


6 


SS 


1.070s 


2.8010 


1.0779 


2.6791 


1.0857 


2.56S1 


1.0939 


2.4666 


5 


55 


1.1026 


2.3736 


1.1118 


2.2880 


I.X2I5 


2.2090 


1.1317 


2.1359 


5 


S6 


1.0707 


2.7989 


1.0780 


2.6772 


1.0858 


2.5663 


1.0941 


2.4650 


4 


56 


1. 1028 


2.3721 


1.1120 


2.2866 


1.12x7 


2.2077 


1.1319 


2.1347 


4 


57 


1.0708 


2.7968 


1.0781 


2.6752 


1.0859 


2.5646 


1.0942 


2.4634 


3 


57 


1.1029 


2.3706 


1.1121 


2.2853 


1.12X8 


2.2065 


1.1320 


2.1335 


3 


58 


1.0709 


2.7947 


1.0783 


2.6733 


1. 086 1 


2.5628 


1.0943 


2.4618 


2 


58 


1.1031 


2.3691 


I.XI23 


2.2839 


1.1220 


2.2052 


1.1322 


2.1324 


2 


59 


1.0710 


2.7925 


1.0784 


2.6714 


1.0862 


2.56x0 


1.0945 


2.4602 


I 


59 


1.X032 


2.3677 


1. 1124 


2.2825 


1.1222 


2.2039 


1.1324 


2.1312 


I 


60 


1.0711 


2.7904 


1.0785 


2.669s 


1.0864 


2.5593 


1 .0946 


2.4586 





60 


1.1034 


2.3662 


1.1126 


2.2812 


1.1223 


2.2027 


1.1326 


2.1300 





"7" 


Co-SEC. 


Sec. 


Co-SEC. 


Sec. 


Co-SEC. 


Sec. 


Co-SEC 


Sec. 


/ 


' 


Co-SEC. 


Sec 


Co-SEC. 


Sec. 


Co-SEC. 


Sec. 


Co-SEC. 


Sec. 


• 




6< 


3° 


6i 


i° 


6 


r° 


1 6 


5° 






6; 


i° 


6^ 


t° 


6i 


i° 


62 


jO 





MATHEMATICAL TABLES 



521 









1 


■B 


Table 5. — 


STATURAL Trigonometric Functions— 


•(Continued) 














28° 1 


29° 1 


30° 1 


31° 1 






32° 1 


33° i 


34° 1 


35° 






t 


Sec. 


Co-sec. 


Sec. 


Co-SEC. 


Sec. 


Co-SEC. 


Sec. 


Co-SEC. 


/ 
60 


' 


Sec. 


Co-SEC. 


Sec. 


Co-SEC 


Sec. 


Co-SEC. 


Sec. Cc 


l-SEC. 


/ 


o 


1.1326 


2.1300 


1-1433 


2.0627 


1-IS47 


2.0000 


1.1666 


1.9416 





1.1792 


1.8871 


1.1924 


1.8361 


1.2062 


I.78S3 


1.2208 1 


7434 


60 


I 


1.1327 


2.1289 


1-1435 


2.0616 


1-1549 


1.9990 


1. 1668 


1.9407 


59 


I 


1-1794 


1.8862 


1.1926 


1-8352 


1.2064 


1.7875 


1.2210 1 


7427 


59 


2 


1.1329 


2.1277 


1. 1437 


2.0605 


1-1551 


1 .9980 


1.1670 


1-9397 


58 


2 


1.1796 


1-8853 


1. 1928 


1.8344 


1.2067 


1.7867 


1.2213 I 


7420 


58 


3 


1-1331 


2.1266 


1. 1439 


2.0594 


1-1553 


1-9970 


1.1672 


1-9388 


57 


3 


1.1798 


1.S844 


I -1930 


1.8336 


1.2069 


1.7860 


1.221s I 


7413 


57 


4 


I-I333 


2.1254 


1.1441 


2.0583 


1-1555 


1.9960 


1.1674 


1-9378 


56 


4 


1.1800 


1.8836 


1-1933 


1.8328 


1.2072 


1.7852 


1.2218 1 


740s 


S6 


S 


1-1334 


2.1242 


1-1443 


2.0573 


I.I557 


1.9950 


1.1676 


1.9369 


55 


S 


1.1802 


1.8827 


1-1935 


1.8320 


1.2074 


1.7844 


1,2220 1 


7398 


55 


6 


1.1336 


2.1231 


1-I44S 


2.0562 


1-1559 


1.9940 


1.1678 


1.9360 


54 


6 


I. 1805 


1.8818 


1-1937 


I.831I 


1.2076 


1-7837 


1.2223 I 


7391 


54 


7 


1.1338 


2.1219 


1.1446 


2.0551 


1-1561 


1-9930 


1.1681 


I -93 SO 


S3 


7 


1.1807 


1.8809 


1-1939 


1.8303 


1.2079 


1.7829 


1.2225 I 


7384 


53 


8 


1.1340 


2.1208 


1.1448 


2.0540 


1.1562 


1 .9920 


1.1683 


I -9341 


53 


8 


1.1809 


1.8801 


1.1942 


1-8295 


1.2081 


1.7821 


1.2228 1 


7377 


52 


9 


1.1341 


2.1 196 


1. 1450 


2.0530 


1-1564 


1.9910 


1. 1685 


19332 


SI 


9 


1. 1811 


1.8792 


1.1944 


1.8287 


1.2083 


1.7814 


1.2230 1 


7369 


51 


lo 


1-1343 


2.1185 


1.1452 


2.0519 


1.1566 


1.9900 


1.1687 


1-9322 


SO 


10 


1.1813 


1.8783 


1.1946 


1.8279 


1.2086 


1.7806 


1.2233 I 


7362 


SO 


11 


I-I34S 


2.1173 


1. 1454 


2.0508 


1.1568 


1.9890 


1.1689 


1-9313 


49 


11 


1. 1815 


1.8785 


1.1948 


I.8271 


1.2088 


1.7798 


1.2235 I 


7355 


49 


12 


I-I347 


2.1162 


1.1456 


2.0498 


1.1570 


1.9880 


1.1691 


1.9304 


48 


12 


1.1818 


1.8766 


I- 1 95 1 


1.8263 


1.2091 


1-7791 


1.2238 1 


7348 


48 


13 


1-1349 


2.1150 


1.1458 


2.0487 


1-1572 


1.9870 


1.1693 


1-9295 


47 


13 


1.1820 


I-87S7 


I-I953 


1-8255 


1-2093 


1-7783 


1.2240 1 


7341 


47 


14 


1-1350 


2.1139 


1-1459 


2.0476 


1-1574 


1.9860 


1.169s 


1-9285 


46 


14 


1.1822 


1.8749 


I-I9SS 


1.8246 


1-2095 


1-7776 


1.2243 1 


7334 


46 


IS 


1-1352 


2.1127 


1.1461 


2.0466 


1.1576 


1.9850 


1.1697 


1.9276 


45 


15 


1.1824 


1.8740 


1-1958 


1.8238 


1.2098 


1.7768 


1.2245 I 


7327 


45 


i6 


1-1354 


2.1116 


1-1463 


2 .045 5 


1.1578 


1.9840 


1.1699 


1.9267 


44 


16 


1.1826 


1.8731 


H960 


1.8230 


1.2100 


1.7760 


1.2248 1 


7319 


44 


17 


1-1356 


2.1104 


1.1465 


2.0444 


1.1580 


1.9830 


1.1701 


1-9258 


43 


17 


1.1828 


1.8723 


1.1962 


1.8222 


1.2103 


1-7753 


1.2250 1 


7312 


43 


i8 


1-1357 


2.1093 


1. 1467 


2.0434 


1.1582 


1.9820 


1. 1703 


1.9248 


42 


18 


1.1831 


1.8714 


1.1964 


1.8214 


1.2105 


1-7745 


I-22S3 1 


7305 


42 


19 


1-1359 


2.1082 


I. 1469 


2.0423 


1.1584 


1.9811 


1-1705 


1.9239 


41 


19 


1-1833 


1.8706 


1.1967 


1.8206 


1.2107 


1-7738 


1-2255 1 


7298 


41 


lO 


I.1361 


2.1070 


1.1471 


2.0413 


1.1586 


1.9801 


1. 1 707 


1.9230 


40 


20 


1.1835 


1.8697 


1. 1969 


1.8198 


1.2110 


1-7730 


1.2258 I 


7291 


40 


21 


1.1363 


2.1059 


I-I473 


2.0402 


1.1588 


1.9791 


1.1709 


1.9221 


39 


21 


1.1837 


1.8688 


1. 1971 


1.8190 


1.2112 


1-7723 


1.2260 I 


7284 


39 


22 


I-136S 


2.1048 


1.1474 


2.0392 


1.1590 


1.9781 


1.1712 


1.9212 


38 


22 


1-1839 


1.8680 


1.1974 


I.8182 


1.2115 


1-771S 


1.2263 1 


7277 


38 


23 


1.1366 


2.1036 


1. 1476 


2.0381 


1.1592 


1.9771 


1.1714 


1.9203 


37 


23 


1.1841 


1.8671 


1.1976 


1.8174 


1.2117 


1-7708 


1.2265 1 


7270 


37 


24 


1.1368 


2.1025 


1.1478 


2.0J70 


1.IS94 


I.976I 


1. 1716 


1-9193 


36 


24 


1.1844 


1.8663 


1.1978 


1.8166 


1.2119 


1.7700 


1.2268 1 


7263 


36 


25 


1.1370 


2.1014 


1.1480 


2.0360 


1.1596 


1.9752 


1. 1718 


1.9184 


35 


25 


1.1846 


1.8654 


1.1980 


1.8158 


1.2122 


1.7693 


1.2270 1 


7256 


35 


26 


1.1372 


2.1002 


1.1482 


2.0349 


1.1598 


1.9742 


1.1720 


1-9175 


34 


26 


1.1848 


1.8646 


1.1983 


1.8150 


I.2I24 


1.7685 


1.2273 I 


7249 


34 


27 


I-I373 


2.0991 


1.1484 


2.0339 


1.1600 


1-9732 


1.1722 


1.9166 


33 


27 


1.1850 


1-8637 


1.198s 


1.8142 


1.2127 


1.7678 


1.2276 I 


7242 


33 


28 


1-I37S 


2.0980 


1.1486 


2.0329 


1.1602 


1.9722 


1.1724 


1-9157 


3» 


28 


1-1852 


1.8629 


1.1987 


1-8134 


1.2129 


1.7670 


1.2278 I 


7234 


32 


29 


1-1377 


2.0969 


1. 1488 


2.0318 


1.1604 


I-97I3 


1.1726 


1.9148 


31 


29 


1-1855 


1.8620 


1.1990 


1.8126 


1.2132 


1.7663 


1.2281 1 


7227 


31 


3° 


1-1379 


2.09S7 


1.1489 


2.0308 


1.1606 


1-9703 


1.1728 


1-9139 


30 


30 


1.1857 


I.861I 


1.1992 


1.8118 


1.2154 


1.7655 


1.2283 1 


7220 


30 


31 


1-1381 


2.0946 


1.1491 


2.0297 


1. 1608 


1.9693 


1.1730 


1. 9130 


29 


31 


i.i8s9 


1.8603 


1.1994 


1.8110 


1.2136 


1.7648 


1.2286 1 


7213 


29 


32 


1.1382 


2.0935 


I-I493 


2.0287 


i.i6io 


1-9683 


1.1732 


1.9121 


28 


32 


i.i86i 


1.8595 


1.1997 


1.8102 


1.2139 


1.7640 


1.2288 I 


7206 


28 


33 


1.1384 


2.0924 


I-I495 


2.0276 


1.1612 


1.9674 


1-1734 


1.9112 


27 


33 


1.1863 


1.8586 


1.1999 


1.8094 


1.2141 


1-7633 


1.2291 I 


7199 


27 


34 


1.1386 


2.0912 


1.1497 


2.0266 


1.1614 


1.9664 


1-1737 


1.9102 


26 


34 


1.1866 


1.8578 


1.2001 


1.8086 


1.2144 


1.7625 


1.2293 1 


7192 


26 


35 


1.1383 


2.0901 


1.1499 


2.0256 


1.1616 


1-9654 


1.1739 


1.9093 


25 


35 


1.1868 


1.8569 


1.2004 


1.8078 


1.2146 


1.7618 


1.2296 1 


7185 


25 


36 


1.1390 


2.0890 


1.1501 


2.024s 


1.1618 


1.9645 


1.1741 


1.9084 


24 


36 


1. 1870 


1.8561 


1.2006 


1 .8070 


1.2149 


1.7610 


1.2298 1 


7178 


24 


37 


I. I 391 


2.0879 


1-1503 


2.0235 


1.1620 


1-9635 


I -1 743 


1-9075 


23 


37 


1.1872 


1.8552 


1.2008 


1.8062 


1.2151 


1-7603 


1.2301 1 


7171 


23 


38 


1.1393 


2.0868 


1.1505 


2.0224 


1.1622 


1-9625 


1-1745 


1.9066 


22 


38 


1.1874 


1.8544 


1.2010 


1.8054 


I-21S3 


1-7596 


1.2304 1 


7164 


22 


39 


1-I39S 


2.0857 


1.1507 


2.0214 


1.1624 


1.9616 


1.1747 


1-9057 


21 


39 


1.1877 


I-8S3S 


1.2013 


1.8047 


1-2156 


1.7588 


1.2306 I 


7157 


21 


40 


I-I397 


2.0846 


1.1508 


2.0204 


1.1626 


1.9606 


1-1749 


1.9048 


20 


40 


1.1879 


1.8527 


1.2015 


1.8039 


1.2158 


1.7581 


1.2309 I 


7151 


20 


41 


I -1 399 


2.0835 


1.1510 


2.0194 


1.1628 


1.9596 


1-1751 


1.9039 


19 


41 


1.1881 


1.8519 


1.2017 


1. 803 1 


1.2161 


1-7573 


I.231I I 


7144 


19 


42 


1.1401 


2.0824 


1.1512 


2.0183 


1.1630 


19587 


1.1753 


1.9030 


18 


42 


1.1883 


1.8510 


1.2020 


1.8023 


1.2163 


1.7566 


1-2314 1 


7137 


18 


43 


1.1402 


2.0812 


1.1514 


2.0173 


1.1632 


1-9577 


1.1756 


1.9021 


17 


43 


1.1886 


1.8502 


1.2022 


1.801S 


1.2166 


1-7559 


1.2316 1 


7130 


17 


44 


1.1404 


2.o8oi 


1.1516 


2.0163 


1-1634 


1.9568 


1.1758 


1. 9013 


16 


44 


1.1888 


1.8493 


1.2024 


1.8007 


1.2168 


I-75SI 


1.2319 I 


7123 


16 


45 


1.1406 


2.0790 


1.1518 


2.0152 


1.1636 


1-9558 


1.1760 


1.9004 


IS 


45 


1.1890 


1.8485 


1.2027 


1-7999 


1.2171 


1-7544 


1.2322 I 


7116 


15 


46 


1.1408 


2.0779 


1. 1520 


2.0142 


1.1638 


1-9549 


1. 1762 


1.8995 


14 


46 


1.1892 


1.8477 


1.2029 


1-7992 


1.2173 


1-7537 


1.2324 I 


7109 


14 


47 


1.1410 


2.0768 


1.1522 


2.0132 


1.1640 


1-9539 


1. 1764 


1.8986 


13 


47 


1. 1894 


1.8468 


1.2031 


1.7984 


1.2175 


1-7529 


1.2327 1 


7102 


13 


48 


1.1411 


2.0757 


1.1524 


2.0122 


1.1642 


1.9530 


1.1766 


1.8977 


12 


48 


1.1897 


1.8460 


1.2034 


1.7976 


1.2178 


1-7522 


1-2329 1 


7095 


12 


49 


1.1413 


2.0746 


1.1526 


2.0111 


1.1644 


1.9520 


1. 1768 


1.8968 


II 


49 


1.1899 


1.8452 


1.2036 


1.7968 


1.2180 


I-75I4 


1-2332 1 


7088 


11 


SO 


1.1415 


2.0735 


1.1528 


2.0101 


1.1646 


1-9510 


1. 1770 


1.8959 


10 


50 


1.1901 


1.8443 


1.2039 


1 .7960 


1.2183 


1-7507 


1.2335 I 


7081 


10 


SI 


1.1417 


2.0725 


I-IS30 


2.0091 


1.1648 


I-950I 


1.1772 


1.8950 


9 


51 


I -1 903 


1-8435 


1.2041 


1-7953 


1.2185 


1-7500 


1-2337 I 


7075 


9 


52 


1.1419 


2.0714 


1-1531 


2.0081 


1.1650 


1.9491 


1-1775 


1. 8941 


8 


52 


1. 1906 


1.8427 


1.2043 


1-7945 


1.2188 


1-7493 


1-2340 I 


7068 


8 


53 


1.1421 


2.0703 


1-1533 


2.0071 


1.1652 


1.9482 


1-1777 


1-8932 


7 


53 


1. 1908 


1.8418 


1.2046 


1-7937 


1.2190 


1-7485 


1-2342 I 


7061 


7 


54 


1. 1422 


2.0692 


I-1535 


2.oo6i 


1.1654 


1-9473 


1.1779 


1.8924 


6 


54 


1.1910 


1.8410 


1.2048 


1.7929 


1.2193 


1.7478 


1-2345 1 


7054 


6 


55 


1.1424 


2.0681 


1.1537 


2.0050 


1. 1656 


1.9463 


1.1781 


1.891S 


S 


55 


1.1912 


1.8402 


1.2050 


1.7921 


1.219s 


1-7471 


1-2348 1 


7047 


5 


S6 


1.1426 


2.0670 


1-1539 


2.0040 


I.i6s8 


1-9454 


1.1783 


1.8906 


4 


56 


1.1915 


1.8394 


1.2053 


1.7914 


1.2198 


1-7463 


1-2350 I 


7040 


4 


57 


1.1428 


2.0659 


1.1541 


2.0030 


1.1660 


1-9444 


1.178s 


1.8897 


3 


57 


1.1917 


1.838s 


I-2055 


1.7906 


1.2200 


1-7456 


1-2353 I 


7033 


3 


58 


1.1430 


2.0648 


1-1543 


2.0020 


1.1662 


1-9435 


1.1787 


1.8888 


2 


S8 


1.1919 


1.8377 


1.2057 


1.7898 


1.2203 


1-7449 


1-2355 I 


7027 


2 


SO 


1.1432 


2.0637 


I-I545 


2.0010 


1.1664 


1-9425 


1.1790 


1.8879 


X 


59 


1.1921 


1.8369 


1.2060 


1.7891 


1.2205 


1-7442 


1-2358 I 


7020 


I 


60 


1-1433 


2.0627 


1-1547 


2.0000 


1.1666 


1.9416 


1.1792 


1.887s 




* 


60 


1.1922 


1.8361 


1.2062 


1.7883 


1.2208 


1-7434 


1.2361 1 


7013 





/ 


CO-SEC. 


Sec. 


Co-SEC. 


Sec. 


Co-sec. 


Sec. 


Co-SEC. 


Sec 


/ 


Co-sec. 


Sec. 


Co-SEC 


Sec. 


Co-SEC. 


Sec. 


Co-SEC. 


Sec. 


T 




6 


1° 


6 


IP 


5 


9° 


5 


S° 






5 


7° 


5 


5° 


5 


5° 


54° 







522 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 













Table 5. — 


Natural Trigonometric Functions— 


{Continued) 














36° 1 


37" ( 


38° 1 


39° 1 






40° 1 


41° 1 


42° 1 


43° 




/ 


Sec. 
1.2361 


Co-sec. 


Sec. 


Co-sec. 


Sec. 


Co-sec. 


Sec. 


Co-sec. 


/ 

60 


' 


Sec. 


Co-sec. 


Sec. 


Co-SEC. 


Sec. 


Co-SEC. 


Sec 


Co-SEC. 


' 


o 


1.7013 


1.2521 


1.6616 


1.2690 


1.6243 


1.2867 


1.5890 





1-3054 


1-5557 


1.3250 


1.5242 


1-3456 


1-4945 


1-3673 


1.4663 


60 


I 


1.2363 


1.7006 


1.2524 


1.6610 


1.2693 


1.6237 


1.2871 


1.5884 


59 


1 


1-3057 


1-5552 


1-3253 


1-5237 


1.3460 


1.4940 


1.3677 


1.4658 


59 


2 


1.2366 


1.6999 


1-2527 


1.6603 


1.2696 


1.6231 


1.2874 


1-5879 


58 


2 


1.3060 


1.5546 


1-3257 


1-5232 


1-3463 


1-4935 


1.3681 


1-4654 


S8 


3 


1.2368 


1.6993 


1-2530 


1-6597 


1.2699 


1.6224 


1.2877 


1-5873 


57 


3 


1-3064 


1-5541 


1-3260 


1.5227 


1-3467 


1.4930 


1.3684 


1.4649 


57 


4 


1.2371 


1.6986 


1-2532 


1-6591 


1.2702 


1.6218 


1.2880 


1.5867 


56 


4 


1-3067 


1-5536 


1-3263 


1.5222 


1-3470 


1-4925 


1.3688 


1.4644 


56 


5 


1-2374 


1.6979 


1-2535 


1-6584 


1.2705 


1.6212 


1.2883 


1.5862 


55 


5 


1.3070 


I -5530 


1.3267 


I-5217 


1-3474 


1.4921 


1.3692 


1.4640 


55 


6 


1.2376 


1.6972 


1-2538 


1.6578 


1.2707 


1.6206 


1.2886 


1.5856 


54 


6 


1-3073 


1-5525 


1-3270 


I.5212 


1-3477 


1.4916 


1-3695 


1-4635 


54 


7 


1.2379 


1.6965 


1.2541 


1-6572 


1.2710 


1.6200 


1.2889 


1.5850 


53 


7 


1-3076 


1-5520 


1-3274 


1.5207 


I -3481 


1.49H 


1.3699 


1.4631 


53 


8 


1.2382 


1-6959 


1.2543 


1-6565 


1.2713 


1.6194 


1.2892 


1.5845 


52 


8 


1 .3080 


1-5514 


1-3277 


1.5202 


1-3485 


1.4906 


1-3703 


1.4626 


52 


9 


1.2384 


1.6952 


1-2546 


1-6559 


1.2716 


1.6188 


1.2895 


1-5839 


SI 


9 


1.3083 


1-5509 


1-3280 


I.5197 


1-3488 


1. 490 1 


1-3707 


1.4622 


51 


lo 


1.2387 


1-6945 


1.2549 


1-6552 


1.2719 


1.6182 


1.2898 


1-5833 


50 


10 


1.3086 


1-5503 


1.3284 


1.5192 


1.3492 


1-4897 


1.3710 


I.46I7 


SO 


11 


1-2389 


1.6938 


1-2552 


1.6546 


1.2722 


1. 61 76 


1.2901 


1.5828 


49 


" 


1.3089 


1.5498 


1.3287 


1.5187 


1-3495 


1.4892 


1-3714 


1.4613 


49 


12 


1.2392 


1-6932 


1-2554 


1.6540 


1.2725 


1.6170 


1.2904 


1.5822 


48 


12 


1.3092 


1-5493 


1-3290 


I-5182 


1-3499 


1.4887 


1-3718 


1.4608 


48 


13 


1-2395 


1-6925 


I-25S7 


1-6533 


1.2728 


1.6164 


1.2907 


1.5816 


47 


13 


1.3096 


1-5487 


1-3294 


1-5177 


1-3S02 


1.4882 


1.3722 


1 .4604 


47 


14 


1-2397 


1. 6918 


1-2560 


1-6527 


1.2731 


1.6159 


1.2910 


1.5811 


46 


14 


1-3099 


1-5482 


1-3297 


I-5171 


1-3506 


1.4877 


1-3725 


1-4599 


46 


IS 


1.2400 


1.6912 


1-2563 


1.6521 


1-2734 


1-6153 


1.2913 


1.5805 


45 


15 


1.3102 


1-5477 


1-3301 


1-5166 


1-3509 


1.4873 


1-3729 


1-4595 


45 


i6 


1.2403 


1. 690 5 


1.2565 


1.6514 


1-2737 


1.6147 


1. 2916 


1-5799 


44 t 


16 


1-3105 


I-S47I 


1-3304 


I.5161 


1-3513 


1.4868 


1-3733 


1.4590 


44 


17 


1. 2405 


^6898 


1-2568 


1.6508 


1-2739 


1.6141 


1.2919 


1-5794 


43 


17 


1-3109 


1-5466 


1-3307 


1.5156 


1-3517 


1.4863 


1-3737 


1.4586 


43 


i8 


1.2408 


1. 6891 


1-2571 


1.6502 


1-2742 


1-6135 


1.2922 


1-5788 


42 


18 


1.3112 


1.5461 


I-33II 


1-5151 


1-3520 


1.4858 


1-3740 


1-4581 


42 


19 


1.2411 


1.6885 


1-2574 


1.6496 


1-2745 


1-6129 


1.2926 


1.5783 


41 


19 


1-3115 


1-5456 


1-3314 


1-5146 


1-3524 


1-4854 


1-3744 


I-4S77 


41 


20 


1-2413 


1.6878 


1-2577 


1.6489 


1.2748 


1. 6123 


1.2929 


I-S777 


40 


20 


1.3118 


1-5450 


1-3318 


1.5141 


1-3527 


1.4849 


1-3748 


1-4572 


40 


21 


1.2416 


1.6871 


1.2579 


1-6483 


1.2751 


1.6117 


1.2932 


I-577I 


30 


21 


1.3121 


1-5445 


1-3321 


I-5136 


1-3531 


1.4844 


I -3752 


1.4568 


39 


22 


1.2419 


1.6865 


1.2582 


1.6477 


1.2754 


1. 611 1 


1-2935 


1.5766 


38 


22 


1-3125 


1.5440 


1-3324 


I-5131 


1-3534 


1-4839 


1-3756 


1-4563 


38 


23 


1.2421 


1.6858 


1-2585 


1.6470 


1-2757 


1.6105 


1.2938 


1.5760 


37 


23 


1.3128 


1.5434 


1-3328 


1.5126 


1-3538 


1-4835 


1-3759 


1-4559 


37 


24 


1.2424 


1. 6851 


1.2588 


1 .6464 


1.2760 


1.6099 


1.2941 


1-5755 


36 


24 


1-3131 


1-5429 


1-3331 


1.5121 


1-3542 


1.4830 


1-3763 


I-4SS4 


36 


25 


1.2427 


1.6845 


1.2591 


1.6458 


1.2763 


1.6093 


1.2944 


1-5749 


35 


25 


1-3134 


1.5424 


1-3335 


1.5116 


1-3545 


1-4825 


1-3767 


1-4550 


35 


26 


1-2429 


1.6838 


1.2593 


1-6452 


1.2766 


1.60S7 


1.2947 


1-5743 


34 


26 


1-3138 


1-5419 


1.3338 


I-511I 


1-3549 


1-4821 


I-377I 


1-4545 


34 


27 


1-2432 


1. 6831 


1.2596 


1.6445 


1.2769 


1.6081 


1.2950 


1.5738 


33 


27 


1-3141 


1-5413 


1-3342 


I. 5106 


1-3552 


1-4816 


1-3774 


I -4541 


33 


28 


1-2435 


1.6825 


I.2S99 


1.6439 


1.2772 


1.6077 


1.2953 


1-5732 


32 


28 


1.3144 


1.5408 


1-3345 


I. 5101 


1-3556 


I.4811 


1-3778 


1-4536 


32 


29 


1-2437 


1. 6818 


1.2602 


1-6433 


1-2775 


1.6070 


1.2956 


1-5727 


31 


29 


1.3148 


1-5403 


1-3348 


1-5096 


1-3560 


1.4806 


1.3782 


1-4532 


31 


30 


1.2440 


1. 6812 


1.2605 


1.6427 


1.2778 


1.6064 


1.2960 


I.5721 


30 


30 


1-3151 


1-5398 


1-3352 


1-5092 


1-3563 


1.4802 


1.3786 


1-4527 


30 


31 


1.2443 


1. 6805 


1.2607 


1.6420 


1.2781 


1.6058 


1.2963 


1.5716 


29 


31 


1-3154 


1-5392 


1-3355 


1-5087 


1-3567 


1-4797 


1-3790 


1-4523 


29 


32 


1-2445 


1.6798 


1. 2610 


1.6414 


1.2784 


1.6052 


1.2966 


1.5710 


28 


32 


1-3157 


1-5387 


I-33S9 


1.5082 


1-3571 


1.4792 


1-3794 


1.4518 


28 


33 


1-2448 


1.6792 


I.2613 


1.6408 


1.2787 


1.6046 


1.2969 


1-5705 


27 


33 


1.3161 


1-5382 


1-3362 


1-5077 


1-3574 


1-4788 


1-3797 


1-4514 


27 


34 


1.2451 


1.6785 


I.2616 


1.6402 


1.2790 


1.6040 


1.2972 


1.5699 


26 


34 


1.3164 


1-5377 


1-3366 


1.5072 


1-3578 


1-4783 


1. 3801 


1.4510 


26 


35 


1-2453 


1.6779 


1.2619 


1.6396 


1-2793 


1.6034 


1.2975 


1.5694 


2S 


35 


1.3167 


1-5371 


1-3369 


1.5067 


1-3581 


1-4778 


1-3805 


I -450s 


25 


36 


1-2456 


1.6772 


1.2622 


1-6389 


1-2795 


1.6029 


1.2978 


1.5688 


24 


36 


1.3170 


1-5366 


1-3372 


1.5062 


1-3585 


1-4774 


1-3809 


1-4501 


24 


H 


1-2459 


1.6766 


1.2624 


1.6383 


1.2798 


1.6023 


1. 298 1 


1.5683 


23 


37 


1-3174 


1-5361 


1-3376 


1-5057 


1-3589 


1.4769 


1-3813 


1-4496 


23 


38 


1. 2461 


1-6759 


1.2627 


1-6377 


1.2801 


1.6017 


1.2985 


1-5677 


22 


38 


1.3177 


1-5356 


1-3379 


1-5052 


1-3592 


1.4764 


1-3816 


1.4492 


22 


39 


1.2464 


1-6752 


1.2630 


1-6371 


1.2804 


1. 601 1 


1.2988 


1.5672 


21 


39 


1. 3180 


1-5351 


1-3383 


1-5047 


1-3596 


1.4760 


1-3820 


1.4487 


21 


40 


1.2467 


1.6746 


1-2633 


1.636s 


1.2807 


1.600S 


1.2991 


1.5666 


20 


40 


1.3184 


1-5345 


1.3386 


1.5042 


1.3600 


1-4755 


1-3824 


1-4483 


20 


41 


1.2470 


1.6739 


1.2636 


1-6359 


1.2810 


1.6000 


1.2994 


1.5661 


19 


41 


1.3187 


1-5340 


1-3390 


1-5037 


1.3603 


I-47SO 


1.3828 


1-4479 


19 


42 


1.2472 


1-6733 


1.2639 


1-6352 


1.2813 


1-5994 


1-2997 


1.5655 


18 


42 


1.3190 


1-5335 


1-3393 


1.5032 


1-3607 


1.4746 


1-3832 


1-4474 


18 


43 


1-2475 


1.6726 


I.2641 


1.6346 


1.2816 


1-5988 


1-3000 


1.5650 


17 


43 


1-3193 


I -5330 


1-3397 


1.5027 


1-3611 


1-4741 


1-3836 


1-4470 


17 


44 


1-2478 


1.6720 


1.2644 


1.6340 


1.2819 


1.5982 


1-3003 


1.5644 


16 


44 


1-3197 


1-5325 


1.3400 


1'5022 


1-3614 


1-4736 


1-3839 


1-4465 


16 


45 


1.2480 


1-6713 


1.2647 


1-6334 


1.2822 


1-5976 


1-3006 


1-5639 


IS 


45 


1.3200 


1-S319 


1.3404 


1. 5018 


I -36 18 


1-4732 


1-3843 


1.4461 


15 


46 


1.2483 


1.6707 


1.2650 


1.6328 


1.2825 


1-5971 


1. 3010 


1-5633 


14 


46 


1-3203 


1-5314 


1-3407 


1-5013 


1.3622 


1.4727 


1-3847 


1-4457 


14 


^l 


1.2486 


1.6700 


1-2653 


1.6322 


1.2828 


1-5965 


1-3013 


1-5628 


13 


47 


1.3207 


1-5309 


1.3411 


1.5008 


1.362s 


1-4723 


I -385 1 


1-4452 


13 


48 


1.2488 


1.6694 


1.2656 


1.6316 


1.2831 


1-5959 


1.3016 


1.5622 


12 


48 


1.3210 


1-5304 


1-3414 


1-5003 


1.3629 


1-4718 


1-3855 


1.4448 


12 


49 


1.2490 


1.66S7 


1-2659 


1.6309 


1.2834 


1-5953 


1.3019 


1.5617 


II 


49 


1-3213 


1-5299 


1-3418 


1.4998 


1-3633 


1-4713 


1-3859 


1.4443 


II 


50 


1.2494 


1.6681 


1. 2661 


1-6303 


1-2837 


1-5947 


1.3022 


1.5611 


10 


SO 


1.3217 


1-5294 


1.3421 


1-4993 


1.3636 


1.4709 


1-3863 


1-4439 


10 


SI 


1.2497 


1 .6674 


1.2664 


1.6297 


1-2840 


1.5942 


1.3025 


1.5606 


9 


SI 


1.3220 


1.5289 


1.3425 


1.4988 


1.3640 


1.4704 


1-3867 


1-4435 


9 


52 


1.2499 


1.6668 


1.2667 


1. 6291 


1-2843 


1.5936 


1.3029 


1.5600 


8 


52 


1.3223 


1.5283 


1-3428 


1.4983 


1-3644 


1.4699 


1-3870 


1.4430 


8 


S3 


1.2502 


1.6661 


1.2670 


1.6285 


1.2846 


1.5930 


1-3032 


1-5595 


7 


S3 


1.3227 


1.5278 


1-3432 


1.4979 


1-3647 


1.4695 


1-3874 


1.4426 


7 


S4 


1-2505 


1.6655 


1-2673 


1.6279 


1.2849 


1-5924 


1-3035 


1-5590 


6 


54 


1-3230 


1-5273 


1-3435 


1-4974 


1-3651 


1.4690 


1-3878 


1.4422 


6 


55 


1.2508 


1.6648 


1-2676 


1.6273 


1.2852 


1-5919 


1-3038 


1-5584 


S 


55 


1-3233 


1.5268 


1-3439 


1.4969 


1-3655 


1.4686 


1-3882 


I -441 7 


5 


56 


I.2510 


1 .6642 


1.2679 


1.6267 


1.285s 


1-5913 


I -3041 


1-5579 


4 


S6 


1-3237 


1-5263 


1-3442 


1.4964 


1-3658 


1.4681 


1.3886 


1-4413 


4 


57 


1-2513 


1.6636 


1.2681 


1. 6261 


1.2858 


1-5907 


1.3044 


I-5S73 


3 


57 


1-3240 


1-5258 


1-3446 


1-4959 


1.3662 


1.4676 


1-3890 


1.4408 


3 


S8 


I.2516 


1.6629 


1.2684 


1.625s 


1. 2861 


1. 5901 


1.3048 


1-5568 


2 


58 


1-3243 


1-5253 


1-3449 


1-4954 


1.3666 


1.4672 


1-3894 


1.4404 


2 


S9 


1-2519 


1.6623 


1.2687 


1.6249 


1.2864 


1.5896 


1-3051 


1-5563 


I 


59 


1-3247 


1-5248 


1-3453 


1-4949 


1.3669 


1.4667 


1-3898 


1.4400 


I 


60 


1-2521 


1.6616 


1.2690 


1-6243 


1.2867 


1.5890 


1-3054 


1-5557 





60 

/ 


1.3250 


1-5242 


1-3456 


1-4945 


1-3673 


1.4663 


1-3902 


1-4395 





/ 


CO-SEC. 


Sec. 


CO-SEC. 


Sec. 


CO-SEC. 


Sec. 


Co-sec. 


Sec. 


/ 


Co-sec. 


Sec. 


Co-SEC. 


Sec. 


Co-SEC. 


Sec. 


Co-SEC. 


Sec 


/ 




5 


3° 


5. 


2° 


5 


L° 


5 


0° 






49° 


48° 


4 


r° 


4 


3° 





MATHEMATICAL TABLES 



623 



T 


4BLE S. 


— Nattjeal Trigonometric Functions — {Continued) 


44° 






44° 






44° 




/ 


Sec. 


Co-sec. 


/ 


' 


Sec. 


Co-sec. 


f 


t 


Sec. 


Co-sec. 


/ 


K ° 


1.3902 


1-439S 


60 


21 


1.3984 


1-4305 


39 


41 


1-4065 


1.4221 


19 


■ I 


1-3905 


1-4391 


59 


22 


1.3988 


1-4301 


38 


42 


1 .4069 


1-4217 


iH 


W ' 


1.3909 


1-4387 


58 


23 


1-3992 


1.4297 


37 


43 


1-4073 


1.4212 


17 


m 3 


1-3913 


1-4382 


57 


24 


1.3996 


1.4292 


36 


44 


14077 


1.4208 


16 


■ 4 


1. 391 7 


1-4378 


56 


25 


1.4000 


1.4288 


35 


45 


1. 4081 


1.4204 


15 


B 5 


1. 392 1 


1-4374 


55 


2b 


1 .4004 


1.4284 


34 


4b 


1.4085 


1.4200 


14 


■ 6 


1-3925 


I-4370 


54 


27 


1 .4008 


1.4280 


33 


47 


1 .4089 


I. 41 96 


13 


B 7 


1-3929 


1-4365 


53 


28 


1. 4012 


1.4276 


32 


48 


1-4093 


1.4192 


12 


B s 


1-3933 


I-436I 


52 


29 


1.4016 


1-4271 


31 


49 


1.4097 


1.4188 


II 


B "^ 


1-3937 


1-4357 


51 


30 


1 .4020 


1.4267 


30 


50 


I. 410 I 


1.4183 


10 


P '° 


I -3941 


1.4352 


50 


31 


1.4024 


1.4263 


29 


51 


I. 4105 


1.4179 


9 


II 


I -3945 


1.4348 


49 


32 


1.4028 


1.4259 


28 


52 


I. 4109 


1-4175 


8 


12 


1-3949 


1-4344 


48 


33 


1.4032 


1-4254 


27 


53 


1-4113 


1.4171 


7 


13 


1-3953 


1.4339 


47 


34 


1.4036 


1.4250 


26 


54 


1.4117 


1. 4167 


b 


14 


1-3957 


1-4335 


4b 


35 


1.4040 


1.4246 


25 


55 


1.4122 


1.4163 


5 


15 


1-3960 


I-433I 


45 


3b 


1.4044 


1.4242 


24 


50 


1.4126 


1-4159 


4 


i6 


1.3964 


1-4327 


44 


37 


1.4048 


1.4238 


23 


57 


1.4130 


1-4154 


3 


17 


1.3968 


1.4322 


43 


38 


1.4052 


1.4233 


22 


58 


1-4134 


1.4150 


2 


i8 


1-3972 


1-4318 


42 


39 


1.4056 


1.4229 


21 


59 


1-4138 


1.4146 


I 


19 


1-3976 


1-4314 


41 


40 


1.4060 


1.4225 


20 


bo 


1.4142 


1. 4142 





» 20 


1.3980 


1-4310 


40 














Sec. 




P ' 


CO-SEC. 


Sec. 


' 


' 


Co-sec. 


Sec. 


' 


' 


Co-sec. 


' 




45'^ 






4 


50 






45° 





The factoring method of extracting roots (also called the successive 
approximation method) is applicable to any root and, except for the 
square root, is less laborious than other arithmetical methods. 
It is greatly facilitated by the use of Table 6 of factors. 

To find the square root proceed as follows: Look in the table of 
second powers for the number nearest the given number of which the 
root is desired, and take the factor of that power. Divide the given 
number by that factor. Then divide the number by the half sum 
of the factor and quotient for a second approximation. Divide the 
number again by the half sum of the second approximation and the 
second quotient. 

This process can be continued until the result is obtained to any 
required degree of exactness. Usually two divisions give the root 
to as great a number of places as is necessary. 

To find the cube root proceed in a similar way, as follows: Look 
in the column of third powers for the number nearest the given 
number. Take out the factor. Divide the given number by that 
factor. Divide the quotient also by the factor. Take one-third 
the sum of the two divisors and the last quotient for a second divisor. 
Divide the given number by this second divisor and divide the quo- 
tient also by it. Takeone-third the sum of twice the second divisor 
and the final quotient for a third divisor. It may not be necessary 
to divide the number a third time. This third divisor may be the 



cube root as close as needed. For the fourth root, the new divisor 
will be one-fourth the sum of the last quotient and three times the 
preceding divisor. For the fifth root, one-fifth the sum of the last 
quotient and four times the preceding divisor, and so for any root 
required. 

Whatever the root, to the number of decimal places that the 
divisor and quotient agree, they are correct — comparison of the two 
showing at once the degree of approximation to which the process 
has been carried. 



Table 6. — Factors tor Use in Extracting Roots by The 
Factoring Method 





2d power 


3d power 


4th power 


Sth power 


Factors 


The factor is 


The factor is 


The factor is 


The factor is 




the sq. root 


the third root 


the fourth root 


the fifth root 


I 


I 


I 


I 


I 


2 


4 


8 


16 


32 


3 


9 


27 


81 


243 


4 


16 


64 


256 


1.024 


5 


25 


123 


625 


3.125 


6 


36 


216 


1,296 


7.776 


7 


49 


343 


2,401 


16,807 


8 


64 


512 


4,096 


32,768 


9 


81 


729 


6,s6i 


59.049 


10 


100 


1,000 


10,000 


100.000 


II 


121 


1. 331 


14,641 


161,051 


12 


144 


1,728 


20,736 


248,832 


13 


169 


2,197 


28,561 


371,293 


14 


196 


2,744 


38,416 


537.824 


15 


225 


3,375 


50,625 


759.375 


16 


256 ■ 


4,096 


65.536 


1.048,576 


17 


289 


4,913 


83.521 


1,419.857 


18 


324 


5.832 


104,976 


1,889,568 


19 


361 


6,8S9 


130,321 


2,476,099 


20 


400 


8,000 


160,000 


3,200,000 


21 


441 


9,261 


J94.481 


4,084,101 


22 


484 


10,648 


234.256 


5.153.632 


23 


529 


12,167 


279.841 


6.436.343 


24 


576 


13,824 


331.776 


7.962,624 


25 


62s 


15,625 


390,625 


9.765.625 



Table 7. — Sizes or L'argest Squares (Corners being Sharp) Which can be Obtained from Round Stock 







Diameter 


Diameter 


Decimal 


Diameter 
< — V 






Diameter 
/ — V. 


Diameter 


Decimal 


/''~"\ f 


/ 




) 


Diameter 


Decimal 


/ 




\1 


of stock 


equivalent 


U-i- 


of stock 


equivalent 


( 




of stock 


equivalent 


( 




)i 






^-^ 




^- ^ 




^ 


.125 


.0883-1- 


liV 


1.0625 


I-7551 + 


2 


2 , 000 


1-414 


A 


.1875 


.1325+ 


li 


1. 125 


-7953 + 


2 iV 


2.0625 


I-4581 + 


i 


.250 


.1767-1- 


lA 


1.187s 


•8395 + 


2i 


2. 125 


i.S023-f 


ft 


.312s 


.2209-1- 


li 


1.250 


.8837 + 


2 A 


2.187s 


1.5465 + 


i 


.375 


.2651+ 


ift 


1.312s 


.9279 + 


2i 


2.250 


1.5907 + 


A 


.4375 


.3093+ 


i| 


1-375 


• 9721 + 


2A 


2.312s 


1.6349 + 


3 


.500 


• 3535 


lA 


I-437S 


I.0163-I- 


2| 


2. 375 


1. 6791 + 


IT 


.5625 


.3976-1- 


ih 


1.500 


1.060S 


2 A 


2.437s 


1.7233 + 


1 


.62s 


.44l8-f 


lA 


1.5625 


I. 1046 -|- 


2h 


2.500 


1.7675 


H 


.6875 


.48604- 


i| 


1.625 


I.1488-I- 


2 A 


2.562s 


1.8116 + 


1 


.750 


.5302-f- 


iH 


1.687s 


I.1930 + 


25 


2.62s 


1.8558 + 


H 


.8125 


• 5744 + 


li 


1.750 


1-2372 + 


2H 


2.6875 


1.9000-)- 


1 


-&75 


.6i86-|- 


iH 


1. 8125 


I.2814-I- 


2| 


2.750 


1.9442+ 


k 


-9375 


.6628-1- 


li 


1.875 


13256 + 


2H 


2.812s 


1.9884-t- 


I 


I. 000 


.707 


in 


1-9375 


I. 3698-1- 


2| 


2.875 


2.0326+ 


Rule 






1 




Multiply diameter of 








2H 


2.9375 


2.0768+ 


stock by the constant .707 








3 


3 000 


2. 121 


Example J in.= .750X .707 


=53025 













524 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 8. — Whole and Fractional Inches Reduced to Decimals 
or a Foot 



Ins. 


Ft. 1 Ins. 


Ft. 


Ins. 


Ft. 


Ins. 


Ft. 


Ins. 


Ft. 


Ins. 


Ft. 





.0000 


2 


.1667 


4 


.3333 


6 


• 5000 


8 


.6667 


10 


.8333 


Hi 


.0026 




.1693 




• 3359 




• 5026 




• 6693 




.8359 


H« 


.0052 




.1719 




.3385 




• 5052 




.6719 




.8385 


?62 


.0078 




.1745 




.3411 




.5078 




.6745 




.8411 


H 


.0104 


H 


.1771 


M 


.3438 


H 


.5104 


H 


.6771 


H 


.8438 


H2 


.0130 




.1797 




• 3464 




.5130 




.6797 




.8464 


Me 


.0156 




.1823 




.3490 




.5156 




.6823 




.8490 


J^2 


.0182 




.1849 




.3516 




.5182 




• 6849 




.8516 


V* 


.0208 


M 


.1875 


h7. 


.3542 


H 


• 5208 


H 


• 6875 


H 


.8542 


%i 


.0234 




.1901 




.3568 




• 5234 




• 6901 




.8568 


Ms 


.0260 




.1927 




■ 3594 




.5260 




.6927 




.8594 


1H2 


.0286 




.1953 




.3620 




.5286 




• 6953 




.8620 


H 


• 0313 


% 


.1979 


H 


.3646 


% 


.5313 


H 


• 6979 


H 


.8646 


1?42 


.0339 




.2005 




.3672 




• 5339 




• 700s 




.8672 


Mo 


.0365 




.2031 




.3698 




.5365 




.7031 




.8698 


'H2 


.0391 




.2057 




.3724 




.5391 




• 7057 




.8724 


H 


.0417 


\i 


.2083 


Vi. 


• 3750 


H 


• 5417 


Vi 


.7083 


K2 


.8750 


'^2 


• 0443 




.2109 




.3776 




.5443 




.7109 




.8776 


M« 


.0469 




.2135 




.3802 




• 5469 




• 7135 




.8802 


1^2 


.049s 




.2161 




3828 




• 5495 




.7161 




.8828 


5^ 


.0521 


% 


.2188 


Vs 


• 3854 


% 


• 5521 


5i 


.7188 


H 


.8854 


2H2 


• 0547 




.2214 




.3880 




• 5547 




.7214 




.8880 


>H« 


.0573 




.2240 




.3906 




• 5573 




.7240 




.8906 


2?^2 


.0599 




.2266 




• 3932 




• 5599 




.7266 




.8932 


M 


.0625 


% 


.2292 


M 


• 3958 


% 


.5625 


% 


.7292 


% 


.8958 


25^2 . 


.0651 




.2318 




• 3984 




.5651 




.7318 




.8984 


'Me 


.0677 




• 2344 




.4010 




.5677 




.7344 




.9010 


"/42 


.0703 




.2370 




.4036 




• 5703 




• 7370 




.9036 


li 


.0729 


% 


.2396 


% 


.4063 


% 


• 5729 


% 


• 7396 


'A 


.9063 


"/42 


.0755 




.2422 




• 4089 




• 5755 




• 7422 




.9089 


'Me 


.0781 




.2448 




• 411S 




• 5781 




.7448 




• 911S 


8^2 


.0807 




.2474 




.4141 




• 5807 




• 7474 




.9141 


I 


.0833 


3 


.2500 


5 


• 4167 


7 


• 5833 


9 


.7500 


II 


.9167 


Hj 


.0859 




.2526 




• 4193 




• 5859 




• 7526 




.9193 


H. 


.0885 




.2552 




• 4219 




• 588s 




.7552 




.9219 


?^2 


.0911 




.2578 




•424s 




.5911 




.7578 




.9245 


H 


.0938 


H 


.2604 


H 


.4271 


H 


.5938 


\^ 


.7604 


H 


.9271 


^2 


.0964 




.2630 




.4297 




• 5964 




• 7630 




• 9297 


M. 


.0990 




.2656 




.4323 




• 5990 




• 7656 




.9323 


%2 


.1016 




.2682 




• 4349 




• 6016 




• 7682 




• 9349 


H 


.1042 


H 


.2708 


H 


• 4375 


H 


.6042 


Vi 


.7708 


H 


• 9375 


?42 


.1068 




• 2734 




.4401 




.6068 




• 7734 




• 9401 


M» 


.1094 




.2760 




• 4427 




.6094 




• 7760 




.9427 


1H2 


. I120 




.2786 




•4453 




• 6120 




• 7786 




• 9453 


?^ 


.1146 


H 


.2813 


H 


•4479 


^yi 


.6146 


% 


• 7813 


H 


• 9479 


J ^2 


.1172 




.2839 




• 4505 




• 6172 




• 7839 




• 9505 


Me 


.1198 




.2865 




.4531 




• 6198 




• 7865 




• 9531 


■5^2 


. 1224 




.2891 




• 4557 




.6224 




• 7891 




• 9557 


H 


.1250 


M 


.2917 


Vi 


• 4583 


H 


.6250 


H 


• 7917 


H 


• 9583 


"A2 


. 1276 




• 2943 




• 4609 




• 6276 




• 7943 




.9609 


Me 


.1302 




.2969 




• 4635 




.6302 




• 7969 




• 9635 


'9^2 


.1328 




.2995 




.4661 




.6328 




• 7995 




• 9661 


« 


.1354 


% 


.3021 


^A 


.4688 


^A 


• 6354 


H 


.8021 


% 


.9688 


2H2 


.1380 




.3047 




•4714 




• 6380 




■8047 




• 9714 


'Me 


.1406 




.3073 




• 4740 




• 6406 




• 8073 




• 9740 


»H2 


•1432 




.3099 




.4766 




.6432 




.8099 




• 9766 


M 


.1458 


H 


.3125 


M 


• 4792 


?4 


.6458 


M 


• 8125 


Vi 


• 9792 


25^2 


.1484 




.3151 




• 4818 




.6484 




• 8151 




.9818 


'Me 


. 1510 




■3177 




.4844 




• 6510 




.8177 




• 9844 


2%2 


.1536 




.3203 




.4870 




• 6536 




• 8203 




• 9870 


'/^ 


.1563 


li 


.3229 


H 


.4896 


% 


• 6563 


U 


.8229 


li 


.9896 


2%2 


• 1589 




.3255 




.4922 




• 6589 




• 825s 




.9922 


'Me 


.1615 




.3281 




.4948 




.6615 




• 8281 




• 9948 


3^2 


.1641 




.3307 




.4974 




• 6641 




• 8307 




• 9974 



Table 9. — ^Lengths of Circular Arcs 
From Kent's Mechanical Engineers' Pocket Book 

(Diameter = I. Given the Chord and Height of the Arc.) 
Rule for Use of the Table. — Divide the height by the chord. Find 
in the column of heights the number equal to this quotient. Take out the 
corresponding number from the column of lengths. Multiply this last num- 
ber by the length of the given chord; the product will be length of the arc. 

// the arc is greater than a semicircle, first find the diameter from the formula, 
Diam. = (square of half chord -^ rise) + rise; the formula is true whether 
the arc exceeds a semicircle or not. Then find the circumference. From the 
diameter subtract the given height of arc, the remainder will be height of the 
smaller arc of the circle; find its length according to the rule, and subtract it 
from the circumference. 



Hgts. 1 Lgths. 1 


Hgts. 


Lgths. 1 


Hgts. 


Lgths. 


Hgts. 


Lgths. 1 


Hgts. 


Lgths. 


.001 


I .00002 


• IS 


I .05896 


.238 


1 . 14480 


• 326 


1 .26288 


.414 


1.40788 


.005 


I .00007 


.152 


I .06051 


.24 


1.14714 


.328 


1.26588 


.416 


1-41145 


.01 


I .00027 


■ 154 


I .06209 


.242 


1. 1495 1 


.33 


1.26892 


.418 


I. 41503 


• 015 


I .00061 


.156 


1.06368 


.244 


1.15189 


.332 


1.27196 


• 42 


1.41861 


.02 


I. 00107 


.158 


1.06530 


.246 


1.15428 


.334 


1.27502 


.422 


1.42221 


.025 


I. 00167 


.16 


I . 06693 


.248 


I. 15670 


• 336 


I .27810 


• 424 


1.42583 


.03 


1.00240 


.162 


1.06858 


.25 


1.15912 


• 338 


1.28188 


• 426 


1.4294s 


.035 


I .00327 


• 164 


I .07025 


.252 


I .16156 


• 34 


1.28428 


.428 


1.43309 


.04 


I .00426 


.166 


I. 07 194 


.254 


I . 16402 


• 342 


1.28739 


•43 


1-43673 


• 045 


1.00539 


.168 


1.0736s 


.256 


I. 16650 


•344 


1.29052 


.432 


1.44039 


• OS 


1.00665 


.17 


1.07537 


.258 


1.16899 


• 346 


1.29366 


• 434 


I. 4440s 


.055 


1.00805 


.172 


1.07711 


• 26 


1.17150 


.348 


1.29681 


• 436 


1-44773 


.06 


1.00957 


• 174 


1.07888 


.262 


1.17403 


• 35 


1.29997 


• 438 


1.45142 


.065 


I .01123 


.176 


I .08066 


.264 


1.17657 


• 352 


1.3031S 


•44 


1 -45512 


.07 


I .01302 


.178 


1.08246 


.266 


1.17912 


• 354 


1.30634 


■ 442 


1.45883 


• 075 


I. 01493 


.18 


1.08428 


.268 


1.18169 


.356 


1.30954 


• 444 


1.4625s 


• 08 


I. 01698 


.182 


1.08611 


• 27 


I .18429 


• 358 


I. 31276 


• 446 


1.46628 


.085 


1 .01916 


.184 


1.08797 


.272 


I .18689 


• 36 


1.31599 


• 448 


1.47002 


• 09 


I .02146 


.186 


I .08984 


• 274 


I. 18951 


.362 


I. 31923 


■ 4S 


1.47377 


.095 


1.02389 


.188 


I. 09174 


.276 


1.19214 


■ 364 


1.32249 


.452 


1.47753 


.10 


1.02646 


.19 


1.09365 


.278 


1.19479 


.366 


1.32577 


• 454 


1.48131 


. 102 


1.02752 


. 192 


r. 0955 7 


.28 


1.19746 


.368 


1.32905 


• 456 


1.48509 


. 104 


I .02860 


.194 


1.09752 


.282 


I .20014 


.37 


1.33234 


• 458 


I .48889 


. 106 


I .02970 


.196 


1.09949 


.284 


1 .20284 


.372 


1.33564 


.46 


I .49269 


.108 


1.03082 


.198 


1.10147 


.286 


I. 20555 


.374 


1.33896 


.462 


1. 4965 1 


.11 


I .03196 


.20 


I . 10347 


.288 


1.20827 


.376 


1.34229 


.464 


I • S0033 


.112 


I. 03312 


.202 


I. 10548 


• 29 


I .21102 


• 378 


1.34563 


• 466 


I .S0416 


• 114 


1.03430 


.204 


I. 10752 


• 292 


1.21377 


.38 


1.34899 


• 468 


1.50800 


.116 


I. 03551 


.206 


I. 10958 


• 294 


1.21654 


• 382 


1.35237 


• 47 


1.5118S 


.118 


1.03672 


.208 


1.11165 


.296 


I. 21933 


.384 


1.35575 


• 472 


I.SIS7I 


.12 


1.03797 


.21 • 


1.11374 


.298 


I .22213 


.386 


1.35914 


•474 


I .31958 


. 122 


1.03923 


.212 


1.11584 


.30 


1.22495 


.388 


1.36254 


• 476 


1.52346 


.124 


I .04051 


.214 


1.11796 


.302 


1.22778 


.39 


1.36596 


• 478 


l^52736 


• 126 


I .04181 


.216 


I .12011 


•304 


1 .23063 


.392 


1.36939 


•48 


1.53126 


.128 


1.04313 


.218 


I. 12225 


• 306 


1.23349 


.394 


1.37283 


.482 


1.53518 


•13 


1.04447 


.22 


I. I 2444 


.308 


1.23636 


.396 


1.37628 


.484 


i^S39io 


.132 


1.04584 


.222 


I .12664 


• 31 


1.23926 


.398 


1.37974 


.486 


l^S4302 


•134 


1.04722 


.224 


1.12885 


.312 


I .24216 


.40 


1.38322 


• 488 


I • 54696 


.136 


I .04862 


.226 


1.13108 


• 314 


1.24507 


.402 


I. 38671 


•49 


I. 55091 


.138 


I .05003 


.228 


1.13331 


.316 


I .24801 


.404 


I. 39021 


• 492 


1-55487 


■14 


I. 05147 


.23 


1.I3SS7 


.318 


1.25095 


• 406 


1.39372 


•494 


1-55854 


.142 


1.05293 


• 232 


I. 13785 


.32 


1.25391 


.408 


1.39724 


• 496 


1.56282 


• 144 


1.05441 


■ 234 


I.1401S 


• 322 


1.25689 


.41 


1.40077 


.498 


I. 56681 


• 146 


I. 05591 


• 236 


I. 14247 


.324 


1.25988 


.412 


1.40432 


• 50 


I .57080 


• 148 


I. 05743 



















MATHEMATICAL TABLES 



525 



Table io. — Cutting Speeds and Revolutions pee Minute 
By the Cincinnati Gear Cutting Machine Company 



Diam. 



IS |I7.S| 20 |22.s| 2S |27.sl 30 I 35 



SO 



SS 



6o 



65 



70 



75 



8o I go I 100 I no I 120 I 130 I 140 



Revolutions per minute 



Ms 


917 


1070 


1222 


I37S 


1528 


1681 


1883 


2139 


244s 


2750 


3056 


3361 


3667 


3973 


4278 


4584 


4889 
















M 


458 


535 


611 


688 


764 


840 


917 


1070 


1222 


1375 


1528 


I68I 


1833 


1986 


2139 


2292 


2445 


2750 


3056 


3361 


3667 


3973 


4278 


4584 


Vie 


306 


357 


407 


458 


509 


560 


611 


713 


815 


917 


1019 


II20 


1222 


1324 


1426 


1528 


1630 


1833 


2037 


2241 


244s 


2648 


2852 


3056 


W 


229 


267 


306 


344 


382 


420 


458 


53S 


611 


688 


764 


840 


917 


993 


1070 


1 146 


1222 


1375 


1528 


1681 


1833 


1986 


2139 


2292 


Vie 


183 


214 


244 


275 


306 


336 


367 


428 


489 


SSO 


611 


672 


733 


794 


856 


917 


978 


1 100 


1222 


134s 


1467 


1589 


171I 


1833 


H 


153 


178 


204 


229 


255 


280 


306 


357 


407 


4S8 


509 


s6o 


611 


662 


713 


764 


81S 


917 


IOI9 


II20 


1222 


1324 


1426 


1528 


Via 


131 


153 


I7S 


196 


218 


240 


262 


306 


349 


393 


437 


480 


524 


568 


611 


6S5 


698 


786 


873 


960 


1048 


1135 


1222 


1310 


a 


IIS 


134 


153 


172 


191 


210 


229 


267 


306 


344 


382 


420 


458 


497 


S3S 


573 


611 


688 


764 


840 


917 


993 


1070 


II 46 


5/« 


91.7 


107 


122 


138 


153 


168 


183 


214 


244 


275 


306 


336 


367 


397 


428 ' 


458 


489 


SSO 


611 


672 


733 


794 


856 


917 


H 


76.4 


89.1 


102 


115 


127 


140 


153 


178 


204 


229 


255 


280 


306 


331 


357 


382 


407 


45 8 


509 


560 


611 


662 


713 


764 


■'A 


65.5 


76.4 


87.3 


98.2 


109 


120 


131 


153 


175 


196 


218 


240 


262 


284 


306 


327 


349 


393 


437 


480 


524 


568 


611 


655 


I 


57-3 


66.8 


76.4 


85.9 


95.5 


105 


IIS 


134 


I S3 


172 


191 


210 


229 


248 


267 


287 


306 


344 


382 


420 


4S8 


497 


S3S 


573 


i^i 


SO. 9 


59.4 


67.9 


76.4 


84.9 


93.4 


102 


119 


136 


153 


170 


187 


204 


221 


238 


255 


272 


306 


340 


3,73 


407 


441 


475 


S09 


iH 


45.8 


53.5 


61. I 


68.8 


76.4 


84.0 


91.7 


107 


122 


138 


153 


I'Ss 


183 


199 


214 


229 


244 


275 


306 


336 


367 


397 


428 


458 


i?i 


41.7 


48.6 


55.6 


62.5 


69.5 


76.4 


83.3 


97.2 


III 


125 


139 


153 


167 


181 


194 


208 


222 


250 


278 


306 


333 


361 


389 


417 


iH 


38.2 


44.6 


50.9 


57.3 


63.7 


70.0 


76.4 


89.1 


102 


IIS 


127 


140 


IS3 


166 


178 


191 


204 


229 


255 


280 


306 


331 


357 


382 


i^ 


35-3 


41. 1 


47.0 


52.9 


58.8 


64.6 


70. S 


82.3 


94.0 


106 


118 


129 


141 


153 


i6s 


176 


188 


212 


23s 


259 


282 


306 


329 


353 


1% 


32.7 


38.2 


43.7 


49.1 


54-6 


60.0 


6s. 5 


76.4 


87.3 


98.2 


109 


120 


131 


142 


153 


164 


175 


196 


218 


240 


262 


284 


306 


327 


il.i 


30.6 


35.7 


40.7 


45.8 


50.9 


56.0 


61. 1 


71.3 


81.5 


91.7 


102 


112 


122 


132 


143 


153 


163 


183 


204 


224 


244 


26s 


285 


306 


2 


28.7 


33.4 


38.2 


43.0 


47.7 


52.5 


57.3 


66.8 


76.4 


85.9 


95-5 


105 


IIS 


124 


134 


143 


153 


172 


191 


210 


229 


248 


267 


287 


2H 


25.5 


29.7 


34.0 


38.2 


42.4 


46.7 


50.9 


59.4 


67.9 


76.4 


84.9 


93.4 


102 


no 


119 


127 


136 


153 


170 


187 


204 


221 


238 


255 


2]^ 


22 .9 


26.7 


30.6 


34-4 


38.2 


42.0 


45.8 


53.5 


61. 1 


68.8 


76.4 


84.0 


91.7 


99.3 


107 


IIS 


122 


138 


153 


168 


183 


199 


214 


229 


2% 


20.8 


24.3 


27.8 


31.3 


34.7 


38.2 


41.7 


48.6 


S5.6 


62.5 


69.5 


76.4 


83.3 


90.3 


97.2 


104 


III 


125 


139 


IS3 


167 


181 


194 


208 


3 


19.1 


22.3 


25.5 


28,6 


31.8 


35.0 


38.2 


44.6 


SO. 9 


57.3 


63.7 


70.0 


76.4 


82.8 


89.1 


95.5 


102 


IIS 


127 


140 


153 


166 


178 


191 


3/4 


17.6 


20.6 


23.5 


26 .4 


29.4 


32.3 


35.3 


41. 1 


47.0 


52.9 


S8.8 


64.6 


70.5 


76.4 


82.3 


88.2 


94-0 


106 


118 


129 


141 


153 


16S 


176 


3H 


16.4 


19. 1 


21.8 


24. S 


27.3 


30.0 


32.7 


38.2 


43.7 


49.1 


54-6 


60.0 


65.5 


70.9 


76.4 


81.9 


87.3 


98.2 


109 


120 


131 


142 


IS3 


164 


m 


15-3 


17.8 


20.4 


22 .9 


25.5 


28.0 


30.6 


35-7 


40.7 


45-8 


50.9 


S6.o 


61. 1 


66.2 


71.3 


76.4 


81.5 


91.7 


102 


112 


122 


132 


143 


153 


4 


14.3 


16.7 


19. 1 


21. s 


23.9 


26.3 


28.7 


33.4 


38.2 


43.0 


47.7 


52.5 


57.3 


62.1 


66.8 


71.6 


76.4 


85.9 


95. 5 


105 


IIS 


124 


134 


143 


4h 


12.7 


14.9 


17.0 


19. 1 


21.2 


23.3 


25. 5 


29.7 


340 


38.2 


42.4 


46.7 


SO. 9 


55. 2 


59. 4 


63.6 


67.9 


76.4 


84.9 


93.4 


102 


no 


119 


127 


5 


II. 5 


13.4 


15.3 


17.2 


I9.I 


21 .0 


22.9 


26.7 


30.6 


34-4 


38.2 


42.0 


45.8 


49-7 


53. 5 


57-3 


61. 1 


68.8 


76.4 


84.0 


91.7 


99-3 


107 


IIS 


sH 


10.4 


12.2 


13.9 


15.6 


17.4 


19. I 


20.8 


24.3 


27.8 


31.3 


34.7 


38.2 


41.7 


45.1 


48.6 


52. 1 


5S.6 


62.5 


69. 5 


76.4 


83.3 


90.3 


97.2 


104 


6 


9.5 


II .1 


12.7 


14.3 


15-9 


17.5 


19. 1 


22.3 


25. S 


28.6 


31.8 


35.0 


38.2 


41.4 


44-6 


47.8 


SO. 9 


57.3 


63.7 


70.0 


76.4 


82.8 


89.1 


95. S 


6H 


8.8 


10.3 


II. 8 


13.2 


14-7 


16.2 


17.6 


20.6 


23 5 


26.4 


29.4 


32.3 


35.3 


38.2 


41. 1 


44 I 


47 


52.9 


58.8 


64.6 


70.5 


76.4 


82.3 


88.2 


7 


8.2 


9.S 


10.9 


12.3 


13.6 


ISO 


16.4 


19. 1 


21.8 


24. S 


27-3 


30.0 


32.7 


35. 5 


38.2 


40.9 


43-7 


49.1 


54.6 


60 .0 


65.5 


70.9 


76.4 


81 .9 


7K2 


7.6 


8.9 


10.2 


II. 5 


12.7 


14.0 


IS. 3 


17.8 


20.4 


22.9 


25. 5 


28.0 


30.6 


33.1 


35.7 


38.2 


40.7 


45.8 


SO. 9 


56.0 


61. 1 


66.2 


71.3 


76.4 


8 


7.2 


8.4 


9.S 


10.7 


II. 9 


13. I 


14.3 


16.7 


19. 1 


21. s 


23.9 


26.3 


28.7 


31.0 


33.4 


35.8 


38.2 


43.0 


47.7 


52. 5 


57.3 


62.1 


66.8 


71.6 


8^2 


6.7 


7.9 


9.0 


10. I 


II. 2 


12.4 


135 


15.7 


18.0 


20.2 


22.5 


24.7 


27 .0 


29.2 


31. 5 


33.7 


36.0 


40.4 


44-9 


49.4 


53.9 


58. 4 


62 .9 


67.4 


9 


6.4 


7.4 


8.5 


9.S 


10.6 


II. 7 


12.7 


14.9 


17.0 


19. I 


21 .2 


23.3 


25 S 


27.6 


29.7 


31.8 


34.0 


38,2 


42.4 


46.7 


50.9 


5S.2 


59.4 


63.6 


qH 


6.0 


7.0 


8.0 


9.1 


10. 1 


n .1 


12. I 


14. 1 


16. 1 


18. I 


20. I 


22 . 1 


24.1 


26.1 


28.2 


30.2 


32 .2 


36.2 


40 .2 


44-2 


48.3 


52.3 


S6.3 


60.3 


10 


5-7 


6.7 


7.6 


8.6 


9.5 


10.5 


II. 5 


13.4 


15.3 


17.2 


19. I 


21 .0 


22 .9 


24.8 


26.7 


28.7 


30.6 


34.4 


38.2 


42 .0 


45.8 


49-7 


53-5 


57-3 


II 


5.2 


6.1 


6.9 


7.8 


8.7 


9-5 


10.4 


12 .2 


13.9 


IS. 6 


17.4 


19. 1 


20.8 


22 .6 


24.3 


26.0 


27.8 


31.3 


34.7 


38.2 


■41.7 


45.1 


48.6 


52. 1 


12 


4.8 


5.6 


6.4 


7.2 


8.0 


8.8 


9.5 


II .1 


12.7 


14.3 


IS. 9 


17. 5 


19. 1 


20.7 


22.3 


23.9 


25.5 


28.6 


31.8 


35.0 


38.2 


41.4 


44.6 


47.8 


13 


4-4 


S.I 


5.9 


6.6 


7.3 


8.1 


8.8 


10.3 


II. 8 


13.2 


14.7 


16.2 


17.6 


19.1 


20.6 


22 .0 


23 5 


26.4 


29.4 


32.3 


35.3 


38.2 


41. 1 


44,1 


14 


4.1 


4.8 


55 


6.1 


6.8 


7.5 


8.2 


9.5 


10.9 


12.3 


13.6 


ISO 


16 .4 


17.7 


19.1 


20.5 


21.8 


24.5 


27.3 


30.0 


32.7 


35. 5 


38.2 


40.9 


IS 


3.8 


4.5 


5.1 


5.7 


6.4 


7.0 


7.6 


8.9 


10.2 


II. 5 


12.7 


14.0 


15-3 


16.6 


17.8 


19. I 


20.4 


22.9 


25. 5 


28.0 


30.6 


33-1 


35. 7 


38.2 


16 


3.6 


4.2 


4.8 


5.4 


6.0 


6.6 


7.2 


8.4 


9.5 


10.7 


II. 9 


13. 1 


14.3 


IS. 5 


16.7 


17.9 


19. 1 


21. 5 


23.9 


26.3 


28.7 


31.0 


33.4 


35.8 


17 


3.4 


3.9 


4-5 


S.I 


5.6 


6.2 


6.7 


7.9 


9.0 


10. I 


II. 2 


12.4 


13. S 


14.6 


IS. 7 


16.9 


18.0 


20.2 


22.5 


24.7 


27.0 


29.2 


31.5 


33-7 


18 


3.2 


3.7 


4.2 


4.8 


S.3 


5.8 


6.4 


7.4 


8.5 


9.5 


10.6 


II. 7 


12.7 


13.8 


14.9 


15.9 


17.0 


19. I 


21 .2 


23.3 


25. S 


27.6 


29.7 


31.8 




15 


17. s 


20 


22.5 


25 


27. S 


30 


35 


40 


45 


so 


55 


60 


65 


70 


75 


80 


90 


100 


no 


120 


130 


140 


150 



526 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table ii. — Decimal Equivalents of Prime Number Fractions 

Denominators (Prime Numbers Only) 



1 97 


89 


83 1 79 


73 


71 1 67 


61 59 


53 47 


43 1 41 


37 


31 


29 


23 


19 


17 


13 


II 


7 


S 


3 


I 


.0103 


.0112 


.0120 


.0126 


.0137 


.0141 


• 0149 


.0164 


.0169 


.0189 


.0213 


• 0233 


■ 0244 


.0270 


.0323 


.0345 


.0435 


.0526 


.0588 


.0769 


.0909 


. 1429 


.2000 


■3333 


3 


.0309 


.0337 


.0361 


.0380 


.0411 


.0423 


.0448 


.0492 


.0508 


.0566 


.0638 


.0698 


.0732 


.0811 


.0968 


.1034 


.1304 


• 1579 


.1765 


.2308 


.2727 


.4286 


.6000 




S 


.0515 


.0562 


.0602 


.0633 


.0685 


.0704 


.0746 


.0820 


.0847 


.0943 


.1064 


.1163 


.1220 


.1351 


• I613 


.1724 


.2174 


• 2632 


.2941 


.3846 


•4545 


.7143 






7 


.0722 


.0787 


.0843 


.0886 


.0959 


.0986 


.1045 


.1148 


.1186 


.1321 


.1489 


.1628 


.1707 


.1892 


• 2258 


.2414 


.3043 


.3684 


.4118 


.5385 


.6364 








II 


.1134 


.1236 


.1325 


.1392 


.1507 


.1549 


.1642 


.1803 


.1864 


■ 2075 


.2340 


.2558 


.2683 


.2973 


• 3548 


.3793 


• 4783 


•5789 


.6471 


.8462 










^ 13 


.1340 


.1461 


.1566 


.1646 


.1781 


.1831 


.1940 


.2131 


.2203 


.2453 


.2766 


.3023 


.3171 


.3514 


.4194 


.4483 


• 5652 


.6842 


•7647 












^ 17 


.1753 


.1910 


.2048 


.2152 


.2329 


.2394 


.2537 


.2787 


.2881 


.3208 


.3617 


.3953 


■ 4146 


■ 4595 


.5484 


.5862 


• 7391 


.8947 














O 19 


.1959 


.2135 


.2289 


.240s 


.2603 


.2676 


.2836 


.3115 


.3220 


.3585 


■4043 


.4419 


■4634 


■513s 


.6129 


.6552 


.8261 
















e 23 


.2371 


.2584 


.2771 


.2911 


.3151 


.3299 


• 3433 


.3770 


.3898 


.4340 


.4894 


.5349 


■ 5610 


.6216 


.7419 


.7931 


















;. 29. 

a' 


.2990 


.3258 


•3494 


.3671 


•3973 


.4085 


.4328 


•4754 


.4915 


.5472 


.6170 


• 6744 


■7073 


.7838 


•9355 




















3 

iz; 31 


.3196 


.3483 


.3735 


• 3924 


.4247 


.4366 


.4627 


.5082 


.5254 


• 5849 


.6596 


.7209 


■7561 


.837S 






















^ 37 


.3814 


.4157 


.4458 


.4684 


.5068 


.5211 


.5522 


.6066 


.6271 


.6981 


.7872 


.8605 


.9024 
























•= 41 


.4227 


.4607 


.4940 


.5190 


.5616 


.5775 


.6119 


.6721 


.6949 


.7736 


• 8723 


.9535 


























fc. 43 


■ 4433 


.4831 


.5181 


•5443 


.5890 


.6056 


.6418 


.7049 


.7288 


.8113 


• 9149 




























£ '" 


.484s 


.5281 


.5663 


.5949 


.6438 


.6620 


.7015 


.7705 


.7966 


.8868 






























^^ 53 


■ 5464 


.5955 


.6386 


.6709 


.7260 


.7465 


.7910 


.8689 


.8983 
































S 59 


.6082 


.6629 


.7108 


. 7468 


.8082 


.8310 


.8806 


.9672 


































;^6I 


.6289 


.6854 


.7349 


.7722 


.8356 


.8592 


.9104 






















67 


.6907 


.7528 


.8072 


.8481 


.9178 


• 9437 




















Only those common fractions having prime 




71 


.7320 


.7978 


.8554 


.8987 


.9726 






















numbers for both the numerator and denomi- 
nator are given in this table. Others can be 




73 


.7526 


.8202 


.8795 


.9241 
























found by simple multiplication or division as: 




79 


.8144 


.8876 


.9518 


























*5'^i=5^X2^i = 2X.3239= .6478 




83 


.8557 


.9326 




























»?^3=HX1%1 = .4194-^3= 1398 




89 


.9175 









































•^ 


x 


























Deg. 


Length | Deg. 


1 Length 


Deg. 


Length 


Min. 


Length 




27 


.4712 


87 


1.S184 


147 


2.5656 


27 


.0079 








s 1 


> 








28 


.4887 


88 


I. 5359 


148 


2.5831 


28 


.0081 








\\ ^ 


i^ 








29 


.5061 


89 


1. 5533 


149 


2.6005 


29 


.0084 








5,3'^i 


v^ 








30 


.5236 


90 


1.5708 


150 


2.6180 


30 


.0087 


















31 


.5411 


91 


1.5882 


151 


2.6354 


31 


.0090 


Table 12.— Lengths of Circular Arcs to Radius of 


I In. 


32 
33 
34 
35 


.5585 
.5760 
.5934 
■ 6109 


92 
93 
94 
95 


1.6057 
1.6232 
1.6406 
1.6581 


1S2 

153 

IS4 
155 


2.6529 
2.6704 
2.6878 
2.7052 


32 
33 
34 
35 


.0093 
.0096 
.0099 


Deg. 


Length 


Deg. 


Length 


Deg. 


Length 


Min. 


Length 


I 


.0175 


61 


1.0647 


121 


2.1118 


I 


.0003 


.0102 


2 


■ 0349 


62 


I. 0821 


122 


2.1293 


2 


.0006 


36 


.6283 


96 


1.6755 


156 


2.7227 


36 


.0105 


3 


.0524 


63 


1.0996 


123 


2.1468 


3 


.0009 


37 


.6458 


97 


1.6930 


157 


2.7402 


37 


.0108 


4 


.0698 


64 


I. 1170 


124 


2.1642 


4 


.0012 


38 


.6632 


98 


I. 7 104 


158 


2.7576 


38 


.0111 


5 


• 0873 


65 


I. 1345 


I2S 


2. 1817 


5 


.OOIS 


39 


.6807 


99 


1.7279 


159 


2.7751 


39 


. 1130 


6 


.1047 


66 


1.IS19 


126 


2.1991 


6 


.0017 


40 


.6981 


100 


I^74S3 


160 


2.7925 


40 


. 1160 


7 


. 1222 


67 


1. 1694 


127 


2 . 2166 


7 


.0020 


41 


■ 7156 


lOI 


1.7628 


i6i 


2.8100 


41 


.1190 


8 


• 1396 


68 


I. 1868 


128 


2.2340 


8 


.0023 


42 


.7330 


102 


1 .7802 


162 


2.8274 


42 


.0122 


9 


• 1571 


69 


1.2043 


129 


2.2515 


9 


.0026 


43 


■ 75 05 


103 


1.7977 


163 


2.8449 


43 


.0125 


10 


• 1745 


70 


1. 2217 


130 


2 .2690 


10 


.0029 


44 


■ 7679 


104 


I.8151 


164 


2.8623 


44 


.0128 


II 


.1920 


71 


1.2392 


131 


2.2864 


II 


.0032 


45 


■ 7854 


105 


1.8326 


165 


2.8798 


45 


.0131 


12 


.2094 


72 


I .2566 


132 


2.3038 


12 


.0035 


46 


.8029 


106 


1.8500 


166 


2.8972 


46 


.0134 


13 


.2269 


73 


I. 2741 


133 


2.3132 


13 


.0038 


47 


.8203 


107 


1.8675 


167 


2.9147 


47 


.0137 


14 


■2443 


74 


I. 2915 


134 


2.3387 


14 


.0041 


48 


.8378 


108 


1.8850 


168 


2.9322 


48 


.0140 


IS 


.2618 


75 


I .3090 


13s 


2.3562 


IS 


.0044 


49 


.8552 


109 


1.9024 


169 


2.9496 


49 


.0143 


16 


.2793 


76 


I. 326s 


136 


2.3736 


16 


.0047 


50 


.8728 


HO 


1.9199 


170 


2.9671 


so 


.0145 


17 


.2967 


77 


1.3439 


137 


2.3911 


17 


.0050 


51 


.8901 


III 


1.9373 


171 


2.9845 


51 


.0148 . 


18 


.3142 


78 


1.3614 


138 


2.4086 


18 


.0052 


52 


.9076 


112 


1.9548 


172 


3.0020 


52 


.0151 


19 


.3316 


79 


1.3788 


139 


2.4260 


19 


.0055 


53 


.9250 


113 


1.9722 


173 


3-0194 


S3 


.0154 


20 


.3491 


80 


1.3963 


140 


2.443S 


20 


.0058 


54 


.9425 


114 


1.9897 


174 


3.0369 


54 


.0157 


21 


.3665 


81 


I. 4137 


141 


2.4609 


21 


.0061 


55 


.9599 


IIS 


2.0071 


. 17s 


3.0543 


SS 


.0160 


22 


.3840 


82 


1. 43 1 2 


142 


2.4784 


22 


.0064 


56 


.9774 


116 


2.0246 


176 


3.0718 


56 


.0163 


23 


.4014 


83 


1.4486 


143 


2.4958 


23 


.0067 


57 


.9948 


117 


2.0420 


177 


3.0892 


57 


.0166 


24 


.4189 


84 


I. 4661 


144 


2.5133 


24 


.0070 


58 


I. 0123 


118 


2.059S 


178 


3 1067 


58 


.0169 


25 


.4363 


85 


1.483s 


14s 


2.5307 


25 


.0073 


59 


I . 0297 


119 


2.0769 


179 


3.1241 


59 


.0172 


26 


.4538 


86 


I. 5010 


146 


2.5482 


26 


■ 0076 


60 


1.0472 


120 


2.0944 


180 


3.1416 


60 


.0175 



The squares of mixed numbers not found in Table 13 are most 
conveniently computed by remembering that (a+6)2 = a^+2a&+6'', 
that is to say, add the square of the whole number, the square of 
the fraction and twice the product of the whole number and fraction. 



Squares of binary fractions will be found in Table 22 and decimal 
equivalents in Table 16 which will greatly facilitate the process. 

Example: Required the square of 2Jys 

square of 27 729.0000 

square of ris ■ o°39 

twice the product of 27 and ys 

= 2X27X^o62S i-ilS 



square of 2 7ris 



732^3789 



MATHEMATICAL TABLES 



527 



Table 13. — Squares op Mixed Numbers 
(W. L. and R. E. Tyron, Amer. Mack., Dec. 23, 1909) 



No. 





I 


2 


3 


4 


5 


6 


7 







. 000000 


I . 00000 


4.00000 


9.00000 


16.00000 


25.00000 


36.00000 


49.00000 




^ 


.000244 


I. 03 149 


4.06274 


9-09399 


i6. 12524 


25.15649 


36.18774 


49.21899 




*? 


.000977 


1.06348 


4-12598 


9.18848 


16. 25098 


25-31348 


36.37598 


49.43848 




A 


.002197 


1.09595 


4. 18970 


9.2834s 


16.37720 


25-47095 


36.56470 


49.65845 




-V 


.003906 


I. I289I 


4.25391 " 


9.37891 


16.50391 


25.62891 


36.75391 


49.87891 




X 


.006104 


I. 16235 


4.31860 


9.47485 


16.63110 


25.7873s 


36.94360 


50.0998s 




jfe 


.068789 


I. 19629 


4.38379 


9.57129 


16.75879 


25.94629 


37.13379 


50.32129 




A 


.011963 


I. 23071 


4.44946 


9.66821 


16.88696 


26. 10571 


37.32446 


50.54321 




i 


.015625 


1.26563 


4-51563 


9.76563 


17.01563 


26.26563 


37.51563 


50.76563 




A- 


.019775 


I. 30103 


4-58228 


9.86353 


17.14478 


26-42603 


37.70728 


50.98853 




A 


.024414 


I. 33691 


4.64941 


9.96191 


17.27441 


26.58691 


37.89941 


51.21191 




tt 


.029541 


1.37329 


4-71704 


IO-O6079 


17.40454 


26.74829 


38.09204 


SI. 43579 




A 


.035156 


I.4IOI6 


4.78516 


10. 16016 


17.53516 


26.91016 


38.28516 


51.66016 




ii 


.041260 


I. 44751 


4.85376 


10. 26001 


17.66626 


27.07251 


38.47876 


51.88501 




A 


.047852 


1.48535 


4.92285 


10.36035 


17.79785 


27.23535 


38.67285 


52.1103s 




i5. 


.054932 


1.52368 


4.99243 


10.46118 


17.92993 


27.39868 


38.86743 


52.33618 




i 


.062500 


1.56250 


5. 06250 


10.56250 


18.06250 


27.56250 


39.06250 


52.56250 




H 


.070557 


I.60I8I 


5.13306 


10.66431 


18.19556 


27.72681 


39.25806 


52.78931 




A 


.079102 


I. 64160 


5.20410 


10.76660 


18.32910 


27.89160 


39.45410 


53.01660 




tt 


.08813s 


I. 68188 


5. 27563 


10.85938 


18.46313 


28.05688 


39.65063 


53.24438 




■ft 


.097656 


1.72266 


5.34766 


10.97266 


18.59766 


28.22266 


39.84766 


53-47266 




fl- 


. 107666 


1.76392 


5-42017 


11.07642 


18.73267 


28.38892 


40.04517 


53.70142 




-H 


. 118164 


1.80566 


5- 493 16 


II. 18066 


18.86816 


28.55566 


40.24316 


53-93066 




H 


.129151 


1.84790 


5.56665 


11.28540 


19.00415 


28.72290 


40.44165 


54-16040 




t 


.140625 


1.89063 


5-64063 


11.39063 


19. 14063 


28.89063 


40.64063 


54-39063 




U 


.152588 


1.93384 


5-71509 


11.49634 


19.27759 


29,05884 


40.84009 


54-62134 




a 


.165039 


1-97754 


5-79004 


11.60254 


19-41504 


29.22754 


41 . 04004 


54-85254 




a 


.177979 


2.02173 


5-86548 


11.70923 


19-55298 


29.39673 


41.24048 


SS- 08423 




A 


.191406 


2.06641 


5-94141 


11.81641 


19.69141 


29.56641 


41-44141 


55-31641 


•5 

u 


II 


.205322 


2.II157 


6-01782 


11.92407 


19.83032 


29.73657 


41.64282 


55-54907 





M 


.219726 


2.15723 


6.09473 


12.03223 


19.96973 


29.90723 


41.84473 


SS. 78223 


>. 

^ 


a 


.234619 


2.20337 


6. 17212 


12. 14087 


20. 10962 


30.07837 


42.04712 


56.01587 


t3 


i 


.250000 


2.25000 


6.25000 


12.25000 


20.25000 


30.25000 


42.25000 


56.20500 


M 


.265869 


2.29712 


6-32837 


12.35962 


20.39087 


30.42212 


42.45337 


56.48462 


•a 

<; 


M 


.282227 


2.34473 


6-40723 


12.46973 


20.53223 


30.59473 


42.65723 


56.71973 


a 


.299072 


2.39282 


6-48657 


12-58032 


20.67407 


30.76782 


42.86157 


56-95532 




A 


.316406 


2.44141 


6.56641 


12. 69 141 


12.81641 


30-94141 


43.06641 


57-19141 




i!i 


.334229 


2.49048 


6-64673 


12.80298 


20.95923 


31-11548 


43.27173 


57.42798 




iJ 


.352539 


2.54004 


6.72754 


12-91504 


21. 10254 


31.29004 


43-47754 


57-66504 




-=?^ 


.371338 


2.59009 


6.80884 


13.02759 


21.24634 


31.46509 


43.68384 


57-90259 




i 


.390625 


2.64063 


6.89063 


13. 14063 


21.39063 


31.64063 


43 . 89063 


58.14063 




a 


.410400 


2.69165 


6.97290 


13-25415 


21.53540 


31.81665 


44.09790 


S8-3791S 




a 


.430664 


2.74316 


7.05566 


13.36816 


21.68066 


31.99316 


44.30566 


58.61816 




a 


.451416 


2.79517 


7.13892 


13.48267 


21.82642 


32. 17017 


44.51392 


58.85767 




a 


.472656 


2.84766 


7.22266 


13.59766 


21.97266 


32.34766 


44. 72266 


59.09766 




a 


.494385 


2.90063 


7.30688 


13.71313 


22. 11938 


32.52563 


44.93188 


59.33813 




a 


.516602 


2.95410 


7.39160 


13.82910 


22. 26660 


32.70410 


45. 14160 


59.57910 




n 


. 5393-07 


3.00806 


7.47681 


13.94556 


22.41431 


32.88306 


45.35181 


59.82056 




i 


.562500 


3.06250 


7.56250 


14.06250 


22.56250 


33.06250 


45.56250 


60.06250 




a 


.586182 


3.11743 


7.64868 


14.17993 


22.71118 


33.24243 


45.77368 


60.30493 




u 


.610352 


3.17285 


7.73535 


14-29785 


22.8603s 


33.4228s 


45.98535 


60.5478s 




a 


.635010 


3.22876 


7.82251 


14.41626 


23.01001 


33.60376 


46.19751 


60.79126 




H 


.660156 


3.28516 


7.91016 


14-53516 


23. 16016 


33.78516 


46.41016 


61.03516 




H 


.685791 


3.34204 


7-99829 


14.65454 


23.31079 


33-96704 


46.62329 


61.27954 




a 


.711914 


3.39941 


8.08691 


14.77441 


23.46191 


34-14941 


46.83691 


61.52441 




M 


.738526 


3.45728 


8. 17603 


14-89478 


23.61353 


34-33228 


47.05103 


61.76978 




I 


.765625 


3.51563 


8.26563 


1S-01563 


23.76563 


34-51563 


47.26563 


62.01563 




a 


■793213 


3.57446 


8.35571 


IS. 13696 


23.91821 


34-69946 


47-48071 


62.26196 




a 


.821289 


3-63779 


8.44629 


15-25879 


24.07129 


34-88379 


47.69629 


62-50879 




a 


.849854 


3.69360 


8.5373S 


15.3811O 


24.22485 


35-06860 


47-91235 


62-7S6lO 




i* 


.878906 


3-75391 


8-62891 


IS-50391 


24-37891 


35-25391 


48.12891 


63-00391 




n 


.908447 


3-81470 


8.72095 


15.62720 


24-53345 


35-43970 


48.3459s 


63.25220 




M 


.938477 


3-87598 


8.81348 


lS-75098 


24.68848 


35.62598 


48.56348 


63.50098 




I- H 


.968994 


3.93774 


8.90649 


15.87524 


24.84399 


35-81274 


48.78149 


63-75024 



528 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 13. — Squares of Mixed Numbers — (Continued) 
(W. L. and R. E. Tyron, Amer. Mack., Dec. 23, 1909.) 



No. 


8 


9 


10 


11 


12 


13 


14 


IS 







64.0000 


81.0000 


100. 0000 


121.0000 


144.0000 


169. 0000 


196 . 0000 


225.0000 




A 


64.5010 


81.5635 


100.6260 


121.688s 


144.7510 


169.8135 


196.8760 


225.938s 




A 


65.0039 


82.1289 


loi. 2539 


122.3789 


145.5039 


170.6289 


197.7539 


226.8789 




A 


65.5088 


82.6963 


101.8838 


123.0713 


146.2588 


171.4463 


198.6338 


227.8213 




i 


66.0156 


83.2656 


102.5156 


123.7656 


147 .0156 


172. 2656 


199.5136 


228.7656 




■h 


66.5244 


83.8369 


103.1494 


124.4619 


147.7744 


173.0869 


200.3994 


229.7119 




A 


67.0352 


84.4102 


103.7852 


125. 1602 


148.5352 


173-9102 


201 .2852 


230.6602 




sV 


67.5479 


84-9854 


104.4229 


125.8604 


149.2979 


174-7354 


202. 1725 


231.6104 




i 


68.0625 


85.5625 


105.0625 


126.5625 


150.0625 


175-5625 


203.062s 


232.562s 




A 


68.5791 


86.1416 


105.7041 


127.2666 


ISO. 8291 


176.3916 


203.9541 


233-5166 




A 


69.0977 


86.7227 


106.3477 


127.9727 


151.5977 


177-2227 


204.8477 


234-4727 




Ji 


69.6182 


87.3057 


106.9932 


128.6807 


152.3682 


178.0557 


205.7432 


235-4307 




f 


70. 1406 


87.8906 


107.6406 


129.3906 


153- 1406 


178.8906 


206. 6406 


236.3906 


C 


M 


70.6650 


88.4775 


108. 2900 


130. 1025 


153.9150 


179-7278 


207 - 5404 


237.353s 


i 


A 


71.1914 


89.0664 


108.9414 


130.8164 


154-6914 


180.5664 


208.4414 


238.3164 




M 


71-7197 


89.6572 


109.5947 


131-5322 


155.4697 


181.4072 


209.3447 


239. 2822 




i 


72. 2500 


90.2500 


no. 2500 


132. 2500 


156.2500 


182. 2500 


210. 2500 


240. 2500 


■a 


H 


72.7822 


90.8447 


110.9072 


132.9697 


157.0322 


183.0947 


211. 1572 


241. 2197 


•a 
•a 


A 


73.3164 


91.4414 


III. 5664 


133-6914 


157.8164 


183.9414 


212.0664 


242. I9I4 


< 


M 


73.8525 


92. 0400 


112. 2275 


134.4150 


158.602s 


184.7900 


212.9775 


243. 1650 




1 


74-3906 


92.6406 


112. 8906 


135.1406 


159.3906 


185.6406 


213.8906 


244. 1406 




li 


74-9307 


93.2432 


II3-S557 


135-8682 


160. 1807 


186.4932 


214.8057 


24s. 1182 




ii 


75.4727 


93.8477 


114. 2227 


136.5977 


160.9727 


187.3477 


215-7227 


246.0977 




fl 


76.0166 


94-4541 


114. 8916 


137-3291 


161.7666 


188.2041 


216.6416 


247.0791 




i 


76.5625 


95-0625 


115-5625 


138.0625 


162.5625 


189.0625 


217.5623 


248.062s 




if 


77.1104 


95-6729 


116.2354 


138.7979 


163.3604 


189.9229 


218.4854 


249.0479 




M 


77.6602 


96.2852 


116. 9102 


139.5352 


164. 1602 


190.7852 


219.4102 


250.0352 




fi 


78. 2119 


96.8994 


117-5869 


140.2744 


164.9619 


191.6494 


220.3369 


251.0244 




1 


78.7656 


97-5156 


118.2656 


141.0156 


165.7656 


192. S156 


221.2656 


252.0156 




a 


79-3213 


98-1338 


118.9463 


141.7588 


166.5713 


193-3838 


222. 1963 


253-0088 




a 


79-8789 


98-7539 


119.6289 


142.5039 


167-3789 


194-2539 


223. 1289 


254-0039 




1 1* 


80.438S 


99-3760 


120.3135 


143.2510 


168. 188s 


195- 1260 


224.0635 


255.0010 



No. 


16 


17 


18 


19 


20 


21 


22 


23 







256.000 


289.000 


324.000 


361.000 


400.000 


441.000 


484.000 


529.000 




A 


257.001 


290.063 


325.126 


362.188 


401.251 


442.313 


485.376 


530.438 




A 


258.004 


291. 129 


326.254 


363-379 


402 . SO4 


443 . 629 


486.754 


531.879 




A 


259.009 


292. 196 


327-384 


364-571 


403 - 759 


444 . 946 


488.134 


533. 321 




i 


260.016 


293 . 266 


328.516 


365-766 


405 . 016 


446 . 266 


489. S 16 


534.766 




A 


261.024 


294-337 


329-649 


366.962 


406 . 274 


447 . 587 


490.899 


536.212 




A 


262.035 


295-410 


330.78s 


368.160 


407 - 535 


448.910 


492. 286 


537.660 




A 


263 . 048 


296.48s 


331-923 


369.360 


408.798 


450.235 


493-673 


539-110 




i 


264.063 


297-563 


333-063 


370.563 


410.063 


451.563 


495 - 063 


540.563 




A 


265.079 


298.642 


334-204 


371.767 


411-329 


452.892 


496 . 454 


542-017 




A 


266 . 098 


299.723 


335-348 


372.973 


412.598 


454.223 


497.848 


543 ■ 473 




a 


267. 118 


300.806 


336.493 


374-181 


413.868 


455.556 


499 . 243 


544-931 


■0 


i 


268. 141 


301.891 


337-641 


375-391 


415.141 


456.891 


500.641 


546.391 


e 




a 


269- i6s 


302.978 


338.790 


376.603 


416. 41s 


458.228 


502.040 


547.853 


CJ 


A 


270. 191 


304 . 066 


339-941 


377.816 


417.691 


459. S66 


503.441 


549.316 




M 


271 . 220 


305-157 


341-095 


379-032 


418.970 


460.907 


504.845 


SSO.782 




i 


272.250 


306.250 


342.250 


380.250 


420.250 


462.250 


506.250 


552-250 




a 


273-282 


307.34s 


343.407 


381.470 


421-532 


463.595 


507.657 


S53.720 


•a 

•a 


A 


274-316 


308.441 


344-566 


382.691 


422.816 


464.941 


509 . 066 


SSS.191 


<! 


a 


275-353 


309.540 


345 -728 


383-915 


424-103 


466. 290 


510.478 


556.665 




i 


276.391 


310.641 


346.891 


385-141 


425-391 


467.641 


511.891 


558.141 




n 


277.431 


311-743 


348.056 


386.368 


426.681 


468.993 


513.306 


559.618 




H 


278.473 


312.848 


349-223 


387-598 


427.973 


470.348 


S14.723 


561.098 




H 


279-517 


313-954 


350.392 


388.829 


429.267 


471-704 


516. 142 


562.579 




1 


280.563 


3 IS- 063 


351-563 


390.063 


430.563 


473.063 


5 17. 563 


564.063 




M 


281.610 


316-173 


352.735 


391.299 


431.860 


474-423 


518.985 


565.548 




U 


282.660 


317-285 


353-910 


392.535 


433.160 


475-785 


520.410 


567.035 




U 


283.712 


318.399 


355-087 


393.774 


434 462 


477-149 


521.837 


568.524 




i 


284.766 


319.516 


356. 266 


395.016 


435.766 


478.516 


523. 266 


570.016 




H 


285.821 


320.634 


357-446 


396.259 


437.071 


479.884 


524.696 


571.509 




a 


286.879 


321-754 


358-629 


397-504 


438.379 


481.254 


526. 129 


573.004 




fi 


287-938 


322.876 


359-813 


398.751 


439.688 


482.626 


527.563 


574.501 



MATHEMATICAL TABLES 



529 



Table 13. — Squares of Mixed Numbers — {Continued) 
{W. L. and R. E. Tryon, Amer. Mack., Dec. 23, igog) 



N 


0. 


24 


25 


26 


27 


28 


29 


30 


31 







S76. 000 


625 . 000 


676- 000 


729. 000 


784. 000 


841.000 


900. 000 


961 . 000 




A 


577-501 


626. 564 


677 .626 


730.689 


785.751 


842. 814 


901 .876 


962.939 




A 


579-004 


628. 129 


679-254 


732-379 


787.504 


844.629 


903-754 


964.879 




A 


580.509 


629.696 


680.884 


734-071 


789-259 


846 . 446 


90s - 634 


966.821 




J. 


582.016 


631 .266 


682.516 


735-766 


791 . 016 


848 . 266 


907 -5 16 


968 . 766 




A 


583.524 


632.837 


683.149 


737-462 


792.774 


850.087 


909 - 399 


970. 712 




A 


585.035 


634.410 


685-785 


739-160 


794-535 


851 .910 


91I- 286 


972.660- 




A 


586.548 


635.98s 


687.423 


740.860 


796. 298 


853.73s 


913-173 


974-610 




i 


588.063 


637-563 


689.063 


742-563 


798.063 


855.563 


915-063 


976.563 




A 


589-579 


639-142 


690.704 


744-267 


799-829 


857-392 


916.954 


978-S17 




A 


591 098 


640.723 


692.348 


745-973 


801 . 598 


859-223 


918.848 


980.473 


1 


M 


592.618 


642.306 


693-993 


747-681 


803.368 


861-056 


920.743 


982.431 





i 


594-141 


643-891 


695.641 


749-391 


805. 141 


862.891 


922 . 641 


984-391 


<u 


M 


595-665 


645-478 


697. 290 


751.103 


806.915 


864.728 


924.540 


986.353 




A 


597- 191 


647 - 066 


698.941 


752.816 


808.691 


866.566 


926.441 


988.316 


Ic 


a 


598.720 


648.657 


700.595 


754-532 


810.470 


868.407 


928.34s 


990.282 


•a 


i 


600. 250 


650. 250 


702. 250 


756-250 


812. 250 


870.250 


930.250 


992.250 


:§ 


a 


601 . 782 


651-845 


703.907 


757-970 


814. 032 


872.09s 


932.157 


994- 220 


< 


A 


603.316 


653.441 


705.566 


759.691 


815.816 


873-941 


934.066 


996. 191 




M 


604. 853 


655.040 


707. 228 


761. 41S 


817.603 


875-790 


935-978 


998 . l6s 




5 


606.391 


656.641 


708.891 


763-141 


819-391 


877-641 


937-891 


1000. 141 




k 


607.931 


658.243 


710.556 


764-868 


821. 181 


879-493 


939-806 


1002. 118 




H 


609-473 


659-848 


712.223 


766.598 


822.973 


881.348 


941-723 


1004. 098 




a 


611-017 


661.454 


713.892 


768.329 


824.767 


883.204 


943-642 


1006.079 




3 


612.063 


663 - 063 


7IS-563 


770.063 


826.563 


885.063 


945-563 


1008.063 




k 


614- 110 


664.673 


717-235 


771-798 


828.360 


886.923 


947-486 


loio. 048 




H 


615.660 


666-285 


718 .910 


773-535 


830. 160 


888.785 


949-410 


1012 . 03s 




H 


617 - 212 


667 . 899 


720.587 


77S-274 


831 .962 


890. 650 


951-337 


1014. 024 




1 


618-766 


669-516 


722. 266 


777-016 


833 -766 


892.516 


953-266 


1016. 016 




« 


620.321 


671- 134 


723.946 


778-759 


835.571 


894.384 


9SS-196 


1018. 009 




a 


621.879 


672.754 


725.629 


780-504 


837.379 


896.254 


957- 129 


1020. 004 




u 


623.439 


674-376 


727.314 


782. 251 


839- 188 


898.126 


959-064 


1022. 001 



No. 


32 


33 


34 


35 


36 


37 


38 


39 







1024. 00 


1089.00 


1156-00 


1225 .00 


1296.00 


1369 00 


1444 - 00 


1521 .00 




A 


1026 . 00 


1091 . 06 


1158.13 


1227 . 19 


1298 . 25 


1371-31 


1446-38 


1523-44 




A 


1028 .00 


1093.13 


1160.25 


1229.38 


1300.50 


1373-63 


1448-75 


1525-88 




A 


1030. 01 


1095 . 20 


1162.38 


1231-57 


1302.76 


1375-95 


1451.13 


1528-32 




i 


1032.02 


1097.27 


1164.52 


1233-77 


1305 -02 


1378.27 


1453.52 


1530-77 




A 


1034.02 


1099.34 


1166.65 


1235-96 


1307-27 


1380.59 


I4SS 90 


IS33-2I 




A 


1036.04 


IIOI .41 


1168.79 


1238.16 


1309-54 


1382.91 


1458.29 


1535-66 




A 


1038.05 


1103.49 


1170.92 


1240.36 


I3II . 80 


1385-24 


1460.67 


1538.11 




i 


1040.06 


1105.56 


117306 


1242.56 


1314.06 


1387-56 


1463 . 06 


1540.56 




A 


1042.08 


I I 07 .64 


1175 - 20 


1244.77 


1316.33 


1389-89 


1465.45 


1543-02 




A 


1044. 10 


1109.72 


1177-35 


1246.97 


I318 .60 


1392.22 


1467-85 


IS4S-47 




M 


1046. 12 


nil .81 


1179-49 


1249. 18 


1320.87 


1394 -S6 


1470.24 


1547-93 


1 

8 


1 


1048. 14 


III3-89 


I181-64 


1251-39 


1323-14 


1396.89 


1472.64 


1550.39 


a 


1050. 17 


III5-98 


1183-79 


1253.60 


1325-42 


1399-23 


1475-04 


1552. 85 


u 


A 


1052. 19 


III8.07 


1185.94 


1255.82 


1327-69 


1401.57 


1477.44 


1555-32 




a 


1054. 22 


II20. 16 


I188-09 


1258.03 


1329-97 


1403.91 


1479-84 


1557-78 


3 


i 


1056. 25 


1122.25 


1190. 25 


1260.25 


1332.25 


1406- 25 


1482. 25 


1560. 25 


•2 


H 


1058.28 


1124.34 


1192.41 


1262.47 


1334-53 


1408 -59 


1484.66 


1562.72 


•0 


A 


1060.32 


1126.44 


1194-57 


1264.69 


1336.82 


1410.94 


1487.07 


1565-19 


<: 


¥1 


1062.35 


1128.54 


1196.73 


1266.91 


1339-10 


1413-29 


1489.48 


1567-67 




f 


1064.39 


1130.64 


1198.89 


1269. 14 


1341-39 


1415-64 


1491.89 


1570- 14 




a 


1066.43 


1132.74 


1201.06 


1271-37 


1343-68 


1417-99 


1494.31 


1572.62 




H 


1068.47 


1134.85 


1203. 22 


1273 .60 


1345-97 


1420-35 


1496-72 


1575-10 




¥i 


1070.52 


II3S-9S 


1205.39 


1275-83 


1348-27 


1422.70 


1499- 14 


1577.58 




3 


1072.56 


I 139 06 


1207.56 


1278-06 


1350-56 


1425.06 


1501-56 


1580.06 




k 


1074.61 


I141.17 


1209.74 


1280.30 


1352-86 


1427.42 


1503-99 


1582.55 




■H 


1076.66 


1143-29 


1211.91 


1282.54 


1355-16 


1429-79 


1506.41 


1585-04 




H 


1078. 71 


1145-40 


1214. 09 


1284.77 


1357.46 


1432-15 


1508.84 


1587.52 




1 


1080.77 


1147-52 


1216. 26 


1287 .02 


1359.77 


1434.52 


isn .26 


159O-O2 




¥i 


1082.82 


1149.63 


1218.45 


1289. 26 


1362.07 


1436-88 


1513-70 


1592.51 




15. 


1084. 88 


1151.75 


1220.63 


1291 50 


1364-38 


1439-25 


1516-13 


1595 00 




¥s 


1086.94 


1153-88 


1222. 81 


1293.75 


1366.69 


1441.63 


1S18.56 


1597-50 



530 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 13. — Squares op Mixed Numbers — {Continued) 
{W . L. and R. E. Tryon, Amer. Mach., Dec. 23, 1909) 





f No. 


40 


41 


42 


43 


44 


45 


46 


47 







1600. 00 


1681.00 


1764.00 


1849.00 


1936.00 


2025.00 


2116.00 


2209.00 




i- 


1610.02 


1691.27 


1774-52 


1859.77 


1947 . 02 


2036.27 


2127.52 


2220.77 




1 


1620.06 


1701.56 


1785.06 


1870.56 


1958.06 


2047.56 


2139.06 


2232.56 




i 


1630. 14 


1711.89 


1795.64 


1881.39 


1969. 14 


2058.89 


2150.64 


2244.39 




h 


1640.2s 


1722.25 


1806.25 


1892.25 


1980. 25 


2070. 25 


2162. 25 


2256.25 




i 


1650.39 


1732.64 


1816.89 


1903.14 


1991-39 


2081.64 


2173-89 


2268.14 




\ 


1660.56 


1743.06 


1827.56 


1914.06 


2002.56 


2093 . 06 


2185. S6 


2280.06 




s 


1670.77 


1753-52 


1838.27 


1925.02 


2013.77 


2104.52 


2197-27 


2292.02 



No. 


48 


49 


50 


SI 


52 


53 


54 


55 





2304.00 


2401 .00 


2500.00 


2601 . 00 


2704. 00 


2809 . 00 


2916.00 


3025,00 


i 


2316.02 


2413.27 


2512.52 


2613.77 


2717.02 


2822.27 


2929.52 


3038.77 


\ 


2328.06 


2425.56 


2525.06 


2626.56 


2730.06 


2835.56 


2943 . 06 


3052.56 


% 


2340. 14 


2437 . 89 


2537-64 


2639-39 


2743.14 


2848.89 


2956.64 


3066.39 


\ 


23S2.25 


2450.25 


2550.25 


2652.25 


2756.25 


2862.25 


2970.25 


3080.2s 


1 


2364.39 


2462.64 


2562.89 


2665. 14 


2769.39 


2875.64 


2983.89 


3094-14 


i 


2376.56 


2475.06 


2575-56 


2678.06 


2782.56 


2889.06 


2997.56 


3108.06 


I 


2388.77 


2487-52 


2588.27 


2691.02 


2775.77 


2902.52 


3011.27 


3122.02 



No. 


56 


57 


58 


59 


60 


61 


62 


63 





3136.00 


3249.00 


3364 00 


3481.00 


3600.00 


3721.00 


3844.00 


3969 -00 


"8 


3150.02 


3263.27 


3378.52 


3495-77 


361S-02 


3736.27 


3859-52 


3984-77 


i 


3 164 . 06 


3277.56 


3393-06 


3510-56 


3630.06 


3751.56 


3875-06 


4000. 56 


a 


3178-14 


3291.89 


3407-64 


3525-39 


3645 - 14 


3766.89 


3890.64 


4016.39 


\ 


3192.25 


3306.2s 


3422.2s 


3540-25 


3660.2s 


3782.25 


3906.25 


4032.2s 


1 


3206.39 


3320.64 


3436.89 


3555-14 


3675-39 


3797.64 


3921.89 


4048 . 14 


3 


3220.56 


3335-06 


3451-56 


3570.06 


3690.56 


3813.06 


3937.56 


4064.06 


I 


3234.77 


3349-52 


3466.27 


3585.02 


3705.77 


3828.52 


3953-27 


4080.02 



No. 



65 



66 



67 



68 



69 



70 



71 



4096.00 
4112.02 
4128 .06 
4144.14 
4160. 25 
4176.39 
4192-56 
4208.77 



4225 .00 
4241. 27 
4257.56 
4273.89 
4290.2s 
4306.64 
4323.06 
4339-52 



4356.00 

4372.52 
4389.06 
4405.64 

4422. 2S 
4438.89 
4455.56 
4472.27 



4489.00 
4505.77 
4522.56 

4539-39 
4556.25 
4573.14 
4590.06 
4607.02 



4624.00 
4641.02 
4658.06 
4675.14 
4692.25 
4709.39 
4726.56 
4743-77 



4761.00 
4778.27 
4795.56 
4812.89 
4830.25 
4847.64 
4865.06 
4882.52 



4900.00 
4917.52 
4935-06 
4952.64 
4970.25 
4987.89 
5005.56 
SO23.27 



5041.00 
5058.77 
SO76.56 
5094-39 
5112.25 
5130.14 
5148.06 
5166.02 



No. 



72 



73 
5329.00 
5347.27 
5365-56 
5383.89 
5402.25 
5420.64 
5439-06 
5457.52 



74 



7S 



76 



77 



78 



79 
6241.00 
6260.77 
6280.56 
6300.39 
6320.25 
6340. 14 
6360.06 
6380.02 



5184.00 

S202.02 
5220.06 
5238.14 
5256.25 
5274.39 
5292.56 
5310.77 



5476.00 
5494-52 
5513-06 
5531-64 
5550-25 
5568.89 
5587-56 
5606.27 



5625.00 

5643-77 
5662.56 
5681.39 
5700.25 
5719.14 
5738.06 

5757.02 



5776.00 
5795-02 
5814.06 
5833-14 
5832.25 
5871-39 
5890.56 
5909-77 



5929 -00 
5948-27 
5967-56 
5986.89 
6006.25 
6025 .64 
604s . 06 
6064. 52 



6084.00 
6103.52 
6123.06 
6142.64 
6162.25 
6181.89 
6201.56 
6221. 27 



No. 



80 



82 



83 



84 



85 



86 



87 



6400.00 
6420.02 
6440.06 
6460. 14 
6480.25 
6500.39 
6520.56 
6540.77 



6561.00 
6581.27 
6601.56 
6621 .89 
6642.2s 
6662.64 
6683.06 
6703.52 



6724.00 
6744-52 
6765.06 
6785.64 
6806.25 
6826.89 
6847.56 
6868.27 



6889.00 
6909.77 
6930.36 
6951.39 
6972.25 
6993-14 
7014.06 
7035-02 



7056. 00 
7077.02 
7098.06 
7119.14 
7140.25 
7161.39 
7182.56 
7203.77 



7225.00 
7246. 27 
7267.56 
7288.89 
7310.25 
7331-64 
7353-06 
7374-52 



7396.00 
7417-52 
7439 - 06 
7460.64 
7482.2s 
7503 - 89 
7525-56 
7547-27 



7569.00 
7590.77 
7612.36 
7634-39 
7656.2s 
7678.14 
7700.06 
7722.02 



No. 


88 


89 


90 


91 


92 


93 


94 


95 







7744-00 


7921.00 


8100.00 


8281.00 


8464.00 


8649.00 


8836.00 


9025.00 




\ 


7766.02 


7943-27 


8122.52 


8303.77 


8487.02 


8672.27 


8859.52 


9048.77 




1 


7788.06 


7965-36 


8145.06 


8326.56 


8510.06 


8695.36 


8883.06 


9072.56 




\ 


7810. 14 


7987-89 


8167.64 


8349-39 


8533.14 


8718.89 


8906.64 


9096.39 




\ 


7832-23 


8010. 23 


8190.25 


8372.25 


8556.23 


8742-25 


8930.25 


9120.25 




\ 


7854-39 


8032.64 


8212.89 


8393.14 


8579-39 


8765-64 


8953-89 


9144-14 




3 


7876.56 


8055.06 


823s. S6 


8418.06 


8602.56 


8789.06 


8977.56 


9168.06 




\ 


7898-77 


8077.52 


8258.27 


8441. 02 


8623.77 


8812.52 


9001. 27 


9192.02 





No. 


96 


97 


98 


99 


100 


101 


102 


103 





9216.00 


9409 . 00 


9604.00 


9801.00 


10000.00 


I020I .00 


10404.00 


10609.00 


\ 


9240.02 


9433-27 


9628.52 


9825.77 


10025.02 


10226. 27 


10429.52 


10634-77 


1 


9264.06 


9457-56 


9653.06 


9850.36 


10050.06 


10231.36 


10455.06 


10660.56 


\ 


9288.14 


9481.89 


9677.64 


9873.39 


10075. 14 


10276.89 


10480.64 


10686.39 


\ 


9312.25 


9506.2s 


9702.25 


9900.25 


10100.25 


10302. 25 


10306.23 


10712. 25 


\ 


9336.39 


9330.64 


9726.89 


9925.14 


10125.39 


10327.64 


10331.89 


10738.14 


\ 


9360.56 


9355-06 


9751.56 


9950.06 


10150.36 


10353.06 


10337.36 


10764.06 


5 


9384.77 


9579-52 


9776.27 


9973-02 


10173.77 


10378.32 


10583.27 


10790.02 



MATHEMATICAL TABLES 



531 



Table 14. — Formulas and Constants for the Computation of Regular 
Factors and their Logarithms, and Central Angles, 



Polygons, by W. L. Benitz {Amer. Mach., May 23, 1907) 
for Polygons of from 3 to 25 Sides 



N 


F 


LogF 


M 


Log M 


H 


hosH 


K 


LogK 


B 


Log B 


z 


3 


S.19615 


.715682 


5-I961S 


.715682 


.324760 


1.511562 


.433013 


T. 636501 


.384900 


T. 585348 


120° 


4 


4.00000 


.602060 


5-65685 


.752575 


.500000 


T. 698970 


I . 00000 


.000000 


.353553 


1.548455 


90° 


S 


3.63271 


.560231 


5-87785 


.769219 


.594410 


T. 774086 


1.72048 


.235649 


.340260 


T.S31811 


72° 


6 


3.46410 


.539591 


6 . 00000 


.778151 


.649519 


T. 812592 


2.59808 


.414652 


.333333 


1.522879 


60° 


7 


3-37100 


.527759 


6.07435 


.783500 


.684103 


T. 835122 


3.63393 


.560377 


.329254 


1.517531 


51° 25' 43" 


8 


3-31371 


.520314 


6. 12294 


.786960 


.707107 ■ 


T. 84948s 


4.82843 


.683806 


.326641 


T. 514070 


45° 


9 


3-27573 


.515308 


6.15636 


.789324 


.723136 


T. 859220 


6.18182 


.791117 


.324867 


T.S11706 


40° 


10 


3.24920 


.511776 


6.18034 


.791012 


.734732 


T. 866129 


7.69421 


.886164 


.323607 


T.510018 


36° 


II 


3.22989 


.509187 


6.19811 


.792259 


.743380 


T.871211 


9.36566 


.971538 


.322679 


T.S0877I 


32° 43' 38" 


12 


3.21539 


.507234 


6.21166 


.793207 


.750000 


T. 875061 


II . 1962 


I .049069 


.321975 


T.S07823 


30° 


13 


3.20420 


.505720 


6.22219 


.793943 


.755173 


T. 878047 


13.1858 


1. I20I07 


.321430 


T.S07087 


27° 41' 32" 


14 


3-19543 


.504529 


6. 23062 


.794532 


.759293 


T. 880410 


15.3344 


I. 185667 


.320993 


T. 506499 


25° 42' 51" 


I,^ 


3-I883S 


.503566 


6.23735 


.795000 


.762631 


T.88231S 


17.6424 


1.246557 


.320649 


T. 506030 


24° 


16 


3. 18260 


.502782 


6. 24289 


.795386 


.765367 


T. 883870 


20. 1094 


1-303398 


.320364 


T.S05644 


22° 30' 


17 


3.17788 


.502138 


6.24754 


.795709 


.767636 


T. 88515s 


22.7353 


1- 356700 


.320126 


T. 505321 


21° 10' 35" 


18 


3-17389 


.501591 


6.25133 


.795973 


.76954s 


T. 886234 


25.5208 


1.406894 


.319932 


T. 505057 


20° 


10 


3.17051 


.501130 


6.25455 


.796196 


.771166 


T. 887148 


28.4654 


I. 454318 


.319767 


T. 504834 


18° 56' si" 


20 


3.16769 


. 500743 


6.25738 


.796392 


.772543 


T. 887922 


31.5688 


1.499258 


.319623 


1.504638 


18° 


21 


3.16523 


. 50040s 


6.2S97S 


.796557 


.773729 


T. 888589 


34.8316 


I. 541974 


.319502 


1.504473 


17° 8'.34" 


22 


3.16317 


.500123 


6.26195 


.796710 


.774763 


T. 889169 


38.2527 


1.582663 


•319389 


T. 504320 


16° 21' 49" 


23 


3. 16129 


.499864 


6.26369 


.796830 


.775668 


T. 889676 


41.8342 


1.621532 


.319301 


T.S04200 


15° 39' 8" 


24 


3.15966 


.499640 


6.26526 


.796939 


.776457 


T. 890118 


45-5743 


1.658722 


.319221 


1 .504091 


15° 


25 


3.15824 


•499444 


6.26666 


•797036 


.777156 


T. 890508 


49.4738 


1.694376 


•319149 


T. 503994 


14° 24' 



Symbols and Equations 



z = Angle subtended at center by side. 

P = Perimeter of polygon. 

C = Length of one side. 

A =Area of polygon. 

N — Number of sides. 

d = Diameter of inscribed circle. 

D = Diameter of circumscribed circle. 




(<— C 









Knowing 










P 


A 


c 


D 


d 




p 




p = 2Vfa 


P=CN 


p MD 
2 


P==Fd 


1 


A 


-^ 




A^KC^ 


A = H£>2 


. Fd' 

A = 

4 





C 


c-l 


„ 2VFA 

^^ N 




^ MD 

^ 2N 






D 


D^BP 


D = 2B\/PA 


D=NBC 




D=BFd 




d 


. aKP 


War 

N 


. aKC 

^ N 


, 2MKD 





The following factors are used in the calculations, their values being 
found in Table 1 1 ; 



F^N tan- 



,. „ . 180- 
M = 2iV stn ^j=- 
N 



„ N . 360° 



-, N , 180° „ I 180° 

K — — cot — i7-: .6= rr cosec — ;-=- 
4 N N N 



Table 15. — Diameters and Spacings of Circles With Nearest 
Whole Number of Divisions 
{Jas. Fraser, Amer. Mach., May 14, 1908) 



Diam- 










Distance on 


circumference 










eter 


A" 


A" 


i" 


A" 


i" 


A" 


f" 


A" 


V 


A" 


i" 


W 


3" 


I" 


I" 


1 


25 


12 


6 


























-h 


31 


16 


8 


























I 


38 


19 


9 


6 
























16 


44 


22 


II 


7 


5 






















5. 


SO 
63 


25 
31 


13 
16 


8 
10 


6 
8 


6 
















-( 




a 


73 


38 


19 


13 


9 


8 


6 














j 




8 


88 


44 


22 


IS 


II 


9 


7 


6 
















I 


100 


so 


25 


17 


13 


10 


8 


7 


6 
















126 


63 


31 


21 


16 


13 


10 


9 


8 


7 


6 












150 


73 


38 


23 


19 


15 


13 


II 


10 


8 


7 












176 


88 


44 


29 


22 


18 


15 


13 


II 


10 


9 


8 


7 


6 




2 


200 


100 


so 


34 


25 


20 


17 


14 


12 


II 


10 


9 


8 


7 


6 


\ 


226 


113 


56 


38 


28 


23 


19 


16 


14 


13 


II 


10 


9 


8 


7 


h 


251 


125 


63 


42 


31 


25 


21 


18 


16 


14 


12 


II 


10 


9 


8 


I 


277 


138 


69 


46 


35 


28 


23 


20 


17 


15 


14 


13 


12 


10 


9 


3 


302 


151 


75 


30 


38 


30 


25 


22 


19 


17 


IS 


14 


13 


II 


9 


i 


327 


163 


82 


34 


41 


ii 


27 


23 


20 


18 


16 


15 


14 


12 


10 


1 


352 


176 


88 


59 


44 


35 


30 


25 


22 


20 


18 


16 


15 


13 


II 


1 


378 


189 


94 


63 


47 


38 


31 


27 


24 


21 


19 


17 


16 


14 


12 


4 


402 


201 


100 


67 


50 


40 


34 


29 


25 


22 


20 


18 


17 


IS 


13 


J 


428 


214 


107 


71 


S3 


43 


36 


31 


27 


24 


21 


19 


18 


15 


13 


\ 


454 


227 


114 


76 


37 


43 


38 


32 


28 


23 


23 


21 


19 


16 


14 


a 


478 


239 


119 


79 


60 


48 


40 


34 


30 


27 


24 


22 


20 


17 


IS 


5 


503 


252 


126 


84 


63 


so 


42 


36 


31 


28 


25 


23 


21 


18 


16 


i- 


528 


264 


132 


88 


66 


S3 


44 


38 


33 


29 


26 


24 


22 


19 


16 


h 


554 


277 


138 


92 


69 


35 


46 


40 


35 


31 


27 


25 


23 


20 


17 


I 


579 


289 


145 


96 


73 


38 


48 


41 


36 


32 


28 


26 


24 


21 


I3 


6 


604 


302 


151 


lOI 


76 


61 


30 


44 


38 


34 


30 


27 


25 


22 


19 



532 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



To find 
multiply it 



Table i6. — Areas of Circular Segments 
From Trautwine's Civil Engineer's Pocket Book 
the area of a segment: Divide the height by the diameter. Opposite the result in this table find the area constant and 
by the square of the diameter. 



Height 


Area | 


Height 1 


Area 


Height 


Area 


Height 1 


Area 


Height 


Area 


Height 


Area 


Height 


Area 


.001 


.000042 


• 073 


■025714 


.14s 


070329 


.217 


125634 


• 289 


.188141 


.361 


.255511 


.433 


.325900 


.002 


.000119 


.074 


.026236 


. 146 


071034 


.218 


126459 


. 290 


. 189048 


• 362 


.256472 


• 434 


.326891 


.003 


.000219 


.075 


.026761 


.147 


071741 


.219 


127286 


.291 


. 189956 


.363 


•257433 


• 435 


.327883 


.004 


.000337 


.076 


.027290 


• 148 


072450 


.2 20' 


128114 


.292 


. 19086s 


• 364 


.25839s 


• 436 


.328874 


.005 


.000471 


• 077 


.027821 


• 149 


073162 


.221 


128943 


.293 


.191774 


• 365 


•259358 


• 437 


.329866 


.006 


.000619 


.078 


.028356 


.150 


073875 


.222 


129773 


.294 


.19268s 


• 366 


.260321 


.438 


.330858 


.007 


.000779 


• 079 


.028894 


• 151 


074590 


■223 


130605 


• 295 


•193597 


.367 


.26128s 


• 439 


.331851 


.008 


.0009S2 


.080 


■02943S 


• 152 


075307 


.224 


131438 


• 296 


.194509 


• 368 


.262249 


• 440 


.332843 


.009 


.00113s 


.081 


•029979 


•153 


076026 


• 225 


132273 


.297 


•195423 


• 369 


.263214 


• 441 


.333836 


.010 


.001329 


.082 


•030526 


• 154 


076747 


• 226 


133109 


.298 


•196337 


.370 


.264179 


• 442 


.334829 


.011 


■001533 


.083 


•031077 


•155 


077470 


.227 


133946 


• 299 


.197252 


.371 


.26514s 


• 443 


.335823 


.012 


.001746 


.084 


• 031630 


.156 


078194 


.228 


134784 


.300 


•198168 


.372 


.266111 


• 444 


.336816 ' 


.013 


.001969 


• 085 


.032186 


•157 


078921 


.229 


135624 


.301 


.19908s 


.373 


.267078 


• 445 


.337810 


.014 


.002199 


.086 


•032746 


• 158 


079650 


• 230 


136465 


.302 


. 200003 


.374 


. 268046 


• 446 


.338804 


.015 


.002438 


.087 


.033308 


• 159 


080380 


• 231 


137307 


.303 


.200922 


.375 


.269014 


• 447 


.339799 


.016 


.002685 


.088 


•033873 


.160 


081112 


• 232 


1381S1 


• 304 


.201841 


.376 


.269982 


• 448 


.340793 


.017 


.002940 


.089 


. 034441 


.161 


081847 


•233 


138996 


• 305 


.202762 


.377 


•270951 


.449 


.341788 


.018 


.003202 


.090 


.035012 


.162 


082582 


•234 


139842 


.306 


.203683 


.378 


•271921 


.450 


•342783 


.019 


.003472 


.091 


.03SS86 


.163 


083320 


•235 


140689 


• 307 


.204605 


.379 


•272891 


.451 


.343778 


.020 


.003749 


.092 


.036162 


• 164 


084060 


•236 


141538 


• 308 


.205528 


.380 


.273861 


• 452 


.344773 


.021 


.004032 


• 093 


•036742 


• 165 


084801 


•237 


142388 


• 309 


.206452 


.381 


.274832 


.453 


.345768 


.022 


.004322 


^094 


•037324 


• 166 


085545 


• 238 


143239 


.310 


•207376 


.382 


.275804 


.454 


.346764 


.023 


.004619 


• 09S 


.037909 


• 167 


086290 


•239 


144091 


•311 


•208302 


.383 


.276776 


.455 


.347760 


.024 


.004922 


• 096 


.038497 


• 168 


087037 


• 240 


144945 


.312 


•209228 


.384 


•277748 


.456 


.348756 


.025 


.005231 


.097 


.039087 


.169 


08778s 


• 241 


145800 


• 313 


•2101SS 


.385 


.278721 


• 457 


.349752 


.026 


.005546 


.098 


.039681 


.170 


088536 


.242 


146656 


• 314 


.211083 


.386 


.279695 


• 458 


.350749 


.027 


.005867 


.099 


.040277 


• 171 


089288 


•243 


147513 


• 31s 


.212011 


.387 


.280669 


• 459 


.35174s 


.028 


.006194 


. 100 


.040875 


• 172 


090042 


•244 


148371 


• 316 


.212941 


.388 


.281643 


.460 


•352742 


.029 


.006527 


. lOI 


.041477 


•173 


090797 


■24s 


149231 


• 317 


.213871. 


.389 


.282618 


.461 


.353739 


.030 


.006866 


• 102 


.042081 


• 174 


091555 


.246 


150091 


.318 


.214802 


.390 


.283593 


.462 


•354736 


.031 


.007209 


.103 


.042687 


• 175 


092314 


•247 


150953 


• 319 


.215734 


.391 


.284569 


.463 


.355733 


.032 


•007559 


. 104 


.043296 


.176 


093074 


• 248 


151816 


.320 


.216666 


.392 


.285545 


.464 


•356730 


.033 


.007913 


• 105 


.043908 


• 177 


093837 


.249 


152681 


• 321 


•217600 


.393 


.286521 


.465 


•3S7728 


■ 034 


.008273 


. 106 


•044523 


.178 


094601 


• 250 


153546 


• 322 


.218534 


.394 


■287499 


.466 


•358725 


.035 


.008638 


• 107 


.045140 


.179 


095367 


.251 


154413 


• 323 


•219469 


•39S 


.288476 


• 467 


.359723 


.036 


. 009008 


.108 


•0457S9 


.180 


096135 


.252 


155281 


• 324 


•220404 


•396 


.289454 


.468 


.360721 


.037 


.009383 


. 109 


.046381 


.181 


096904 


•253 


156149 


• 325 


•221341 


• 397 


.290432 


.469 


.361719 


.038 


.009764 


. no 


.047006 


.182 


097675 


• 254 


157019 


• 326 


.222278 


.398 


.291411 


.470 


.362717 


.039 


.010148 


. Ill 


•047633 


• 183 


098447 


• 255 


157891 


■ 327 


•223216 


•399 


.292390 


.471 


.363715 


.040 


.010538 


.112 


.048262 


• 184 


099221 


.256 


158763 


• 328 


•224154 


• 400 


.293370 


.472 


.364714 


.041 


.010932 


•113 


.048894 


• 185 


099997 


• 257 


159636 


• 329 


•225094 


.401 


.294350 


.473 


.365712 


.042 


.011331 


• 114 


.049329 


.186 


100774 


• 258 


160S11 


• 330 


•226034 


.402 


•295330 


.474 


.366711 


.043 


•OII734 


■IIS 


.050165 


• 187 


10ISS3 


• 259 


161386 


• 331 


•226974 


•403 


.296311 


.475 


.367710 


.044 


.012142 


.116 


•050805 


.188 


102334 


• 260 


162263 


• 332 


.227916 


•404 


.297292 


.476 


.368708 


.04s 


.OI255S 


.117 


.051446 


.189 


.103116 


• 261 


163141 


•333 


.228858 


•40s 


.298274 


.477 


.369707 


.046 


.012971 


.118 


.052090 


. 190 


. 103900 


• 262 


164020 


•334 


.229801 


• 406 


.299256 


.478 


.370706 


.047 


•013393 


.119 


•052737 


.191 


. 104686 


• 263 


164900 


•335 


•23074s 


•407 


.300238 


.479 


.371705 


.048 


.013818 


. 120 


•053385 


. 192 


■105472 


.264 


165781 


• 336 


.231689 


• 408 


.301221 


.480 


.372704 


.049 


.014248 


. 121 


•054037 


• 193 


.106261 


.26s 


166663 


•337 


.232634 


.409 


.302204 


.481 


.373704 


.050 


.014681 


. 122 


•054690 


• 194 


.107051 


.266 


167546 


• 338 


.233580 


.410 


.303187 


.482 


.374703 


.osi 


.015119 


■ 123 


•055346 


• 195 


■107843 


.267 


168431 


• 339 


.234526 


.411 


.304171 


.483 


.375702 


.052 


.015561 


.124 


.056004 


. 196 


.108636 


.268 


.169316 


■340 


•235473 


.412 


.305156 


• 484 


.376702 


.053 


.016008 


.125 


.056664 


.197 


■ 109431 


• 269 


. 170202 


.341 


•236421 


•413 


.306140 


• 485 


.377701 


• 054 


.016458 


.126 


•057327 


.198 


.110227 


.270 


. 17 1090 


• 342 


•237369 


■ ^414 


.307125 


.486 


.378701 


• OSS 


.016912 


.127 


•0S799I 


.199 


.III02S 


.271 


•171978 


•343 


•238319 


•415 


.308110 


.487 


•379701 


.056 


.017369 


.128 


.058658 


.200 


.111824 


.272 


•172868 


•344 


.239268 


• 416 


.309096 


.488 


.380700 


.0S7 


.017831 


. 129 


.059328 


.201 


.112625 


.273 


•173758 


•345 


.240219 


• 417 


.310082 


■ 489 


.381700 


.058 


.018297 


.130 


• 059999 


.202 


.113427 


• 274 


•174650 


•346 


.241170 


• 418 


.311068 


.490 


.382700 


• 0S9 


.018766 


• 131 


• 060673 


• 203 


.114231 


• 275 


■I75S42 


• 347 


.242122 


•419 


.312055 


.491 


.383700 


.060 


.019239 


.132 


.061349 


.204 


.115036 


.276 


.176436 


• 348 


.243074 


.420 


.313042 


• 492 


.384699 


.061 


.019716 


• 133 


.062027 


.20s 


. 115842 


• 277 


•177330 


•349 


.244027 


• 421 


.314029 


• 493 


.385699 


.062 


.020197 


• 134 


.062707 


.206 


. II665I 


• 278 


•178226 


• 350 


.244980 


• 422 


.315017 


• 494 


.386699 


.063 


.020681 


• 13s 


.063389 


.207 


• II 7460 


.279 


•179122 


• 351 


.245935 


• 423 


.316005 


• 495 


.387699 


.064 


.021168 


.136 


•064074 


• 208 


.118271 


.280 


• 180020 


• 352 


.246890 


.424 


.316993 


.496 


.388699 


.06s 


.021660 


•137 


• 064761 


• 209 


. I 19084 


• 281 


. 180918 


•353 


.24784s 


■ 425 


.317981 


.497 


•389699 


,066 


.022155 


.138 


. 065449 


. 210 


.119898 


• 282 


•181818 


• 354 


.248801 


.426 


.318970 


.498 


• 390699 


.067 


.022653 


• 139 


• 066140 


.211 


.120713 


.283 


•182718 


• 355 


■249758 


.427 


.319959 


.499 


•391699 


.068 


•02315s 


• 140 


.066833 


.212 


•I21530 


.284 


.183619 


• 356 


.250715 


.428 


.320949 


.500 


.392699 


.069 


.023660 


.141 


.067528 


• 213 


.122348 


.28s 


. 184522 


• 357 


.251673 


.429 


.321938 






.070 


.024168 


.142 


.068225 


.214 


. I23167 


.286 


•185423 


.358 


.252632 


.430 


.322928 






.071 


.024680 


.143 


.068924 


.215 


.123988 


.287 


. 186329 


• 359 


.253591 


.431 


•323919 






.072 


.025196 


.144 


.069626 


.216 


. I2481I 


.288 


•18723s 


.360 


.254551 


.432 


.324909 







MATHEMATICAL TABLES 



533 



Table 17. — Sides and Diagonals of Squares 
Diagonal = Side X 1-4142 



Side 1 


Diagonal | 


Side 


Diagonal 


Side 


Diagonal 


Side 


L iagonal 


^ 


0.044 


iH 


2.475 


3W 


4-949 


SH 


7-425 


} 


0.088 


I'Me 


2.563 


3916 


5 -038 


SMe 


7.513 


H 


0.177 


Jli 


2 .652 


3H 


5- 126 


5% 


7 .601 


r>6 


0.26s 


I'Me 


2.740 


31 Me 


S-215 


SMe 


7.689 


u 


0.354 


2 


2.828 


3% 


S-303 


sH 


7-778 


Me 


0.442 


2M6 


2.917 


3' Me 


5.392 


sMe 


7-866 


H 


0.530 


2M 


3005 


3% 


5.480 


55^ 


7-955 


^16 


0.619 


2^6 


3 094 


3' Me 


5.568 


SH-ie 


8.043 


yi 


0.707 


2H 


3.182 


4 


5.657 


5M 


8.132 


Me 


0.796 


2^6 


3-270 


4Me 


S.74S 


5IM6 


8.220 


H 


0.884 


234 


3.359 


4M 


5-834 


5% 


8.308 


ma 


0.972 


2M6 


3.447 


4Me 


5-922 


S'Jie 


8.396 


H 


1. 061 


2H 


3.53s 


4H 


6.010 


6 


8.48s 


l?i6 


1 .149 


2M6 


3.624 


4Me 


6 .099 


6M6 


8.563 


li 


1.237 


254 


3-712 


AH 


6.187 


6H 


8.662 


'Me 


1 .326 


2m 6 


3.8or 


4M6 


6.275 


6Me 


8.750 


I 


1. 414 


2?4 


3.889 


4H 


6.364 


6H 


8.838 


iMo 


1 .502 


2 13/1 6 


3.977 


4M6 


6.452 


6M6 


8.928 


iH 


1.590 


2^^ 


4.066 


4^4 


6.541 


6H 


9. 015 


iMe 


1.679 


2' Me 


4-154 


4iMe 


6.629 


6M6 


9-103 


iH 


1.768 


3 


4-243 


4^4 


6-718 


61/4 


9.192 


1M6 


1.8S6 


3M0 


4-331 


4IM0 


6.806 


6M6 


9.280 


i?^ 


1.94s 


3'A 


4.419 


4% 


6.894 


6H 


9.369 


1^6 


2.033 


3M6 


4 508 


4iMe 


6.983 


61H6 


9-457 


iH 


2. 121 


3W 


4 - 596 


S 


7-071 


6?4 


9. 535 


me 


2 .210 


3Me 


4-685 


sMe 


7-158 


6iMe 


9-634 


lii 


2 .298 


3'A 


4-773 


SH 


7.248 


eii 


9-722 


iiHe 


2.386 


3Me 


4.861 


SMe 


7-336 


61M6 


9. 811 



Squares and Square Roots of Numbers Other than Those Given 
in the Tables 

Squares and square roots of larger or smaller, whether whole, deci- 
mal or mixed, numbers not given in Table 26 may be found by- 
proper adjustment of the decimal point. 

To find the square of stcch a number: Move the point to right or left 
to give a number of the table (preferably of three figures). Take out 
the square of this number and move its point back again twice as 
many places as it was first moved. If the original number contains 
more than three significant figures those in excess are to be ignored 
unless greater accuracy is required, in which case, interpolate. The 
result of such interpolation is not exact, but sufiiciently so in case 

of decimals. 

(Continued on next page first column) 



Table 19. — Circles and Squares of Equal Area 
From Kent's Mechanical Engineer's Pocket Book 

This table may be greatly extended in range by shifting the decimal point. 
Thus the side of a square equal to a circle of 5 ins. diam. is 4.431 ins. For 
a circle of .5 in. diam., this becomes .4431 ins. So, also, the reading 10.635 
for a circle of 12 ins. diam. becomes 1.0635 for a circle of 1.2 ins. diam. 

Diameter of circle = 1. 128379 X side of square of same area. 

Side of square = 0.886227 X diameter of circle of same area. 



Diam. 


Side of 


Diam. 


Diam. 


Side of 


Diam. 


Diam. 


of circle 


square 


of circle 


of circle 


square 


of circle 


of circle 


or side 


equiva- 


equiva- 


or side 


equiva- 


equiva- 


or side 


of 


lent to 


lent to 


of 


lent to 


lent to 


of 


square 


circle 


square 


square 


circle 


square 


square 


I 


0.886 


1.128 


34 


30.132 


38.36s 


67 


2 


1.772 


2-257 


35 


31 .018 


39-493 


68 


3 


2.659 


3-385 


36 


31 904 


40.622 


69 


4 


3-545 


4.514 


37 


32.790 


41-750 


70 


S 


4-431 


5.642 


38 


33-677 


42.878 


71 


6 


S-317 


6.770 


39 


34-563 


44-007 


72 


7 


6.204 


7.899 


40 


35-449 


45-135 


73 


8 


7.090 


9.027 


41 


36-335 


46 . 264 


74 


9 


7.976 


10.15s 


42 


37-222 


47-392 


75 


10 


8.862 


11 .284 


43 


38.108 


48.520 


76 


II 


9.748 


12 .412 


44 


38-994 


49 . 649 


77 


12 


10.63s 


13.541 


45 


39.880 


50.777 


78 


13 


11 .521 


14.669 


46 


40.766 


51 -905 


79 


14 


12.407 


15.797 


47 


41-653 


53 034 


80 


IS 


13-293 


16.926 


48 


42-539 


54.162 


81 


16 


14.180 


18.054 


49 


43-425 


55.291 


82 


17 


15 .066 


19. 182 


50 


44-311 


56.419 


83 


18 


15.952 


20.311 


51 


45 198 


57-547 


84 


19 


16.838 


21.439 


52 


46.084 


58.676 


85 


20 


17.725 


22.568 


53 


46.970 


59.804 


86 


21 


18. 611 


23.696 


54 


47.856 


60.932 


87 


22 


19.497 


24.824 


55 


48.742 


62 .061 


88 


23 


20.383 


25-953 


56 


49.629 


63.189 


89 


24 


21 .269 


27 .081 


57 


50.515 


64.318 


90 


25 


22.156 


28.209 


58 


51.401 


65-446 


91 


26 


23.042 


29.338 


59 


52-287 


66.574 


92 


27 


23.928 


30 .466 


60 


53-174 


67.703 


93 


28 


24.814 


31-595 


61 


54 ■ 060 


68.831 


94 


29 


25.701 


32.723 


62 


54 - 946 


69.959 


95 


30 


26.587 


33-851 


63 


55 -832 


71.088 


96 


31 


27-473 


34 980 


64 


56.719 


72 .216 


97 


32 


28.359 


36.108 


65 


57.60s 


73-345 


98 


33 


29.245 


37.237 


66 


58.491 


74-473 


99 



Side of 
square 
equiva- 
lent to 
circle 



Diam. 
of circle 
equiva- 
lent to 
square 



59 


377 


60 


263 


61 


ISO 


62 


036 


62 


922 


63 


808 


64.695 1 


65 


S8i 


66 


467 


67 


353 


68 


239 


69 


126 


70 


012 


70 


898 


71 


784 


72 


671 


73 


557 


74 


443 


75 


330 


76 


216 


77 


102 


77 


988 


78 


874 


79 


760 


80 


647 


81 


533 


82 


419 


83 


30s 


84 


192 


85 


078 


85 


964 


86 


850 


87 


736 



75 .601 
76.730 
77.858 
78.987 
80.115 

81.243 
82.372 
83.500 
84.628 
85.757 

86.885 
88.014 
89. 142 
90.270 
91-399 

92.527 
93.655 
94 784 
95912 
97 -041 

98. 169 
99.297 
100 .426 
101.554 
102 .682 

103 .811 
104.939 
106 .068 
107 . 196 
108.324 

109.453 

110. 581 

III .710 









Table i 


8. — Square Roots, and 


Cube Roots of Binary Fractions 








Fraction 


Sq. root 


Cube root 


Fraction 


Sq. root 


Cube root 


Fraction 


Sq. root 


Cube root 


Fraction 


Sq. root 


Cube root 


Hi 


- 1250 


.2500 


1^4 


•5154 


.6428 


3%4 


.7181 


.8019 


^%4 


•8750 


.9148 


H2 


.1768 


-315° 


%2 


-53°3 


•6552 


1^2 


.7289 


-8099 


2%2 


• 8839 


.9210 


Hi 


.2165 


.3606 


1%4 


-5449 


.6671 


^H4 


•7395 


.8178 


^^4 


.8927 


.9271 


He 


- 2500 


.3968 


He 


-5590 


.6786 


He 


.7500 


•8255 


^He 


.9014 


•9331 


%4 


-279s 


-427s 


^H4 


-5728 


.6897 


3%4 


.7603 


•8331 


^Hi 


.9100 


•9391 


%2 


.3062 


■4543 


IH2 


-5863 


.7005 


1%2 


.7706 


•840s 


"/i2 


.9186 


■9449 


K4 


-3307 


.4782 


2%4 


•5995 


.7110 


3%4 


.7806 


■8478 


^Hei 


.9270 


■9507 


Vs 


-3S3S 


.5000 


H 


.6124 


.7211 


H 


.7906 


■8550 


% 


•9354 


•9565 


%A 


-3750 


.5200 


2^4 


.6250 


.7310 


^H4 


.8004 


.8621 


'%4 


•9437 


.9621 


y^i 


•3953 ■ 


.5386 


1%2 


•6374 


.7406 


^yz2 


.8101 


.8690 


^y32 


.9520 


.9677 


^Hi 


.4161 


•5560 


2^4 


•649s 


• 7500 


*%4 


.8197 


•8758 


^%4 


.9601 


•9732 


He 


-4330 


-5724 


He 


.6614 


•7592 


^Ke 


.8292 


.8826 


151 6 


.9682 


.9787 


1^4 


-4507 


-5878 


2%4 


.6732 


.7681 


*%4 


•8385 


.8892 


6J^4 


•9763 


.9841 


■ ys2 


-4677 


.6025 


1%2 


.6847 


•7768 . 


2%2 


.8478 


.8958 


^^2 


■9843 


■9895 


m4 


.4841 


.6166 


33^4 


.6960 


•7853 


^Ki 


-8569 


.9022 


6%4 


.9922 


■9948 


H 


.5000 


.6300 


Vz 


.7071 


•7937 


Va 


.8660 


.9086 


I 


I 


I 



534 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 20. — ^Lengths of Chords por the Division of Circles, By A. Best {Amer. Mach., Oct. g, 1913) 
For Larger Circles the Chords are in Direct Proportion 
Number of Centers on Circle 



1 


3 


4 


5 


6 


8 


10 


12 


14 


16 


18 




iK 


1.2990 


1.0606 


.8817 


•7500 


-5740 


■4635 


.3882 


-3336 


.2927 


.2604 




iVs 


1-4073 


I. 1400 


•9552 


■ 8125 


.6218 


■5021 


.4205 


■3614 


.3172 


.2821 




iM 


1-5155 


1-2374 


1.0287 


.8750 


.6696 


• 5407 


-4529 


-3892 


•3417 


.3028 




1% 


1.6237 


1-3258 


I. 1029 


•9375 


-7174 


•5793 


• 4852 


.4170 


.3668 


-3245 




2 


1.7320 


I. 4142 


I. 1756 


I. 0000 


.7654 


.6180 


-5176 


.4448 


.3902 


-3473 




2^ 


I . 9486 


I. 5910 


1-3225 


I. 1250 


.8610 


•6952 


-5823 


.5004 


•4390 


• 3907 




2^ 


2.1652 


1.7678 


1.4794 


1.2500 


.9566 


•7725 


.6470 


•5560 


.4878 


.4341 




2^ 


2.3817 


1.9446 


I. 6163 


I-37SO 


1.0522 


.8497 


.7117 


.6116 


•5366 


.4775 




3 


2.5981 


2.1213 


1.7633 


I . 5000 


I . 1480 


.9270 


.7764 


.6672 


.5853 


.5209 




sH 


2.8146 


2.2981 


I. 9102 


1.6250 


1.2436 


I .0042 


.8411 


.7228 


.6341 


■5643 




3K 


3-0311 


2.4749 


2.0571 


I . 7500 


1-3392 


, I. 0815 


•9058 


.7884 


.6829 


.6077 




3M 


3.2476 


2.6517 


2 . 2040 


1.8750 


1.4348 


I. 1587 


•9705 


.8440 


•7317 


.6511 




4 


3-4641 


2.8284 


2-3SII 


2.0000 


1-5307 


I . 2360 


1.0352 


.8896 


.7804 


.6945 




4J^ 


3 , 6806 


3.0052 


2 . 4980 


2.1250 


1.6263 


1-3132 


1.0999 


• 9452 


.8292 


.7379 




4^ 


3-8971 


3.1820 


2 . 6449 


2.2500 


I. 7219 


1-3905 


I . 1646 


I . 0008 


.8780 


• 7813 




4M 


4.1136 


3-3588 


2.7918 


2-3750 


I. 8175 


1.4677 


1.2293 


1.0564 


.9268 


.8247 




5 


4-3301 


3-5355 


2-9389 


2 . 5000 


1-9134 


1-5450 


I . 2440 


I .1120 


•9754 


.8682 




5H 


4-5466 


3-7123 


3-0858 


2.6250 


2.0090 


1.6222 


1.3087 


I. 1676 


1.0242 


.9116 


C 


sK 


4-7631 


3 - 8904 


3-2327 


2.7500 


2. 1046 


1.6995 


1-3734 


1.2232 


1.0730 


-9550 


1— 1 


sM 


4-9796 


4.0672 


3-3796 


2.8750 


2.2002 


1.7767 


I. 4381 


1.2788 


1.1218 


.9984 





6 


5.1966 


4.2426 


3-5267 


3 . 0000 


2 . 2961 


1.8540 


1-5528 


1-3344 


I. 1705 


I. 0418 


Q 


6M 


5.4126 


4.4194 


3.6736 


3-1250 


2.3917 


1. 9312 


I. 6175 


1.3900 


I. 2193 


I .0852 


IH 


6H 


S-6291 


4.5962 


3-8205 


3.2500 


2-4873 


2.0085 


1.6822 


1-4456 


I. 2681 


I. 1286 




6M 


5-8456 


4-7730 


3-9674 


3-3750 


2-5829 


2.0857 


I . 7469 


I. 5012 


1-3169 


I. 1720 


Diam 


7 


6.0622 


4-9497 


4.1147 


3 - 5000 


2.6788 


2 . 1630 


1.8116 


1-5568 


1.2756 


I-2I55 


7M 


6.2787 


5-1265 


4.2616 


3.6250 


2.7744 


2 . 2402 


1-8763 


I. 6124 


1.3244 


I - 2584 




7M 


6.4952 


5-3033 


4-4085 


3-7500 


2.8700 


2-3175 


I .9410 


1.6680 


1.3732 


1.3023 




7H 


6.7117 


5-4801 


4-5554 


3-8750 


2,9656 


2-3947 


2.0057 


1.7236 


1.4220 


1-3457 




8 


6.9282 


5-6568 


4.6022 


4 . 0000 


3.0614 


2.4720 


2.0704 


1.7792 


1.5607 


I. 3891 




8^ 


7-1447 


5-8334 


4.7491 


4.1250 


3-1570 


2 - 5492 


2-1351 


1-8348 


I . 609s 


1-4325 




8M 


7.3612 


6.0102 


4.8960 


4.2500 


3-2527 


2 .6265 


2. 1998 


I . 8904 


1.6583 


1-4759 




8M 


7-5777 


6.1870 


4.0429 


4-3750 


3-3483 


2.7037 


2.2645 


I .9460 


I. 7071 


1-S193 




9 


7.7942 


6.3639 


5 - 2900 


4.5000 


3-4440 


2.4810 


2.3292 


2.0016 


1.7558 


1.5627 




qM 


8.0107 


6.5407 


5-4369 


4.6250 


3-5396 


2.5582 


2-3939 


2.0572 


I . 8046 


I .6061 




9^ 


8.2272 


6.7175 


5 -5838 


4.7500 


3-6353 


2-6355 


2.4586 


2.1128 


I - 8534 


1-6495 




9% 


8.4437 


6 - 8943 


5-7307 


4-8750 


3-7309 


2.7127 


2-5233 


2.1684 


1.9022 


1.6829 




10 


8.6603 


7.0710 


5-8778 


5.0000 


3.8268 


3.0901 


2 . 5880 


2. 2240 


1.9509 


1-7364 




loK 


8.8768 


7-2478 


6.0247 


5-1250 


3.9224 


3-1673 


2.6527 


2.2796 


1.9997 


1.7798 




loH 


9-0933 


7.4246 


6.-1716 


5-2500 


4.0181 


3 - 2446 


2.7174 


2-3352 


2.0485 


I .8232 




loM 


9.3098 


7.6014 


6-3185 


5-3750 


4-II37 


3-3218 


2.7821 


2.3908 


2.9730 


1.8666 




II 


9-5263 


7-7781 


6.3656 


5 - 5000 


4.2095 


3 3990 


2 . 8468 


2.4464 


2 . 1460 


1. 9100 




II>^ 


9-9593 


8.1311 


6.6594 


5-7500 


4.4008 


3-5536 


2.9762 


2-5576 


2.2436 


I . 9968 




12 


10.3923 


8.4852 


7.0534 


6 . 0000 


4-5921 


3.7081 


3-1056 


2.6688 


2.3412 


2 . 0836 



Examples: To find the square of 2760. Move the point owe place 
to the left giving 276., the square of which, by the table, is 76,176. 
Move its point two places to the right and we have 7,617,600., the 
square of 2760. 

To find the square of .276. Move the point three places to the 
right giving 276., the square of which is 76,176. Move its point six 
places to the left giving .076176 the square of .276. 

To find the square of 2.76. Move the point two places to the 



right, giving 276., the square of which is 76,176, Move its point 
four places to the left, giving 7.6176, the square of 2.76 

To find the square root of such a number: Move the decimal point 
to right or left an even number of times to give a number of the table. 
Take out the square root of this number and move its point back 
again one-half as many places as it was first moved. If the point 
when first moved gives two figures in the whole number part, inter- 
polation is necessary for more than ordinary accuracy. 



MATHEMATICAL TABLES 



535 



I 



Table 21. — Decimal Equivalenxs or Binary Fractions 

.015625 

• 03125 
.046875 
.0625 
.078125 

•0937s 

.109375 

.125 

.140625 

•15625 

.171875 

.1875 

.203125 

.21875 

•23437s 

.250 

.265625 

.28125 

.296875 

•3125 
.328125 

■3437S 

■3S9375 

■ 375 

•390625 

.40625 

.421875 

• 4375 
.453125 
.46875 
.484375 
.500 
•515625 
•S3125 
•546875 
•5625 
•578125 
■59375 
•609375 
.625 
.640625 
.65625 
.671875 
.6875 
.703125 
■71875 
■734375 
■750 
.765625 
.78125 
.796875 
.8125 
.828125 
.84375 
.859375 
.875 
.890625 
.90625 
.921875 

.9375 
.953125 
.96875 
•984375 



Examples: To find the square root of 5830. Move the point two 
(an even) number of places to the left giving 58.3. The square root 
of 58 is 7.6158. Moving the point one place to the right gives 76.158, 
the approximate square root of 5830. To interpolate add Koths of 
the difference between the square root of 58 and of 59, but note that, 
to the extent that the two roots agree (7.6) both are correct. 

To find the square root of .0583. Move the point four places to 
the right, giving 583., the square root of which is 24.1454. Moving 
its point iic'o places to the left gives .241454, the square root of .0583. 
No interpolation is necessary when the first shift of the point gives 
three places in the whole number part. 

To find the square root of 5.83. Move the point two places to the 





I^- 


1 


61 




32 


3 




61 




3 


5 




61 


1 


32 


7 
61 


8- 


A- 


A- 


9 




64 
11 




61 




7 


13 




61 




32 


1R 


1 


61 


i 


■> 


9 


1 7 




61 




32 


19 




61 




16 


11 


21 




61 


3 


32 


23 
61 


8 


7 


13 


25 




61 




32 


27 




61 




16 


1 S 


29 
61 




32 


31 

61 




9 


1 7 


33 




61 




32 


35J 

61 




16 


19 


37 




61 


5 


3 2 


39 
61 


8 


11 


21 


41 




61 




32 


43 




61 




16 


23 


a— 




32 


47 


3 


61 


i 


H- 


2 5 


49 




61 




32 


51 




64 




27 


S3 




61 




32 


S5 


7. 


61 


8 


15 


29 


57 
64 




32 


M— 




16 


31 


61 
61 



Table 22. — Decimal Equivalents of Other than Binary 
Fractions 



Fr. 


Deci- 
mal. 


Near 
est 
64th 


■p^ Deci- 
^"^ mal. 


Near- 
est 
64th 


■c,^ Deci- 


Near- 
est 
64th 


Fr. 


Deci- 
mal. 


Near- 
est 
64th 


A 


• 0313 
.0323 

■ 0333 

■ 0345 

■ 0357 
.0370 
.0385 
.0400 

■ 0417 

■ 043s 

■ 045s 
.0476 
.0500 
.0526 

■ 05S6 

■ 0589 
.0625 
.0645 
.0667 
.0690 
.0714 
.0740 
.0769 

0800 

■ 0833 
.0870 
.0909 


A 


ft ■ 


0938 
0932 
0968 
1000 
1034 
1053 
1071 
iiii 

IIS4 
1176 
1200 
1250 
1290 
1304 
1333 
1364 
1379 
1429 
1481 
1300 
1338 
1563 
1579 
1600 
1613 
1667 
1724 


ft 


11 

iu ■ 
A 

A ■ 


1739 
1765 
1785 
1818 
1852 
1873 
1903 
1923 
1933 
2000 
2069 
2083 
2105 
2143 
2174 
2188 
2222 
2258 
2273 
2308 
2333 
2333 
2381 
2400 
2414 
2500 
2580 




A 
A 

A 

7 
2 6 

A 
A 
A 
A 
ft 


.2393 
.2609 
.2632 
.2667 
.2692 
.2727 
.2739 
.2778 
.2800 
.2813 
.2857 

■ 2903 
.2917 

■ 2941 
.2963 
.3000 
.3043 

■ 3077 
.3103 
.3125 
.3138 
.3182 
.3200 
.3214 
.3226 

■ 3333 

■ 3438 






A ■ 

A . 
A . 
A . 
A ■ 

S 

A ■ 
A . 

i . 




A 


A 


H 




A 




■fV 


21 

A . 
A 
5 
A 

A . 
A 
A 

A ■ 
ft . 






A 




^h 


a 


ft 


^ 


i 


31 

A 
A 

A 
A 
A 
A 

_5_ 




A 




i 


A 


ft 




iV 


A 


a 


iV 


A 




rhi 


§_ 

A 

A 

A 

A 

A 

A ■ 

A 

A 

i 




^r 




A 


a 




A 


A 


A 


rh 


A 


A 
A 

il 

i 




A 






A 

"fa 


a 


A 




M 


i 




rV 


A 


a 










n 


.3448 

■ 3462 

■ 3478 

■ 3500 
.3529 
■3348 
■3571 
.3600 
■3636 
.3667 
.3684 
■3704 
■3750 
.3793 
.3810 
.3846 

■ 3871 

■ 3889 
.3913 
• 3929 
.4000 
.4063 
.4074 
.4091 
.4118 
.4138 
.4167 




I' 

IB 
I 

7. 


4194 
42 1 1 
4231 
4286 

4333 
4348 
4375 
4400 
4444 
4483 
4500 
4316 
4343 
4383 
4613 
4642 
4667 
4688 
4706 
4737 
4762 
4783 
4800 
4813 
4828 

4839 
5000 


H 


Vi ■ 
¥i ■ 
ij ■ 
U ■ 

a ■ 
\l ■ 

ig 

A . 

a ■ 


3161 
3172 
3185 
5200 
3217 
3238 
3263 
3294 
5313 

3333 
3337 
3383 
3417 
3435 
5484 
5500 
5517 
3356 
3600 
562s 
5632 
3667 
5714 
3769 
3789 
5806 
3833 


if 


U 
1? 

U 

a 

u 

a 
3? 

13 
J? 

ii 

i 


.3862 
.5882 
.5909 
.5926 

■ 3938 
.6000 
.6071 
.6087 
.6111 
.6129 
.6154 
.6190 
.6207 
.6250 
.6296 
.6316 
■6333 
.6364 
.6400 
.6429 
.6452 
• 6471 
.6500 
.6522 
■6538 

■ 6532 
•6563 








A 


a 


A 


M 


« 


a 




A 




f. 


a 

i 

a 

A 

a 
u 

A 

u 

A 


11 


^h 


H 


T5 

}? ■ 
A ■ 
A 
A 

il ■ 
U ■ 

is ■ 




t\ 


i 


, 


To 

1 


M 


f 


A- 


H 


a 
a 
a 

A 

■2 z 

a 




Vh 


fi 




^? 




1^ 


■la 






!■? 

A 

li 

u 
Vi 

n 


a 




a 


Si 


¥i 
33 

ii 
li 
ii 

A 




il 




H 






A 


h 


a 












if 


.6667 

.6774 
.6786 
.6800 
.6818 
.6S42 
.6873 
.6897 
.6923 
.6937 
.7000 

■ 7037 

■ 7059 
.7083 

■ 7097 

■ 7143 
.7188 
.7200 
.7222 
.7241 

■ 7273 

■ 7308 
■7333 
■7368 

■ 7391 
■7407 
•7419 


a 


a 
n 
■;? 
1? 

A 


.7500 
7586 
7600 
.7619 
.7647 
.7667 
.7692 
.7727 

■ 7742 
.7778 
■7813 
.7826 
.7857 
.7895 
.7917 

■ 7931 
.8000 
.8065 
.8077 
.8095 
.8125 
.8148 
.8182 
.8214 
■8235 
.8261 
.8276 


3 


St 

Si 


8334 
8387 
8400 
8421 
8438 
8462 
8500 
8319 
8371 
8621 
8636 
8667 
8696 
8710 
8750 
8800 
8824 
8846 
8889 
8929 
8947 
8966 
9000 
9032 
9048 
9063 
9091 




u 
is 

1; 

T 

if 


• 9130 

■ 9167 
.9200 
■•9231 
■9259 
.9286 

■ 9310 

• 9333 
■9335 
■9375 

• 9412 
■9444 
.9474 
.9500 
■9524 
■9543 

■ 9565 

■ 9383 
.9600 
.9615 
.9630 
.9643 

9633 
.9667 
.9678 
.9688 

I .0000 




IS 


ii 


11 


a 


H 




1 
I 

IS 

tI 

3 






a 




H 




l"3 


M 


jf 


23 


il 
18 

^! 






5i 
IS 

n 
h 

I 


i 




If 




Si 




!! 

13 

fl 
15 
il 
A 

1 
1? 


H 


a 




H 


il 




■A 


H 




19. 


A 




Tl 


a 






A_a 


li 


ii 




y 





right giving 583., the square root of which is 24.1454. Moving its 
point one place to the left gives 2.41454 the square root of 5.83. 

The method is not satisfactory when applied to cubes and cube 
roots because of the large errors of interpolation. 



536 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 23. — Decimal Eqtjivalents of other than Binary 
Fractions 



Table 25. — Surfaces and Volumes of Spheres — (Continued) 

(Prom Trautwine's Civil Engineers Pocket-Book) 



Thirds, sixths, twelfths and 
twenty-fourths 

^ij 041666 

tV 083333 

2T 125 

1/6 166666 

2? 208333 

A 250 

^% 291666 

1/3 333333 

^\ 375 

■j\ 416666 

ii 458333 

3/6 500 

if 541666 

tV 583333 

ii 625 

2/3 666666 

il 708333 

15 750 

H 791666 

5/6 ^33333 

H 875 

i| 916666 

If 958333 

1 1 . 000 



Sevenths, fourteenths and 
twenty-eighths 

I's .035714 

iV 071429 

2^ , 107143 

1/7 142857 

■ij 178571 

t\ 214286 

■j'j 250 

2/7 285714 

/s 321429 

t\ 357143 

H 392857 

3/7 428571 

ii 464286 

T% 500 

a 535714 

4/7 571429 

il 607143 

r% 642857 

4! 678571 

5/7 714286 

li 7SO 

H 785714 

If 821429 

6/7 857143 

If 892857 

H 928571 

11 964286 

1 1 . 000 



Table 24. — Surfaces and Volumes of Spheres 

(Prom Trautwine's Civil Engineers Pocket-Book) 





1* ro 

rt *-' 

"C . 

3 □• 
CO M 


Volume, 
cu. ins. 


.2 c 


i s- 


a c 

3 ■" 


a „■ 
.2 c 
Q •" 


3 CT 


a.s 
3 . 


A 


.00077 




A 


4.2000 


.80939 


^ 


17.258 


6.7412 


A 


.00307 


.00002 


ft 


4.4301 


.87681 


I 


17.721 


7.0144 


A 


.00690 


.00005 


A 


4.6664 


•94786 


H 


18. 190 


7 . 2949 


A 


.01227 


.00013 


i 


4.9088 


1^0227 


ft 


18.666 


7.5829 


A 


.02761 


.00043 


A 


5. 1573 


I. 1013 


M 


19.147 


7.8783 


i 


.04909 


.00102 


ft 


5.4119 


I. 1839 


\ 


19.635 


8.1813 


A 


.07670 


.00200 


a 


5.6728 


1.2704 


M 


20. 129 


8.4919 


A 


.1104s 


.0034s 


f 


5^9396 


1.3611 


ft 


20.629 


8.8103 


A 


.15033 


.00548 


M 


6.2126 


I. 4561 


H 


21.135 


9. 1366 


i 


■1963s 


.00818 


ft 


6.4919 


I. 5553 


1 


21.648 


9.4708 


A 


.24851 


.01165 


a 


6.7771 


1.6590 


B 


22. 166 


9.8131 


ft 


.30680 


.01598 


i 


7.0686 


I. 7671 


H 


22.691 


10. 164 


a 


.37123 


.02127 


u 


7.3663 


1.8799 


M 


23.222 


10.522 


i 


■44179 


.02761 


ft 


7 ■ 6699 


1.9974 


i 


23.758 


10.889 


a 


.51848 


■03SII 


a 


7.9798 


2. 1196 


f* 


24.302 


11.265 


ft 


.60132 


■04385 


i 


8.2957 


2.2468 


ii 


24.850 


11.649 


H 


.69028 


.05393 


a 


8^6i8o 


2.3789 


H 


25.405 


12.041 


h 


•78540 


.06545 


a 


8.9461 


2.5161 


I 


25.967 


12.443 


u 


.88664 


.07850 


fi 


9.2805 


2.6586 


If 


26.535 


12.853 


la 


■99403 


•09319 


3. 


9.6211 


2.8062 


if 


27. 109 


13.272 


1} 


I. 1075 


. 10960 


*f 


9.9678 


2.9592 


^ 


27.688 


13.700 


1 


I. 2272 


.12783 


a 


10.321 


3.1177 


3- 


28.274 


14-137 


IV 


1-3530 


.14798 


a 


10.680 


3.2818 


ft 


29.465 


15-039 


H 


1.4849 


•I7014 


i 


I I . 044 


3.4514 


i 


30.680 


1S^979 


!} 


1.6230 


•19442 


E 


II .416 


3.6270 


ft 


31.919 


16.957 


2 


I. 7671 


.22089 


M 


11-793 


3.8083 


i 


33.183 


17.974 


Vs 


I. 9175 


• 24967 


a 


I2^I77 


3-9956 


ft 


34.472 


19.031 


u 


2.0739 


. 28084 


2. 


12.566 


4-1888 


1 


35.784 


20.129 


¥1 


2.236s 


■31451 


A 


12.962 


4-3882 


ft 


37-122 


21.268 


5 


2.4053 


•35077 


ft 


13.364 


4-5939 


h 


38-484 


22.449 


H 


2.5802 


.38971 


A 


13.772 


4.8060 


ft 


39-872 


23.674 


fl 


2.7611 


.43143 


i 


14.186 


S-0243 


f 


41.283 


24.942 


ii 


2.9483 


.47603 


A 


14.607 


5-2493 


ii. 


42.719 


26.254 


I. 


3.1416 


.52360 


ft 


15.033 


5 -4809 


a 


44.179 


27.611 


A 


3.3410 


.57424 


A 


15.466 


S-7190 


H 


45.664 


29.016 


ft 


3^5466 


.62804 


i 


15.904 


5.9641 


1 


47.173 


30.466 


A 


3.7583 


.68511 


■h 


16.349 


6.2161 


if 


48 . 708 


31.965 


i 


3.9761 


■74551 


ft 


16.800 


6.4751 


4- 


so. 265 


33-510 



a - 



i 
ft 



a.s 

"o 3 



a ^ 

2 c 



3 5* 



a.s 
3 . 



sl-848 


35-106 


53-456 


36.751 


55-089 


38.448 


56.745 


40.195 


58.427 


41.994 


60.133 


43 . 847 


61.863 


45.752 


63-617 


47.713 


65-397 


49.729 


67-201 


51.801 


69.030 


53.929 


70.883 


56.116 


72.759 


58.359 


74-663 


60.663 


76.589 


63.026 


78.540 


65.450 


80.516 


67.935 


82.516 


70.482 


84.541 


73.092 


86.591 


75.767 


88.664 


78.505 


90.763 


81.308 


92.887 


84.178 


95-033 


87.113 


97-205 


90.118 


99-401 


93.189 


101.62 


96.331 


103.87 


99-541 


106. 14 


102.82 


108.44 


106.18 


110.75 


109.60 


113- 10 


113-10 


117-87 


120.31 


122.72 


127.83 


127.68 


135.66 


132.73 


143-79 


137.89 


152-25 


143.14 


161.03 


148.49 


170.14 


153-94 


179-59 


159-49 


189-39 


165-13 


199-53 


170.87 


210.03 


176-71 


220.89 


182.66 


232.13 


188.69 


243.73 


194-83 


255.72 


201.06 


268.08 


207.39 


280.85 


213.82 


294.01 


220.36 


307.58 


226-98 


321.56 


233-71 


335-95 


240-53 


350-77 


247.45 


366.02 


254-47 


381.70 


261.59 


397.83 


268.81 


414.41 


276. 12 


431.44 


283.53 


448.92 


291.04 


466.87 


298.65 


485.31 


306.36 


504.21 


314-16 


523.60 


322.06 


543.48 


330.06 


563.86 


338.16 


584.74 


346.36 


606. 13 


354-66 


628.04 


363 -OS 


650.46 


371-54 


673-42 


380.13 


696.91 


388.83 


720.95 


397.61 


745.51 


406 . 49 


770.64 



16. 



18. 



19. 



415.48 
424-56 

433^73 
443^01 
452.39 
461.87 
471.44 
481. II 
490.87 
500.73 
510.71 
520.77 
530.93 
S4I.19 
551.55 
562.00 
572.55 
583.20 
593.95 
604.80 
615.75 
626.80 
637.95 
649.17 
660.52 
671.95 
683.49 
■695.13 
706.8s 
718.69 
730.63 
742.6s 
754.77 
767.00 
779.32 
791.73 
804.2s 
816.8s 
829.57 
842.40 
855.29 
868.31 
881.42 
894^63 
907^93 
921.33 
934.83 
948.43 

962. 12 

975-91 
989-80 
1003.8 
1017.9 
1032.1 
1046.4 
1060.8 
1075.2 
1089.8 
II04.S 
1119.3 
1134-I 
I149-I 

1 1 64 - 2 
1179-3 
1194-6 
1210.0 
1225.4 
I24I.O 
1256.7 
1272.4 
1288.3 
1304.2 
1320.3 
1336.4 
1352.7 



796.33 
822.58 
849.40 
876.79 
904.78 
933.34 
962.52 
992. 28 
1022.7 
1053.6 
1085.3 

1117-5 
1150.3 
1183.8 
1218.O 
1252.7 
1288.3 
1324.4 
1361.2 
1398.6 
1436.8 
1475-6 
1515-1 
1555-3 
1596-3 
1637-9 
1680.3 
1723.3 
1767.2 
1811.7 
1857-0 
1903-0 
1949.8 
1997^4 
2045^7 
2094^8 
2144-7 
2195-3 
2246-8 
2299. I 
2352. I 
2406.0 
2460.6 

2516. 1 
2572.4 
2629.6 
2687.6 
2746. 5 
2806.2 
2866.8 

2928 .2 
2990.5 
3053.6 
3117-7 
3182.6 
3248.5 
3315.3 
3382.9 

3451. 5 
3521.0 
3S9I-4 
3662.8 
3735-0 
3808.2 



3882. S 
3957-6 
4033-7 
4110.8 
4188.8 
4267.8 
4347-8 
4428.8 
4510-9 
4593-9 
4677-9 



(5 - 



i 

i 
i 
I 
i 
i 
24. 



2S. 



26. 



27. 



28. 



29. 



30. 



1369 -o 

1385.5 

1402 -O 

1418.6 
1435.4 
1452.2 

1469. 2 
1486.2 
1503.3 

1520. S 

1537.9 
1555.3 
1572.8 
1590.4 
1608.2 
1626.0 
1643-9 

1661 -9 
1680.0 
1698.2 
1716.5 
1735-0 
1753-5 
1772-I 
1790.8 
1809.6 
1828.5 
1847.5 
1866.6 
188s. 8 
1905.1 
1924.4 
1943.9 
1963.5 
1983.2 
2002 .9 
2022.9 
2042. 8 
2062.9 
2083.0 
2103.4 
2123.7 
2144.2 
2164.7 
2185.5 

2206. 2 
2227. I 
2248.0 

2269. 1 
2290.2 
2311.5 
2332.8 
2354-3 

237S-8 
2397 -5 

2419. 2 

2441. I 

2463 . 
2485.1 
2507-2 
2529-s 
2551-8 
2574-3 
2596.7 
2619.4 

2642. 1 
2665.0 
2687.8 

2710.9 

2734-0 
2757-3 
2780.5 
2804.0 
2827.4 



MATHEMATICAL TABLES 



537 





Table 26.— 


SQU.4RES, Cubes, Square Roots 


, Cube Roots, Reciprocals, Circumferences and Circular Areas 






Square 


Cube 


Sq. root 


Cu. 
root 


loooX 
recip. 


No. 


= Dia. 


No. 


Square 


Cube 


Sq. 
root 


Cu. 
root 


lOooX 
recip. 


No 


= Dia. 


No. 


Circum. 


Area 


Circum. 


Area. 


I 


I 


I 


I . 0000 


I . 0000 


1000.000 


3-142 


.7854 


70 


4900 


343,000 


8.3665 


4-1213 


14.2857 


219.91 


3848.45 


2 


4 


8 


1.4142 


1.2599 


500.000 


6.283 


3.1416 


71 


5041 


357.911 


8.4261 


4. 1408 


14-0845 


223.05 


3959-19 


3 


9 


27 


1.7321 


1.4422 


333-333 


9-524 


7.0686 


72 


5184 


373.248 


8.4853 


4. 1602 


13-8889 


226. 19 


4071.50 


4 


16 


64 


2.0000 


I-S874 


250.000 


12.566 


12.5664 


73 


5329 


389.017 


8 . 5440 


4-1793 


13-6986 


229.34 


4185-39 


5 


25 


125 


2.2361 


1.7100 


200.000 


15.708 


19-6350 


74 


5476 


40s. 224 


8.6023 


4-1983 


13-S135 


232.48 


4300.84 


6 


36 


216 


2.4495 


I-8171 


166.667 


18.850 


28.2743 


75 


562s 


421,875 


8.6603 


4.2172 


13.3333 


235.62 


4417-86 


7 


49 


343 


2.6458 


1. 9 1 29 


142.857 


21.991 


38.4845 


76 


5776 


438,976 


8.7178 


4-2358 


13-1579 


238.76 


4536-46 


8 


64 


512 


2.8284 


2.0000 


125. 000 


25.133 


50.2655 


77 


5929 


456,533 


8.7750 


4-2543 


12,9870 


241.90 


4656.63 


9 


81 


729 


3.0000 


2.0801 


III . Ill 


28.274 


63-6173 


78 


6084 


474,552 


8.8318 


4.2727 


12.8205 


24s . 04 


4778.36 


10 


100 


1,000 


3-1623 


2.1544 


100.000 


31-416 


78.5398 


79 


6241 


493,039 


8.8882 


4.2908 


12.6582 


248. 19 


4901,67 


11 


121 


1,331 


3-3166 


2.2240 


90.9091 


34-558 


95-0332 


80 


6400 


512,000 


8.9443 


4.3089 


12.5000 


251.33 


5026,55 


12 


144 


1,728 


3.4641 


2.2894 


83.3333 


37-699 


113-097 


81 


6561 


531.441 


9 . 0000 


4.3267 


12.3457 


254-47 


5153-00 


13 


169 


2,197 


3-6056 


2.3513 


76.9231 


40.841 


132.732 


82 


6724 


551.368 


9-0554 


4.3445 


12.1951 


257-61 


5281.02 


14 


196 


2,744 


3.7417 


2.4101 


71.4286 


43.982 


153.938 


83 


6889 


571.787 


9- I 104 


4.3621 


12.0482 


260.7s 


5410,61 


IS 


225 


3,375 


3.8730 


2.4662 


66-6667 


47.124 


176.71s 


84 


7056 


592,704 


9.1652 


4-3795 


II 9048 


263.89 


5541-77 


I6 


2S6 


4,096 


4-0000 


2.5198 


62.5000 


50.265 


201.062 


85 


7225 


614,125 


9.219s 


4.3968 


11-7647 


267 . 04 


5674-50 


J7 


289 


4.913 


4.1231 


2.5713 


58-8235 


53.407 


226.980 


86 


7396 


636,056 


9.2736 


4.4140 


II-6279 


270. 18 


5808.80 


I8 


324 


5,832 


4.2426 


2.6207 


55.5556 


56.549 


254-469 


87 


7569 


658,503 


9.3274 


4.4310 


11-4943 


273.32 


5944-68 


19 


361 


6,8S9 


4.3589 


2.6684 


52.6316 


59-690 


283. S29 


88 


7744 


681,472 


9.3808 


4.4480 


II -3636 


276.46 


6082. 12 


20 


400 


8,000 


4-4721 


2.7144 


50.0000 


62.832 


314-159 


89 


7921 


704,969 


9.4340 


4-4647 


II -2360 


279.60 


6221. 14 


21 


441 


9,261 


4-5826 


2.7589 


47.6190 


65.973 


346-361 


90 


8100 


729,000 


9.4868 


4.4814 


II. IIII 


282.74 


6361.73 


22 


484 


10,648 


4-6904 


2. 8020 


4S.4545 


69-115 


380.133 


91 


8281 


753,571 


9-5394 


4-4979 


10.9890 


285.88 


6503,88 


23 


529 


) 2,167 


4.7958 


2.8439 


43-4783 


72-257 


415-476 


92 


8464 


778,688 


9-5917 


4-5144 


10.8696 


289 . 03 


6647.61 


24 


576 


;3.824 


4.8990 


2.8845 


41.6667 


75-398 


452.389 


93 


8649 


804,357 


9-6437 


4-5307 


10.7527 


292.17 


6792.91 


25 


62s 


;5,625 


5 . 0000 


2.9240 


40.0000 


78.540 


490.874 


94 


8836 


830,584 


9-6954 


4-5468 


10.6383 


295.31 


6939.78 


26 


676 


17,576 


5 • 0990 


2.962s 


38.4615 


81.681 


530.929 


95 


9025 


857,375 


9-7468 


4.5629 


10.5263 


298.45 


7088.22 


27 


729 


19,683 


5-1962 


3 . 0000 


37-0370 


84.823 


572.555 


96 


9216 


884,736 


9-7980 


4-5789 


10.4167 


301.59 


7238.23 


28 


784 


21,952 


5-2915 


3 - 0366 


35-7143 


87-965 


615.752 


97 


9409 


912,673 


9-8489 


4-5947 


10.3093 


304.73 


7389.81 


29 


841 


24.389 


5.3852 


3-0723 


34-4828 


91. 106 


660.520 


98 


9604 


941.192 


9.899s 


4.6104 


10. 2041 


307.88 


7542.96 


30 


900 


27,000 


5.4772 


3-1072 


33-3333 


94-248 


706.858 


99 


9801 


970,299 


9.9499 


4.6261 


10. lOIO 


311.02 


7697.69 


31 


961 


29.791 


5.5678 


3-1414 


32.2581 


97.389 


754-768 


100 


10,000 


1,000,000 


10.0000 


4.6416 


10, 0000 


314-16 


7.853-98 


32 


1024 


32,768 


5-6569 


3.1748 


31.2500 


100. S3I 


804.248 


101 


10,201 


1,030,301 


10. 0499 


4-6570 


9-90099 


317-30 


8,011.85 


33 


1089 


35,937 


5-7446 


3.2075 


30.3030 


103-673 


855.299 


102 


10,404 


1,061,208 


10.0995 


4-6723 


9.80392 


320-44 


8,171.28 


34 


1156 


39,304 


5.8310 


3.2396 


29.4118 


106.814 


907.920 


103 


10,609 


1,092,727 


10. 1489 


4-6875 


9.70874 


323-58 


8,332.29 


35 


1225 


42,875 


5.9161 


3.2711 


28.5714 


109.956 


962. 113 


104 


10,816 


1,124,864 


10. it)8o 


4.7027 


9.61538 


326.73 


8,494-87 


36 


1296 


46,656 


6 . 0000 


3.3019 


27-7778 


113.097 


1017.88 


105 


11,025 


1,157.625 


10.2470 


4-7177 


9.52381 


329.87 


8,659.01 


37 


1369 


50,653 


6.0828 


3-3322 


27 -O270 


116.239 


1075.21 


106 


11,236 


1,191.016 


10. 2956 


4-7326 


9.43396 


333-01 


8,824.73 


38 


1444 


54.872 


6. 1644 


3.3620 


26.3158 


119.381 


1134-11 


107 


11,449 


1,225,043 


10.3441 


4-7475 


9-34579 


336-15 


8,992.02 


39 


IS2I 


59,319 


6.2450 


3-3912 


25.6410 


122.522 


1194-59 


108 


11,664 


1.259,712 


10.3923 


4.7622 


9-25926 


339-29 


9.160.88 


40 


1600 


64,000 


6.3246 


3.4200 


25.0000 


125.66 


1256-64 


109 


11,881 


1,295,029 


10.4403 


4.7769 


9-17431 


342-43 


9.331-32 


41 


I68I 


68,921 


6.4031 


3.4482 


24.3902 


128.81 


1320.25 


110 


12,10c 


1,331,000 


10.4881 


4-7914 


9.09091 


345-58 


9.303,32 


42 


1764 


74,088 


6.4807 


3-4760 


■23 . 8095 


131.95 


1385.44 


III 


12,321 


1,367.631 


10.5357 


4-8059 


9.00901 


348.72 


9,676.89 


43 


1849 


79.507 


6.5574 


3.5034 


23-2558 


135.09 


1452.20 


112 


12,544 


1,404,928 


10.5830 


4-8203 


8.92857 


351-86 


9,852.03 


44 


1936 


85,184 


6.6332 


3.5303 


22.7273 


138.23 


1520.53 


113 


12,769 


1,442,897 


10.6301 


4-8346 


8.84956 


355-00 


10,028.7 


45 


2025 


91.125 


6.7082 


3.5569 


22. 2222 


141-37 


1590.43 


114 


12,996 


1,481,544 


10.6771 


4.8488 


8.77193 


358.14 


10,207.0 


46 


2I16 


97.336 


6.7823 


3.5830 


21-7391 


144-51 


1661 .90 


IIS 


13,225 


1,520,875 


10.7238 


4.8629 


8.69565 


361.28 


10,386.9 


47 


2209 


103,823 


6.8557 


3.6088 


21.2766 


147-65 


1734-94 


116 


13.456 


1,560,896 


10.7703 


4-8770 


8.62069 


364-42 


10,568.3 


48 


2304 


110,592 


6.9282 


3-6342 


20.8333 


150.80 


1809-56 


117 


13,689 


1,601,613 


10.8167 


4-8910 


8.5+701 


367-57 


10,751.3 


49 


2401 


117.649 


7.0000 


3.6593 


20.4082 


153-94 


1885.74 


118 


13,924 


1,643,032 


10.8628 


4-9049 


8.47458 


370.71 


10,935-9 


SO 


2500 


125,000 


7.0711 


3-6840 


20.0000 


157.08 


1963-50 


119 


14,161 


1.685,159 


10.9087 


4.9187 


8.40336 


373.8s 


11,122-0 


SI 


2601 


132,651 


7.1414 


3.7084 


19.6078 


160.22 


2042.82 


120 


14,400 


1,728,000 


10.954s 


4-9324 


8.33333 


376-99 


11.309-7 


52 


2704 


140,608 


7.2111 


3-7325 


19.2308 


163.36 


2123.72 


121 


14,641 


1,771,561 


i I . 0000 


4-9461 


8.26446 


380.13 


11,499,0 


53 


2809 


148.877 


7.2801 


3-7563 


18.8679 


166.50 


2206. 18 


122 


14,884 


1,815,848 


11.0454 


4-9597 


8. 19672 


383.27 


11,689-9 


54 


2916 


157.464 


7.3485 


3-7798 


18.5185 


169.65 


2290. 22 


123 


15,129 


1,860.867 


11.0905 


4-9732 


8.13008 


386.42 


11,882-3 


55 


3025 


166,375 


7.4162 


3.8030 


18.1818 


172.79 


2375.83 


124 


15.376 


1,906,624 


11.1355 


4-9866 


8.06452 


389.56 


12,076.3 


56 


3136 


175,616 


7.4833 


3-8259 


17-8571 


175.93 


2463 -01 


125 


15.625 


1,953,125 


II. 1803 


5 . 0000 


8 . 00000 


392.70 


12,271.8 


57 


3249 


185,193 


7-5498 


3.8485 


17.5439 


179.07 


2551.76 


126 


■ 15.876 


2,000,376 


11.2250 


5-0133 


7.93651 


395.84 


12,469.0 


58 


3364 


195,112 


7-6158 


3-8709 


17-2414 


182. 21 


2642.08 


127 


16,129 


2,048,383 


11.2694 


5 -026s 


7.87402 


398.98 


12,667.7 


59 


3481 


205,379 


7.6811 


3.8930 


16.9492 


185.35 


2733.97 


128 


16,384 


2,097.152 


11-3137 


5 - 0397 


7.81250 


402. 12 


12,868.0 


60 


3600 


216,000 


7.7460 


3-9149 


16.6667 


188.50 


2827.43 


129 


16,641 


2,146,689 


11-3578 


5. 0528 


7.75194 


405.27 


13,069.8 


61 


3721 


226,981 


7.8102 


3.936s 


16.3934 


191-64 


2922.47 


130 


16,900 


2,197,000 


11 .4018 


5.0658 


7.69231 


408.41 


13.273.2 


62 


3844 


238,328 


7-8740 


3.9579 


16. 1290 


194.78 


3019.07 


131 


17,161 


2.248,091 


11-4455 


5.0788 


7.63359 


411.55 


13,478.2 


63 


3969 


250,047 


7-9373 


3.9791 


15.8730 


197.92 


3117-25 


132 


17.424 


2,299,968 


11.4891 


5.0916 


7.57576 


414-69 


13.684.8 


64 


4096 


262,144 


8 - 0000 


4.0000 


15.6250 


201.06 


3216.99 


133 


17.689 


2,352,637 


11.5326 


5 ■ 1045 


7-51880 


417-83 


13.892.9 


65 


422s 


274,625 


8.0623 


4.0207 


15.3846 


204.20 


3318.31 


134 


17.956 


2,406,104 


11.5758 


5.I172 


7 .46269 


420-97 


14,102.6 


66 


4356 


287,496 


8. 1240 


4.0412 


15. 1515 


207.3s 


3421-19 


135 


18,225 


2,460,375 


1 1. 6 190 


5.1299 


7.40741 


424.12 


14.313-9 


67 


4489 


300,763 


8.1854 


4.0615 


14-9254 


210.49 


3525-65 


136 


18,496 


2,515,456 


II .6619 


5.1426 


7.35294 


427.26 


14.526.7 


68 


4624 


314.432' 


8.2462 


4.0817 


14-7059 


213.63 


3631.68 


137 


18,769 


2,571.353 


11.7047 


5-I551 


7.29927 


430.40 


14.741- 1 


69 


4761 


328,509 


8.3066 


4. IO16 


14.4928 


216.77 


3739-28 


138 


19.044 


2,628,072 


11-7473 


•5 - 1676 


7.24638 


433.54 


14.957- 1 



538 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 





Table 


26. — Squares, Cubes, Squaice Roots, Cube Roots, Reciprocals, Circumferences and Circular Areas — (Continued) 




Square 


Cube 


Sq. 
root 


Cu. 

root 


1000 X 
recip. 


No. 


= Dia. 


No. 


Square 


Cube 


Sq. 
root 


Cu. 
root 


1000 X 

recip. 


No 


= Dia. 


No. 


Circum. 


Area 


Circum. 


Area 


I3Q 


19.321 


2,685,619 


11.7898 


5.1801 


7-19424 


436.68 


15.174-7 


208 


43.264 


8,998,912 


14.4222 


S-9250 


4.80769 


653-45 


33.979-S 


140 


19,600 


2,744,000 


11.8322 


5-1925 


7.14286 


439-82 


13.393. 8 


209 


43.681 


9.129.329 


14-4568 


5-9343 


4-78469 


656-59 


34.307-0 


141 


19.881 


2,803,221 


11.8743 


5- 2048 


7 .09220 


442.96 


1S.614-3 


210 


44,100 


9.261.000 


14.4914 


5 - 9439 


4.76190 


659.73 


34.636.1 


142 


20.164 


2,863,288 


11.9164 


5. 2171 


7-04253 


446. 11 


13,836.8 


211 


44.521 


9,393.931 


14-5258 


3-9533 


4-73934 


662-88 


34.966.7 


143 


20,449 


2,924,207 


11.9583 


3. 2293 


6.99301 


449-23 


16,060.6 


212 


44.944 


9,528,128 


14.5602 


5-9627 


4.71698 


666.02 


35.298.9 


144 


20,736 


2,985,984 


12.0000 


3-2415 


6.94444 


452-39 


16.286.0 


213 


43,369 


9.663.397 


14-3945 


5-9721 


4.69484 


669.16 


35.632-7 


145 


21,025 


3,048,625 


12.0416 


5-2536 


6.89655 


455.53 


16. 5 13.0 


214 


45.796 


9.800,344 


14-6287 


5-9814 


4.67290 


672.30 


33.968.1 


146 


21,316 


3,112,136 


12.0830 


5-2656 


6.84932 


438.67 


16.741. 3 


215 


46,22s 


9.938,375 


14-6629 


5-9907 


4.65116 


675-44 


36.305-0 


147 


21,609 


3.176,523 


12. 1244 


S.2776 


6.80272 


461.81 


16.971. 7 


216 


46.656 


10,077,696 


14.6969 


6.0000 


4-62963 


678.58 


36.643. 5 


148 


21,904 


3.241.792 


12.1655 


5.2896 


6.75676 


464.96 


17.203.4 


217 


47.089 


10,218,313 


14-7309 


6.0092 


4.60829 


681-73 


36,983-6 


149 


22,201 


3,307.949 


12. 2066 


5 -301s 


6.71141 


468. 10 


17.436.6 


218 


47.324 


10,360,232 


14.7648 


6.018s 


4-58716 


684-87 


37.325.3 


150 


22,500 


3.375.000 


12.2474 


5-3133 


6.66667 


471.24 


17.671. 3 


219 


47.961 


10,503,459 


14.7986 


6.0277 


4.56621 


688.01 


37.668.5 


ISI 


22,801 


3,442.951 


12.2882 


5-3251 


6.62252 


474.38 


17.907.9 


220 


48,400 


10,648,000 


14-8324 


6.0368 


4-34543 


691-15 


38,013.3 


IS2 


23,104 


3.511.808 


12.3288 


5-3368 


6.57893 


477-52 


18.143.8 


221 


48,841 


, 10,793.861 


14-8661 


6.0459 


4-52489 


694-29 


38,339-6 


IS3 


23,409 


3.581.577 


12.3693 


5-3485 


6.33595 


480.66 


18.385.4 


222 


49,284 


10.941.048 


14.8997 


6.0550 


4-50450 


697-43 


38,707.6 


IS4 


23,716 


3,652,264 


12.4097 


5-3601 


6.49351 


483.81 


18,626.5 


223 


49,729 


11.089. 567 


14-9332 


6.0641 


4-48431 


7OO-S8 


39,057-1 


ISS 


24,025 


3.723.875 


12.4499 


5-3717 


6.45161 


486.95 


18,869.2 


224 


50,176 


11,239,424 


14-9666 


6.0732 


4-46429 


703-72 


39,408.1 


IS6 


24.336 


3.796,416 


12.4900 


5-3832 


6.41026 


490.09 


19.I13-4 


225 


50,625 


11,390.625 


15-0000 


6.0822 


4-44444 


706-86 


39.760. S 


157 


24,649 


3.869.893 


12.5300 


5-3947 


6.36943 


493.23 


19,359.3 


226 


Si.076 


11. 543. 176 


13-0333 


6.0912 


4.42478 


710.00 


40. 115. 


IS8 


24,964 


3.944.312 


12.5698 


5-4061 


6.32911 


496.37 


19.606.7 


227 


51. 529 


11.697,083 


15.0665 


6.1002 


4-40529 


713.14 


40.470-8 


139 


25,281 


4,019.679 


12.6095 


S-4175 


6.28931 


499.31 


19.855.7 


228 


51.984 


11.852,352 


15-0997 


6.1091 


4.38596 


716.28 


40.828. 1 


160 


25,600 


4.096.000 


12.6491 


5-4288 


6.25000 


502.6s 


20.106. 2 


229 


52.441 


12,008,989 


15-1327 


6.1180 


4.36681 


719.42 


41,187.1 


161 


25,921 


4.173. 281 


12.6886 


3.4401 


6.2HI8 


505-80 


20,358.3 


230 


52,900 


12,167,000 


15-1658 


6.1269 


4-34783 


722.57 


41.547-6 


162 


26,244 


4.251.528 


12.7279 


5.4514 


6. 17284 


308.94 


20,612.0 


231 


53.361 


12,326,391 


IS-1987 


6.1358 


4.32900 


725-71 


41.909.6 


163 


26,569 


4.330,747 


12.7671 


5-4626 


6.13497 


512.08 


20.867. 2 


232 


53.824 


12,487,168 


15-2315 


6 . 1446 


4-31034 


728.85 


42.273.3 


164 


26,896 


4,410,944 


12.8062 


5.4737 


6.09756 


513.22 


21,124. I 


233 


54.289 


12,649,337 


15-2643 


6.1534 


4-29185 


731-99 


42,638.5 


i6s 


27,225 


4,492,125 


12.8452 


5. 4848 


6.06061 


518.36 


21.382. 5 


234 


54.756 


12,812,904 


15-2971 


6.1622 


4-27350 


735-13 


.43,005.3 


166 


27,SS6 


4,574,296 


12.8841 


5-4959 


6.02410 


521. SO 


21,642.4 


23s 


55.225 


12,977.87s 


iS-3297 


6.1710 


4-25332 


738-27 


43.373.6 


167 


27.889 


4,657.463 


12.9228 


5-5069 


5.98802 


524.65 


21,904.0 


236 


55.696 


13,144,256 


15-3623 


6.1797 


4-23729 


741-42 


43.743.5 


168 


28,224 


4,741.632 


12.9615 


3-5178 


5.95238 


527.79 


22,167. I 


237 


56,169 


13.312.053 


15-3948 


6.1885 


4-21941 


744-56 


44.II5-0 


169 


28,561 


4,826,809 


13.0000 


5.5288 


5-91716 


530.93 


22,431.8 


238 


56.644 


13,481,272 


15-4272 


6.1972 


4-20168 


747-70 


44,488 . 1 


170 


28,900 


4,913,000 


13.0384 


5-5397 


5 -88235 


534-07 


22,698.0 


239 


57,121 


13.651,919 


15-4596 


6.2058 


4. 18410 


730-84 


44,862.7 


171 


29,241 


5,000,211 


13-0767 


5.5305 


5.84795 


537.21 


22,965.8 


240 


57.600 


13,824,000 


15-4919 


6.2145 


4 -16667 


753-98 


43.238.9 


172 


29,584 


5,088,448 


13-1149 


5-5613 


5-81395 


340.35 


23.235.2 


241 


58.081 


13.997. 521 


15-5242 


6.2231 


4.14938 


757-12 


43.616.7 


173 


29,929 


5.177,717 


13.1329 


5-5721 


■5.78035 


543.50 


23,506.2 


242 


58,564 


14.172.488 


15-5563 


6.2317 


4.13223 


760-27 


45.996.1 


174 


30,276 


5,268,024 


13-1909 


5.5828 


5.74713 


546.64 


23.778.7 


243 


59.049 


14,348.907 


15-5885 


6.2403 


4.11523 


763-41 


46.377.0 


17S 


30,625 


5.359.375 


13.2288 


5-3934 


5-71429 


549-78 


24,052.8 


244 


59.536 


14.526.784 


15-6205 


6.2488 


4 09836 


766-55 


46.759-3 


176 


30,976 


5,451.776 


13-2665 


5 -6041 


5.68182 


552.92 


24,328.5 


245 


60,025 


14.706.125 


15.6525 


6.2573 


4.08163 


769-69 


47.143-5 


177 


31,329 


5.545.233 


13.3041 


5-6147 


5-64972 


336.06 


24,605.7 


246 


60,516 


14.886,936 


15.6844 


6.2658 


4.06504 


772-83 


47.529.2 


178 


31,684 


5.630.752 


13.3417 


5-6252 


5-61798 


559-20 


24,884.6 


247 


61,009 


IS. 069. 223 


15.7162 


6.2743 


4.04858 


775-97 


47.916.4 


179 


32,041 


5.735.339 


13.3791 


5-6357 


5-58659 


562.35 


25.164.9 


248 


61,304 


15.252,992 


15.7480 


6.2828 


4.03226 


779-12 


48,305.1 


180 


32,400 


5,832,000 


13-4164 


5.6462 


5-55356 


565-49 


25.446.9 


249 


62,001 


15,438,249 


15-7797 


6.2912 


4.01606 


782-26 


48.695.3 


181 


32,761 


5.929.741 


13-4536 


5-6567 


5-52486 


568-63 


23.730-4 


250 


62,500 


15,625,000 


15.8114 


6.2996 


4.00000 


785-40 


49.087-4 


182 


33.124 


6,028,568 


13-4907 


5-6671 


3-49451 


571.77 


26.013-5 


251 


63,001 


IS. 813.231 


15-8430 


6.3080 


3.98406 


788-54 


49.480.9 


183 


33,489 


6,128,487 


13.5277 


S-6774 


3 - 46448 


574.91 


26,302.2 


252 


63.504 


16.003,008 


15-8745 


6.3164 


3-96825 


791-68 


49.873.9 


184 


33.856 


6,229.504 


13.3647 


S-6877 


3-43478 


578.0s 


26.590.4 


253 


64,009 


16,194.277 


IS .9060 


6.3247 


3-93237 


794.82 


50,272.6 


l8s 


34,225 


6,331.625 


13-6013 


5-6980 


5 -40541 


581.19 


26,880-3 


254 


64,516 


16,387,064 


15-9374 


6.3330 


3.93701 


797.96 


50,670.7 


186 


34.S96 


6,434.856 


13-6382 


5 -7083 


5-37634 


584.34 


27,171.6 


255 


65,02s 


16,581,375 


15-9687 


6-3413 


3.92157 


80I-II 


SI. 070. 5 


187 


34.969 


6,539.203 


13 - 6748 


5.7185 


5-34759 


587.48 


27,464.6 


256 


65.336 


16,777.216 


16 -0000 


6 - 3496 


3-90625 


804.2s 


SI.471.9 


188 


35.344 


6,644,672 


13-7113 


3-7287 


5-31915 


590.62 


27.759.1 


257 


66,049 


16,974.393 


16-03I2 


6-3579 


3-89105 


807.39 


51.874-8 


189 


35.721 


6,751,269 


13-7477 


5-7388 


5-29101 


593-76 


28,055.2 


238 


66,564 


17. 173. 512 


16.0624 


6.3661 


3-87597 


810-53 


52,279-2 


190 


36.JOO 


6,859,000 


13.7840 


5-7489 


5-26316 


596.90 


28.352.9 


259 


67,081 


17.373.979 


16.0935 


6.3743 


3-86100 


813-67 


52,685-3 


191 


36.481 


6,967.871 


13.8203 


3-7590 


5-23560 


600 . 04 


28,652.1 


260 


67,600 


17,576.000 


16. 1245 


6.382s 


3 -8461s 


816-81 


53.092-9 


192 


36,864 


7,077.888 


13.8564 


3-7690 


5-20833 


603. 19 


28,952.9 


261 


68.121 


17.779.381 


16.1555 


6-3907 


3-83142 


819.96 


53.502. 1 


193 


37.249 


7,189.057 


13-8924 


5-7790 


5-18135 


606.33 


29.255.3 


262 


68,644 


17.984.728 


16.1864 


6.3988 


3.81679 


823.10 


33.912-9 


194 


37.636 


7,301.384 


13.9284 


5.7890 


3-13464 


609.47 


29.559-2 


263 


69,169 


18. 191. 447 


16.2173 


6.4070 


3-80228 


826.24 


34.325-2 


195 


38,025 


7,414.875 


13-9642 


5-7989 


5-12821 


612.61 


29,864.8 


264 


69,696 


18.399.744 


16.2481 


6. 4151 


3-78788 


829-38 


54.739- 1 


196 


38,416 


7,529.536 


14.0000 


5-8088 


5. 10204 


615-75 


30.171. 9 


265 


70.225 


18,609.625 


16.2788 


6.4232 


3-77358 


832-52 


55.154-6 


197 


38,809 


7.64S.373 


14-0357 


5-8186 


3-07614 


618.89 


30.480. 5 


266 


70.756 


18.821.096 


16.3095 


6.4312 


3-73940 


835 -66 


55.571-6 


198 


39.204 


7,762,392 


14.0712 


5.8285 


3-03031 


622.04 


30,790.7 


267 


71.289 


19,034.163 


16.3401 


6.4393 


3-74532 


838-81 


55.990-3 


199 


39.601 


7,880,599 


14. 1067 


5-8383 


5.02513 


62s- 18 


31,102.6 


268 


71.824 


19,248.832 


16.3707 


6.4473 


3-73134 


841.9s 


56,410-4 


200 


40.000 


8,000,000 


14.1421 


5.8480 


5. 00000 


628.32 


31.413-9 


269 


72,361 


19,465,109 


16.4012 


6.4533 


3-71747 


845-09 


56,832.2 


201 


40,401 


8,120,601 


14.1774 


5.8578 


4-97512 


631.46 


31.730-9 


270 


72,900 


19,683,000 


16.4317 


6.4633 


3-70370 


848.23 


57,255-3 


202 


40,804 


8,242,408 


14-2127 


5. 8675 


4-93050 


634.60 


32.047.4 


271 


73.441 


19.902,511 


16.4621 


6.4713 


3-69004 


831-37 


57.680.4 


203 


41,209 


8,365.427 


14.2478 


5-8771 


4.92611 


637.74 


32.365-5 


272 


73.984 


20,123,648 


16.4924 


6.4792 


3-67647 


834-51 


58,106.9 


204 


41,616 


8,489,664 


14.2829 


5-8868 


4-90196 


640-89 


32.685.1 


273 


74.529 


20,346,417 


16.5227 


6.4872 


3-66300 


857-66 


58,334-9 


205 


42,025 


8,615,125 


14.3178 


5.8964 


4-87805 


644.03 


33,006.4 


274 


75,076 


20,570,824 


16.5529 


6.4951 


3-64964 


860 - 80 


58,964.6 


206 


42,436 


8,741.816 


14-3527 


5 -9039 


4-83437 


647-17 


33.329-2 


275 


75,625 


20,796,87s 


16.5831 


6.5030 


3-63636 


863-94 


59.393-7 


207 


42.849 


8,869,743 


14-3875 


5-9I5S 


4-83092 


650.31 


33.653-3 


276 


76,176 


21,024,576 


16.6132 


6.S108 


3-62319 


867.08 


59.828-5 



I 



MATHEMATICAL TABLES 



539 





Table 


26. — Squares, Cubes, Square Roots, Cube Roots, Reciprocals, Circumferences and 


Circular Areas — {Continued) 


No. 


Square 


Cube 


Sq. 


Cu. 


loooX 


No. 


= Dia. 


No. 


Square 


Cube 


Sq. Cu. 


loooX 


No. 


=.Dia. 
















root 


root 


recip. 


Circum. 


Area 








root root 


recip. 


Circum. 


Area 


277 


76,729 


21,253.933 


16.6433 


6.5187 


3.61011 


870.22 


60,262.8 


346 


119,716 


41,421,736 


18.6011 


7-0203 


2.89017 


1087.0 


94.024.7 


278 


77,284 


21,484,952 


16.6733 


6.5265 


3.59712 


873-36 


60,698 . 7 


347 


120,409 


41,781,923 


18.6279 


7-0271 


2.88184 


1090. 1 


94.569.0 


279 


77,841 


21,717,639 


16.7033 


6.5343 


3-58423 


876. so 


61,136.2 


348 


121,104 


42,144,192 


18.6548 


7-0338 


2.87356 


1093-3 


95.114-9 


280 


74.400 


21,952,000 


16.7332 


6.5421 


3.57143 


879.65 


61,573-2 


349 


121,801 


42,508,549 


18.6815 


7 . 0406 


2.86S33 


1096.4 


95.662.3 


281 


78,961 


22,188,041 


16.7631 


6 . 5499 


3.55872 


882.79 


62,015.8 


350 


122,500 


42,875,000 


18.7083 


7 - 0473 


2.85714 


1099.6 


96,211.3 


282 


79,524 


22,425,768 


16.7929 


6.5577 


3.54610 


885.93 


62,458.0 


'351 


123,201 


43.243.551 


18.7350 


7-0540 


2.84900 


1102.7 


96,761.8 


283 


80,089 


22,665,187 


16.8226 


6.5654 


3-53357 


889.07 


62,901.8 


352 


123,904 


43,614,208 


18.7617 


7.0607 


2.84091 


1105.8 


97,314.0 


284 


80,656 


22,906,304 


16.8523 


6.5731 


3-52113 


892.21 


63,347.1 


353 


124,609 


43.986,977 


18.7883 


7.0674 


2.83286 


1109.0 


97,867.7 


28s 


81,255 


23,149,125 


16.8819 


6.5808 


3-50877 


895.35 


63,794.0 


354 


125,316 


44,361,864 


18.8149 


7-0740 


2.82486 


1112.I 


98,423.0 


286 


81,796 


23.393.656 


16.911S 


6.588s 


3-49650 


898-50 


64,242.4 


333 


126,025 


44,738,875 


18.8414 


7.0807 


2.81690 


iiis-a 


98,979.8 


287 


82,369 


23,639,903 


16.9411 


6.5962 


3-48432 


901.64 


64,692.5 


336 


126,736 


45,118,016 


18.8680 


7-0873 


2.80899 


1118.4 


99.538.2 


288 


82,944 


23,887,872 


16.9706 


6.6039 


3-47222 


904.78 


65,144-1 


357 


127,449 


45,499,293 


18.8944 


7 . 0940 


2.80112 


1121. s 


100,098 


289 


83,521 


24,137,569 


17.0000 


6.6115 


3.46021 


907.92 


65,597-2 


358 


128,164 


45,882,712 


18.9209 


7.1006 


2.79330 


1124.7 


100,660 


290 


84,100 


24,389,000 


17.0294 


6.6191 


3-44828 


911 .06 


66,052.0 


359 


128,881 


46,268,279 


18.9473 


7.1072 


2.78532 


1127.8 


101,223 


291 


84,681 


24,642,171 


17-0587 


6.6267 


3-43643 


914.20 


66,508.3 


360 


129,600 


46,656,000 


18.9737 


7.1138 


2.77778 


1131.0 


101,788 


292 


85,264 


24,897,088 


17.0880 


6.6343 


3.42466 


917.35 


66,966.2 


361 


130,321 


47,045,881 


19.0000 


7.1204 


2.77008 


1134-1 


102,354 


293 


85,849 


25,153,757 


17.1172 


6.6419 


3-41297 


920.49 


67,425-6 


362 


131.044 


47,437,928 


19.0263 


7.1269 


2.76243 


1137.3 


102,922 


294 


86,436 


25,412,184 


17.1464 


6 . 6494 


3-40136 


923.63 


67,886.7 


. 363 


131.769 


47.832,147 


19.0526 


7.1333 


2.75482 


1140.4 


103,491 


295 


87,025 


25.672,375 


17-1756 


6.6569 


3-38983 


926.77 


68,349.3 


364 


132,496 


48,228,544 


19.0788 


7.1400 


2.74725 


1143.5 


104,062 


296 


87,616 


25.934.336 


17.2047 


6.6644 


3-37838 


929.91 


68,813.5 


365 


133.225 


48,627,125 


19.1050 


7.1466 


2.73973 


1146.7 


104,635 


297 


88,209 


26,198.073 


17-2337 


6.6719 


3.36700 


933.05 


69,279.2 


366 


133.956 


49,027,896 


19.1311 


7.1331 


2.73224 


1149.8 


105,209 


298 


88,804 


26,463,592 


17.2627 


6.6794 


3.35570 


936.19 


69,746.5 


367 


134.689 


49,430,863 


19.1572 


7-1596 


2.72480 


1133-0 


105,785 " 


299 


89,401 


26,730,899 


17.2916 


6.6869 


3.34448 


939.34 


70,215.4 


368 


135.424 


49.836,032 


19-1833 


7.1661 


2'. 71739 


1156.1 


106,362 


300 


90,000 


27,000,000 


17-320S 


6.6943 


3-33333 


942.48 


70,685.8 


369 


136,161 


50,243,409 


19.2094 


7.1726 


2.71003 


1159.2 


106,941 


301 


90,601 


27,270,901 


17.3494 


6.7018 


3.32226 


943-62 


71,157.9 


370 


136,900 


50,653,000 


19-2354 


7.1791 


2.70270 


1162.4 


107,521 


302 


91,204 


27,543.608 


17-3781 


6.7092 


3.31126 


948.76 


71,631.5 


371 


137.641 


51,064,811 


19.2614 


7.1855 


2.69542 


1165.5 


108,103 


303 


91,809 


27,818,127 


17.4069 


6.7166 


3.30033 


951.90 


72,106.6 


372 


138.384 


51,478,848 


19-2873 


7.1920 


2.68817 


1168.7 


108,687 


304 


92,416 


28,094,464 


17.4356 


6.7240 


3.28947 


955-04 


72,583.4 


373 


139.129 


51,895.117 


19-3132 


7-1984 


2.68097 


1171.8 


109,272 


305 


93,025 


28,372,625 


17-4642 


6.7313 


3-27869 


958.19 


73,061.7 


374 


139.876 


52,313.624 


19.3391 


7.2048 


2.67380 


1173. 


109,858 


306 


93,636 


28,652,616 


17.4929 


6.7387 


3.26797 


961.33 


73,341.3 


375 


140,625 


32,734.375 


19-3649 


7.2112 


2.66667 


1178.1 


110,447 


307 


94,249 


28,934.443 


17.5214 


6 . 7460 


3-25733 


964.47 


74,023.0 


376 


141.376 


53.157.376 


19-3907 


7.2177 


2.6S9S7 


1181.2 


111,036 


308 


94,864 


29,218,112 


17-5499 


6.7533 


3-24675 


967.61 


74,506.0 


377 


142,129 


53.582,633 


19-4165 


7.2240 


2.65252 


1184.4 


111,628 


309 


95.481 


29,503,629 


17-5784 


6.7606 


3.23625 


970.75 


74,990.6 


378 


142,884 


54,010,152 


19-4422 


7-2304 


2.64550 


1187.5 


112,221 


310 


96,100 


29,791,000 


17.6068 


6.7679 


3-22581 


973.89 


75,476.8 


379 


143.641 


54,439.939 


19-4679 


7-2368 


2.63852 


I 190. 7 


112,81s 


311 


96,721 


30,080,231 


17-6352 


6.7752 


3-21543 


977-04 


73,964-3 


380 


144,400 


54,872,000 


19-4936 


7.2432 


2.63158 


1 193 -8 


113,411 


312 


97,344 


30,371.328 


17-6635 


6.7824 


3-20513 


980.18 


76,433-8 


381 


145,161 


55,306,341 


19-5192 


7-2495 


2 .62467 


1196.9 


114,009 


313 


97,969 


30,664,297 


17.6918 


6.7897 


3 - 19489 


983-32 


76,944.7 


382 


145,924 


55,742,968 


19-3448 


7.2558 


2.61780 


1200.1 


114,608 


314 


98,596 


30,959,144 


17.7200 


6.7969 


3-18471 


986.46 


77,437.1 


383 


146,689 


56,181,887 


19-3704 


7.2622 


2.61097 


1203.2 


115,209 


31S 


99.225 


31,255.87s 


17.7482 


6.8041 


3-17460 


989.60 


77,931-1 


384 


147,456 


56,623,104 


19-5959 


7.2685 


2 .60417 


1206.4 


115.812 


316 


99,856 


31,554.496 


17.7764 


6.8113 


3-16456 


992.74 


78,426.7 


38s 


148,225 


57,066,625 


19.6214 


7-2748 


2.59740 


1209.5 


116,416 


317 


100,489 


31.855.013 


17-8045 


6.8185 


3-15457 


995.88 


78,923-9 


386 


148,996 


57.512,456 


19.6469 


7.2811 


2.59067 


1212.7 


117,021 


318 


101,124 


32,157,432 


17-8326 


6.8256 


3.14465 


999.03 


79,422.6 


387 


149,769 


57,960,603 


19-6723 


7-2874 


2.58398 


1215.8 


117.628 


319 


101,761 


32,461,759 


17.8606 


6.8328 


3-13480 


1002.2 


79,922.9 


388 


130,544 


58,411,072 


19-6977 


7-2936 


2.57732 


1218.9 


118,237 


320 


102,400 


32,768,000 


17-8885 


6.8399 


3.12500 


1005.3 


80,424.8 


389 


151.321 


58,863,869 


19.7231 


7-2999 


2.57069 


1221.1 


118,847 


321 


103,041 


33,076,161 


17.9165 


6.8470 


3-11527 


1008.5 


80,928.2 


390 


152,100 


59.319.000 


19-7484 


7-3061 


2.56410 


1225.2 


119,459 


322 


103,684 


33.386,248 


17.9444 


6.8541 


3-10559 


1011.6 


81.433-2 


391 


152,881 


59,776,471 


19-7737 


7-3124 


2-55753 


1228.4 


120,072 


323 


104,329 


33,698,267 


17-9722 


6.8612 


3.09598 


1014.7 


81,939-8 


392 


153,664 


60,236,288 


19.7990 


7-3186 


2.55102 


I23I-5 


120,687 


324 


104,976 


34,012,224 


18.0000 


6.8683 


3.08642 


1017.9 


82,448.0 


393 


154.449 


60,698,457 


19.8242 


7.3248 


2.54453 


1234.6 


121,304 


325 


105,625 


34,328,125 


18.0278 


6.8753 


3-07692 


1021.0 


82,957-7 


394 


155,236 


61,162,984 


19-8494 


7.3310 


2.53807 


1237.8 


121,922 


326 


106,276 


34,645,976 


18.0555 


6.8824 


3-06749 


1024.2 


83,469.0 


393 


156,025 


61,629,875 


19.8746 


7.3372 


2.5316s 


1240.9 


122,542 


327 


106,929 


34,965,783 


18.0831 


6.8894 


3-05810 


1027.3 


83,981.8 


396 


156,816 


62,099,136 


19-8997 


7.3434 


2.52525 


1244.1 


123,163 


328 


107.584 


35.287,552 


18.1108 


6.8964 


3-04878 


1030.4 


84,496.3 


397 


157.609 


62,570,773 


19.9249 


7-3496 


2.51889 


1247.2 


123,786 


329 


108,241 


35,611,289 


18.1384 


6.9034 


3-03951 


1033.6 


85,012.3 


398 


158,404 


63,044,792 


19-9499 


7-3558 


2.51256 


1250.4 


124,410 


330 


108,900 


35.937.000 


18.1659 


6.9104 


3-03030 


1036.7 


85,529-9 


399 


159.201 


63,521,199 


19-9750 


7-3619 


2.50627 


1253-5 


125,036 


331 


109,561 


36,264,691 


18.1934 


6.9174 


3-02115 


1039.9 


86,049.0 


400 


160,000 


64,000,000 


20.0000 


7-3681 


2 . 50000 


1256.6 


125,664 


332 


110,224 


36,594.368 


18.2209 


6.9244 


3-01205 


1043.0 


86,569.7 


,401 


160,801 


64,481,201 


20.0250 


7-3742 


2.49377 


1259.8 


126,293 


333 


110,889 


36,926,037 


18 . 2483 


6.9313 


3.00300 


1046.2 


87,092 .0 


402 


161,604 


64,964,808 


20.0499 


7-3803 


2.48756 


1262.9 


126,923 


334 


111,556 


37,259.704 


18.2757 


6.9382 


2 .99401 


1049.3 


87,615-9 


403 


162,409 


65,450,827 


20.0749 


7-3864 


2.48139 


1266.1 


127.556 


335 


112,225 


37,595,375 


18.3030 


6.94SI 


2.98507 


1052.4 


88,141.3 


404 


163,216 


65,939.264 


20.0998 


7-3925 


2.47525 


1269.2 


128,190 


336 


112,896 


37,933,056 


18.3303 


6.9521 


2 .97619 


1055.6 


88,668.3 


405 


164,025 


66,430,125 


20.1246 


7.3986 


2 .46914 


1272.3 


128,82s 


337 


113.369 


38,272,753 


18.3576 


6.9589 


2.96736 


1058.7 


89,196.9 


406 


164,836 


66,923.416 


20.1494 


7.4047 


2.46305 


1275-5 


129,462 


338 


114.244 


38,614,472 


18.3848 


6.9658 


2.95858 


1061.9 


89,727-0 


407 


165,649 


67,419.143 


20.1742 


7.4108 


2.45700 


1278.6 


130,100 


339 


114.921 


38,958,219 


18.4120 


6.9727 


2.94985 


1065.0 


90,258.7 


408 


166,464 


67,917.312 


20.1990 


7-4169 


2.45098 


1281.8 


130,741 


340 


115,600 


39,304,000 


18.4391 


6.9795 


2.94118 


1068.1 


90,792.0 


409 


167,281 


68,417,929 


20.2237 


7.4229 


2 . 44499 


1284.9 


131.382 


341 


116,281 


39,651,821 


18.1662 


6.9864 


2.93255 


1071.3 


91,326.9 


410 


168,100 


68,921,000 


20.248s 


7.4290 


2.43902 


128S.1 


132,025 


342 


116,964 


40,001,688 


18.4932 


6.9932 


2.92398 


1074-4 


91,863.3 


411 


168,921 


69,426,531 


20.2731 


7-4350 


2.43309 


1291.2 


132,670 


343 


117,649 


40,353,607 


18.5203 


7 . 0000 


2.91545 


1077-6 


92,401-3 


412 


169,744 


69.934.528 


20.2978 


7.4410 


2 .42718 


1294-3 


133.317 


344 


118,336 


40,707,584 


18.5472 


7.0068 


2 .90698 


1080.7 


92,940.9 


413 


170,569 


70,444.997 


20.3224 


7-4470 


2.42131 


1297. S 


133,965 


345 


119.025 


41,063,625 


18.5742 


7.0136 


2.8985s 


1083.8 


93,482.0 


414 


171,396 


70,957.944 


20.3470 


7.4530 


2.41546 


1300.6 


134.614 



540 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 





Table 


26. — Squares, Cub 


ES, Square Roots, Citbe Roots, Reciprocals, Circumferences and 


Circular Areas — {Continued) 


No. 


Square 


Cube ^'J; 

root 


Cu. 


loooX 


No. 


= Dia. 


No. 


Square 


Cube 


Sq. 


Cu. 


loooX 


No 


. = Dia. 


root 


recip. 


Circum. 


Area 


root 


root 


recip. 


Circum. 


Area 


41S 


172,225 


71.473,375 


20.3715 ■ 


^•4590 


' . 2 .40964 


1303.8 


135,265 


484 


234.256 


113.379.904 


22.0000 


7-8514 


2.06612 


1520. 5 


183,984 


416 


173.056 


71,991,296 


20.3961 


^.4650 


2.4038s 


1306.9 


135.918 


485 


235.225 


114,084,125 


22.0227 


7-8568 


2.06186 


1523.7 


184,74s 


417 


173,889 


72,511,713 


20.4206 


?.'47io 


2.39808 


1310.0 


136,572 


486 


236,196 


114,791,256 


22.0454 


7-8622 


2.05761 


1526.8 


185,508 


418 


174.724 


73,034.632 


20.4450 


7-4770 


2.39234 


1313-2 


137,228 


487 


237,169 


115,501,303 


22.0681 


7.8676 


2.05339 


1530-0 


186,272 


419 


175,561 


73.560,059 


20.4695 


7.4829 


2.38664 


1316.3 


137,885 


488 


238,144 


116,214,272 


22.0907 


7.8730 


2.04918 


1533-1 


187,038 


420 


176,400 


74,088,000 


20.4939 


7.4889 


2.38095 


1319.S 


138,544 


489 


239,121 


116,930,169 


22.1133 


7.8784 


2 . 04499 


1536.2 


187,80s 


421 


177,241 


74,618,461 


20.5183 


7 • 4948 


2.37530 


1322.6 


139.205 


490 


240,100 


117,649,000 


22.1359 


7.8837 


2.04082 


1539.4 


188,574 


422 


178,084 


75,151.448 


20.5426 


7.5007 


2.36967 


1325.8 


139.867 


491 


241,081 


118,370,771 


22.1585 


7.8891 


2.03666 


1542.5 


189,345 


423 


178,929 


75,686,967 


20.5670 


7.5067 


2.36407 


1328.0 


140,531 


492 


242,064 


119,095,488 


22. 1811 


7 • 8944 


2.03252 


1545.7 


190,117 


424 


179,776 


76,225,024 


20.5913 


7.5126 


2.35849 


1332.0 


141,196 


493 


243,049 


119,823,157 


22.2036 


7.8998 


2.02840 


1548.8 


190,890 


42s 


180,625 


76,765,625 


20.6155 


7.5185 


2.35294 


1335.2 


141,863 


494 


244,036 


120,553,784 


22.2261 


7.9051 


2.02429 


1551.9 


191,665 


426 


181,476 


77,308,776 


20.6398 


7.5244 


2.34742 


1338.3 


142,531 


495 


245.025 


121,287,375 


22.2486 


7-910S 


2.02020 


IS55-I 


192,442 


427 


182,329 


77.854.483 


20.6640 


7-5302 


2.34192 


1341-5 


143,201 


496 


246,016 


122,023,936 


22.2711 


7.9158 


2.01613 


1558-2 


193,221 


428 


183,184 


78,402,752 


20.6882 


7.5361 


2.33645 


1344.6 


143,872 


497 


247,009 


122,763,473 


22.2935 


7.9211 


2.01207 


1561-4 


194,000 


429 


184,041 


78,953.589 


20.7123 


7.5420 


2-33100 


1347.7 


144.545 


498 


248,004 


123.505.992 


22.3159 


7.9264 


2.00803 


1564-5 


194.782 


430 


184,900 


79,507,000 


20.7364 


7.5478 


2-32558 


1350.9 


145,220 


499 


249,001 


124,251,499 


22.3383 


7-9317 


2.00401 


1567.7 


195.565 


431 


185,761 


80,062,991 


20.7605 


7-5537 


2.32019 


1354-0 


145,896 


500 


250,000 


125,000,000 


22.3607 


7-9370 


2.00000 


1570.8 


196,350 


432 


186,624 


80,621,568 


20.7846 


7-5595 


2.31482 


1357-2 


146.574 


501 


251,001 


125,751.501 


22.3830 


7-9423 


I. 99601 


1573-9 


197.136 


433 


187,489 


81,182,737 


20.8087 


7.5654 


2.30947 


1360.3 


147,254 


502 


252,004 


126,506,008 


22.4054 


7.9476 


1.99203 


1577-1 


197,923 


434 


188,356 


81,746,504 


20.8327 


7.5712 


2.3041s 


1363.5 


147,934 


503 


253,009 


127,263,527 


22.4277 


7.9528 


1.98807 


1580.2 


198,713 


435 


189,225 


82,312,875 


20.8567 


7.5770 


2.2988s 


1366.6 


148,617 


504 


254,016 


128,024,064 


22.4499 


7.9581 


I. 98413 


1583.4 


199,504 


436 


190,096 


82,881,856 


20.8806 


7-5828 


2.29358 


1369.7 


149,301 


505 


255,025 


128,787,625 


22.4722 


7.9634 


I .98020 


1586.5 


200,296 


437 


190,969 


83,453.453 


20.9045 


7-5886 


2-28833 


1372.9 


149,987 


506 


256,036 


129,554.216 


22.4944 


7.9686 


1.97629 


1589.7 


201,090 


438 


191,844 


84,027,672 


20.9284 


7 - 5944 


2-28311 


1376.0 


150,674 


507 


257,049 


130,323.843 


22.5167 


7-9739 


1.97239 


1592.8 


201,886 


439 


192,721 


84,604,519 


20.9523 


7.6001 


2.27790 


1379-2 


151,363 


508 


258,064 


131.096,512 


22.5389 


7.9791 


1.96850 


1595-9 


202,683 


440 


193,600 


85,184,000 


20.9762 


7.6059 


2.27273 


1382.3 


152,053 


509 


259,081 


131,872,229 


22.5610 


7.9843 


1.96464 


1599-1 


202,482 


441 


194,481 


85,766,121 


21 .0000 


7.6117 


2.26757 


1385.4 


152,745 


510 


260,100 


132,651,000 


22.5832 


7.9896 


I .96078 


1602.2 


204,282 


442 


195,364 


86,350,888 


21.0238 


7.6174 


2.26244 


1388.6 


153,439 


511 


261,121 


133,432,831 


22.6053 


7.9948 


1.95695 


1605.4 


205,084 


443 


196,249 


86,938,307 


21.0476 


7.6232 


2.25734 


1391.7 


154.134 


512 


262,144 


134.217,728 


22.6274 


8.0000 


1.95312 


1608.5 


205,887 


444 


197,136 


87,528,384 


21.0713 


7.6289 


2.25225 


1394-9 


154,830 


513 


263,169 


135,005,697 


22.6495 


8.0052 


1.94932 


1611 .6 


206,692 


445 


198,025 


88,121,125 


21.0950 


7.6346 


2.24719 


1398.0 


155,528 


SI4 


264,196 


135,796,744 


22 .6716 


8.0104 


1-94553 


1614.8 


207,499 


446 


198,916 


88,716,536 


21, II87 


7.6403 


2.2421s 


1401 .2 


156,228 


S15 


265,225 


136,590,875 


22.6936J8.0156 


1-94175 


1617.9 


208,307 


447 


199,809 


89,314.623 


21 . 1424 


7 . 6460 


2.23714 


1404.3 


156,930 


516 


266,256 


137,388,096 


22.7156 


8.0208 


1-93798 


1621.1 


209,117 


448 


200,704 


89.915.392 


21. 1660 


7.6517 


2.23214 


1407.4 


157,633 


517 


267,289 


138,188,413 


22.7376 


8.0260 


1-93424 


1624.2 


209,928 


449 


201,601 


90,518,849 


21.1896 


7.6574 


2.22717 


1410.6 


158,337 


518 


268,324 


138,991,832 


22.7596 


8.0311 


1.930SO 


1627.3 


210,741 


450 


202,500 


91,125,000 


21.2132 


7.6631 


2.22222 


1413-7 


159,043 


519 


269,361 


139,798,359 


22.7816 


8.0363 


I .92678 


1630.5 


211,556 


451 


203,401 


91,733.851 


21.2368 


7.6688 


2.21730 


1416-9 


159,751 


520 


270,400 


140,608,000 


22.8035 


8.0415 


1.92308 


1633.6 


212,372 


452 


204,304 


92,345.408 


21 .2603 


7.6744 


2.21239 


1420.0 


160,460 


521 


271,441 


141,420,761 


22.8254 


8 . 0466 


1.91939 


1636.8 


213.189 


453 


205,209 


92,959,677 


21.2838 


7.6801 


2.20751 


1423-1 


161,171 


522 


272,484 


142,236,648 


22.8473 


8.0517 


1-91571 


1639.9 


214.008 


454 


206,116 


93.576.664 


21.3073 


7-6857 


2.20264 


1426.3 


161,883 


523 


273.529 


143,055.667 


22 .8692 


8.0569 


I.912OS 


1643 . I 


214,829 


455 


207,025 


94.196,375 


21.3307 


7.6914 


2.19780 


1429.4 


162,597 


524 


274.576 


143.877,824 


22.8910 


8.0620 


I .90840 


1646.2 


215,651 


456 


207,936 


94,818,816 


21.3542 


7.6970 


2.19298 


1432.6 


163,313 


52s 


275,625 


144,703,125 


22.9129 


8.0671 


I -90476 


1649 - 3 


216,475 


457 


208,849 


95,443.993 


21.3776 


7.7026 


2.I8818 


1435-7 


164,030 


526 


276,676 


145.531.576 


22.9347 


8.0723 


I-90114 


1652-5 


217,301 


458 


209,764 


96,071,912 


21 .4009 


7.7082 


2.18341 


1438.9 


164,748 


527 


277,729 


146.363.183 


22.9565 


8.0774 


1.89753 


1655-6 


218,128 


459 


210,681 


96,702,579 


21.4243 


7-7138 


2.17865 


1442.0 


165,468 


528 


278,784 


147,197,952 


22.9783 


8.0825 


I . 89394 


1658.8 


218,956 


460 


211,600 


97,336,000 


21.4476 


7.7194 


2.17391 


1445 . I 


166,190 


529 


279,841 


X48,035,889 


23.0000 


8.0876 


I .89036 


1661 .9 


219.787 


461 


212,521 


97,972,181 


21.4709 


7.7250 


2.16920 


1448.3 


166,914 


530 


280,900 


148,877,000 


23.0217 


8.0927 


I .88679 


1665.0 


220,618 


462 


213,444 


98,611,128 


21.4942 


7.7306 


2.164SO 


1451-4 


167,639 


431 


281,961 


149,721,291 


23-0434 


8.0978 


1-88324 


1668.2 


221,452 


463 


214,369 


99,252,847 


21.5174 


7.7362 


2.15983 


1454-6 


168,365 


532 


283,024 


150,568,768 


23-0651 


8.1028 


1.87970 


1671-3 


222,287 


464 


215,296 


99,897,344 


21.5407 


7.7418 


2.15517 


1457-7 


169,093 


533 


284,089 


151,419,437 


23.0868 


8.1079 


I. 87617 


1674-5 


223,123 


465 


216,225 


100,544,625 


21.5639 


7.7473 


2.15054 


1460.8 


169,823 


534 


285,156 


152,273,304 


23.1084 


8- I130 


1.87266 


1677.6 


223,961 


466 


217,156 


101,194,696 


21.5870 


7.7529 


2.14592 


1464.0 


170,554 


535 


286,225 


153,130,375 


23.1301 


8.I180 


I .86916 


1680.8 


224,801 


467 


218,089 


101,847,563 


21 .6102 


7.7584 


2.14133 


1467. I 


171.287 


536 


287,296 


153.990,656 


23.1517 


8.1231 


1.86567 


1683.9 


225,642 


468 


219,024 


102,503,232 


21.6333 


7-7639 


2.13675 


1470.3 


172,021 


537 


288,369 


154.8s4.153 


23.1733 


8.1281 


I .86220 


1687.0 


226,484 


469 


219,961 


103,161,709 


21.6564 


7.7695 


2 . 13220 


1473-4 


172,757 


538 


289,444 


155.720,872 


23-1948 


8.1332 


1.85874 


1690.2 


227,329 


470 


220,900 


103,823,000 


21.6795 


7.7750 


2.12766 


1476.5 


173.494 . 


539 


290,521 


156,590,819 


23.2164 


8.1382 


1.85529 


1693.3 


228,17s 


471 


221,841 


104,487,111 


21.7025 


7.7805 


2.I23I4 


1479-7 


174.234 


540 


291,600 


157,464.000 


23-2379 


8.1433 


1-85185 


1696.5 


229,022 


472 


222,784 


105,154.048 


21.7256 


7.7860 


2.I1864 


1482.8 


174.974 


541 


292,681 


158.340.421 


23-2594 


8.1483 


1-84843 


1699-6 


229,871 


473 


223,729 


105,823,817 


21.7486 


7.7915 


2.II417 


1486.0 


175.716 


542 


293.764 


159.220.088 


23.2809 


8.1533 


I -84502 


1702.7 


230,722 


474 


224,676 


106,496,424 


21.7715 


7.7970 


2. 10971 


1489. I 


176,460 


543 


294.849 


160,103,007 


23-3024 


8.1583 


I .84162 


1705.9 


231,574 


475 


225,625 


107,171,875 


21.7945 


7.8025 


2.10526 


1492.3 


177.20s 


544 


295.936 


160,989,184 


23-3238 


8.1633 


1.83824 


1709.0 


232,428 


476 


226,576 


107,850,176 


21.8174 


7.8079 


2.10084 


1495.4 


177.952 


545 


297,025 


161,878,625 


23-3452 


8.1683 


1.83486 


1712.2 


233.283 


477 


227,529 


108,531,333 


21.8403 


7.8134 


2 . 09644 


1498.5 


178.701 


546 


298,116 


162,771,336 


23-3666 


8.1733 


I .83150 


1715.3 


234.140 


478 


228,484 


109,215,352 


21.8632 


7.8188 


2.09205 


1501.7 


179.451 


547 


299,209 


163,667,323 


23-3880 


8-1783 


I. 82815 


1718.5 


234.998 


479 


229,441 


109,902,239 


21 .8861 


7.8243 


2.08768 


1504-8 


180,203 


548 


300,304 


164,566,592 


23-4094 


8-1833 


1.82482 


1721 .6 


235.858 


480 


230,400 


110,592,000 


21.9089 


7-8297 


2.08333 


1508.0 


180,956 


549 


301,401 


165,469,149 


23.4307 


8.1882 


I. 82149 


1724-7 


236,720 


481 


231,361 


111,284,641 


21.9317 


7-8352 


2.07900 


ISIl-I 


181,711 


550 


302,500 


166,375,000 


23.4521 


8.1932 


I.81818 


1727-9 


237,583 


482 


232,324 


111,980,168 


21.9545 


7 . 8406 


2.07469 


1514-3 


182,467 


551 


303.601 


167,284,151 


23-4734 8.1982 


I-81488 


1731-0 


238,448 


483 


233,289 


112,678,587 


21.9773 


7 . 8460 


2.07039 


1517.4 


183,225 


5S2 


304,704 


168,196,608 


23.4947'8.203i 


1.81159 


1734-2 


239.314 



MATHEMATICAL TABLES 



541 





Table 


26. — Squares, Cubes, Square Roots, Cube Roots, Reciprocals, Circumferences and 


Circular Areas — {C07 


itimied) 


No. 


Square 


Cube 


Sq. 


Cu. 


loooX 


No. 


= Dia. 


No. 


Square 


Cube 


Sq. 


Cu. 


loooX 


No 


. = Dia. 




























root 


root 


recip. 


Circum. 


Area 








root 


root 


recip. 


Circum. 


Area 


553 


305,809 


169,112,377 


23.5160 


8.2081 


1.80832 


1737.3 


240,182 


622 


386,884 


240.641,848 


24-9399 


8.5362 


1 .60772 


1954-1 


303,858 


554 


306,916 


170,031,464 


23.5372 


8.2130 


1 .80505 


1740.4 


241,051 


623 


388,129 


241,804,367 


24.9600 


8 


5408 


1.60514 


1957.2 


304.836 


555 


308,02s 


170,953.875 


23.5584 


8.2180 


1.80180 


1743.6 


241,922 


624 


389.376 


242,970,624 


24.9800 


8 


5453 


1.60256 


1960.4 


305.815 


556 


309,136 


171,879,616 


23.5797 


8.2229 


1.79856 


1746.7 


242.795 


62s 


390,62s 


244,140,625 


25.0000 


8 


5499 


I . 60000 


1963-5 


306,796 


557 


310,249 


172,808,693 


23.6008 


8.2278 


1-79533 


1743-9 


243,669 


626 


391.876 


245,314.376 


25.0200 


8 


5544 


1-59744 


1966.6 


307,779 


558 


311.364 


173. 741. 112 


23 .6220 


8.2327 


1.79211 


1759-0 


244.545 


627 


393.129 


246,491,883 


25.0400 


8 


5590 


I - 59490 


1969.8 


308,763 


559 


312,481 


174.676,879 


23.6432 


8.2377 


I. 78891 


1756.2 


245,422 


628 


394.384 


247,673.152 


25.0599 


8 


563s 


1-59236 


1972.9 


309,748 


s6o 


313,600 


175.616,000 


23.6643 


8.2426 


1-78571 


1759-3 


246,301 


629 


395.641 


248,858,189 


25.0799 


8 


5681 


1.58983 


1976.1 


310,736 


561 


314.721 


176,558,481 


23.6854 


8.2475 


1.78253 


1762.4 


247,181 


630 


396,900 


250,047,000 


25.0998 


8 


5726 


1.58730 


1979.2 


311.72s 


562 


315.844 


177.504.328 


23.706s 


8.2524 


1.77936 


1765.6 


248,063 


631 


398,161 


251,239,591 


25.1197 


8 


5772 


1.58479 


1982.4 


312.715 


S63 


316,969 


178.453.547 


23.7276 


8.2573 


1.77620 


1768.7 


248,947 


632 


399,424 


252,435.968 


25.1396 


8 


5817 


1.58228 


1985.5 


313.707 


564 


318,096 


179,406,144 


23.7487 


8.2621 


I. 7730s 


1771.9 


249,832 


633 


400,689 


253.636,137 


25.1595 


8 


5862 


1.57978 


1988.6 


314.700 


565 


319,225 


180,362,125 


23.7697 


8.2670 


I. 7699 I 


1775-0 


250,719 


634 


401,956 


254,840,104 


25.1794 


8 


5907 


1.57729 


1991.8 


315.696 


S66 


320,356 


181,321,496 


23.7908 


8.2719 


1.76678 


1778. I 


251,607 


635 


403.225 


256,047,875 


25.1992 


8 


5952 


1.57480 


1994-9 


316,692 


567 


321,489 


182,284,263 


23.8118 


8.2768 


1.76367 


1781.3 


252.497 


636 


404.496 


257,259.456 


25.2190 


8 


5997 


1.57233 


1998. I 


317.690 


568 


322,624 


183,250,432 


23.8328 


8.2816 


1.76056 


1784.4 


253.388 


637 


405.769 


258,474.853 


25.2389 


8 


6043 


1.56986 


2001.2 


318,690 


569 


323.761 


184,220,009 


23.8537 


8.2865 


1.75747 


1787-6 


254.281 


638 


407.044 


259,694,072 


25.2587 


8 


6088 


1.56740 


2004.3 


319.692 


570 


324,900 


185,193.000 


23.8747 


8.2913 


1.75439 


1790.7 


255.176 


639 


408,321 


260,917,119 


25.2784 


8 


6132 


1.5649s 


2007. s 


320,96s 


S7I 


326,041 


186,169,411 


23.8956 


8.2962 


1.75131 


1793-9 


256,072 


640 


409,600 


262,144,000 


25 .2982 


8 


6177 


1.56250 


2010.6 


321,699 


S72 


327,184 


187,149,248 


23.9165 


8.3010 


1.7482s 


1797-0 


256,970 


641 


410,881 


263,374.721 


25.3180 


8 


6222 


1.56006 


2013.8 


322,7ors 


573 


328,329 


188,132,517 


23-9374 


8.3059 


1.74520 


1800. I 


257.869 


642 


412,164 


264,609,288 


25.3377 


8 


6267 


1.55763 


2016.9 


323.713 


574 


329,476 


189,119.224 


23-9583 


8.3107 


I. 74216 


1803.3 


258,770 


643 


413.449 


265,847,707 


25.3574 


8 


6312 


1.55521 


2020.0 


324.722 


575 


330,625 


190,109,375 


23.9792 


8.3155 


1.73913 


1806.4 


259,672 


644 


414.736 


267,089,984 


25.3772 


8 


6357 


1.55280 


2023.2 


325.733 


S76 


331.776 


191,102,976 


24.0000 


8.3203 


1.73611 


1809.6 


260,576 


645 


416,02s 


268,336,12s 


25.3969 


8 


6401 


1.55039 


2026.3 


326.74s 


577 


332,929 


192,100,033 


24.0208 


8.3251 


I. 73310 


1812.7 


261,482 


646 


417.316 


269,586,136 


25.4165 


8 


6446 


I. 54799 


2029. s 


327.7S9 


578 


334.084 


193,100,552 


24.0416 


8.3300 


I. 73010 


1815.8 


262,389 


647 


418,609 


270,840,023 


25.4362 


8 


6490 


1.54560 


2032.6 


328,77s 


S79 


335,241 


194.104.539 


24.0624 


8.3348 


1.72712 


1819.0 


263,298 


648 


419,904 


272,097.792 


25.4558 


8 


653s 


1.54321 


2035.8 


329,792 


580 


336,400 


195,112,000 


24.0832 


8.3396 


I. 72414 


1822. I 


264,208 


649 


421,201 


273.359.449 


25.4755 


8 


6579 


1.54083 


2038.9 


330,810 


S81 


337.561 


196,122,941 


24.1039 


8.3443 


1.72117 


1825.3 


265,120 


650 


422,500 


274,625,000 


25.4951 


8 


6624 


1.53846 


2042.0 


331.831 


582 


338,724 


197,137,368 


24.1247 


8.3491 


I .71821 


1828.4 


266,033 


651 


423.801 


275.894.451 


25.5147 


8 


6668 


1.53610 


2045.2 


332,853 


S83 


339,889 


198,155,287 


24.1454 


8.3539 


I. 71527 


1831.6 


266,948 


652 


425.104 


277,167,808 


25-5343 


8 


6713 


1.53374 


2048.3 


333.876 


584 


341,056 


199,176,704 


24.1661 


8.3587 


I-71233 


1834-7 


267,865 


653 


426,409 


278,445,077 


25.5539 


8 


6757 


1.53139 


2051. 5 


334.901 


S85 


342,225 


200,201,625 


24.1868 


8.3634 


I . 70940 


1837.8 


268,783 


654 


427,716 


279,726,264 


25.5734 


8 


6801 


1.5290s 


2054.6 


335.927 


586 


343.396 


201,230,056 


24.2074 


8.3682 


1 . 70649 


1841.O 


269,701 


65s 


429.025 


281,011,37s 


25.5930 


8 


6845 


1.52672 


2057.7 


336,955 


587 


344.569 


202,262,003 


24.2281 


8.3730 


1.70358 


1844. I 


270,624 


656 


430.336 


282,300,416 


25.6125 


8 


6890 


1.52439 


2060.9 


337,985 


S88 


345.744 


203,297,472 


24.2487 


8.3777 


1.70068 


1847.3 


271.547 


657 


431.649 


283,593,393 


25.6320 


8 


6934 


1.52207 


2064.0 


339.016 


589 


346.921 


204,336.469 


24.2693 


8.3825 


1-69779 


1850.4 


272,471 


658 


432.964 


284,890,312 


25-6515 


8 


6978 


1.51976 


2067.2 


340.049 


590 


348,100 


205,379.000 


24.2899 


8.3872 


I . 69492 


1853.5 


273.397 


659 


434.281 


286,191,179 


25.6710 


8 


7022 


1.51745 


2070.3 


341.084 


591 


349.281 


206,425,071 


24.31OS 


8-3919 


1.69205 


1856.7 


274.325 


660 


435.600 


287,496,000 


25.6905 


8 


7066 


1.5151S 


2073-5 


324.119 


592 


350,464 


207,474,688 


24-3311 


8.3967 


I. 68919 


1859.8 


275.254 


661 


436.921 


288,804,781 


25.7099 


8 


7110 


1.51286 


2076.6 


343.157 


593 


351.649 


208,527,857 


24-3516 


8.4014 


1.68634 


1863.0 


276,184 


662 


438,244 


290,117,528 


25.7294 


8 


7154 


1.51057 


2079.7 


344.196 


594 


352,836 


209,584,584 


24.3721 


8.4061 


1.68350 


1866. I 


277,117 


663 


439,569 


291,434.247 


25.7488 


8 


7198 


1.50830 


2082.9 


345.237 


595 


354.025 


210,644,875 


24-3926 


8.4108 


1.68067 


1869.3 


278,051 


664 


440,896 


292,754.944 


25.7682 


8 


7241 


1.50602 


2086.0 


346,279 


596 


355.216 


211,708,736 


24.4131 


8.4155 


1.67785 


1872.4 


278,986 


665 


442,22s 


294,079.625 


25.7876 


8 


7285 


1.50376 


2089.2 


347.323 


597 


356.409 


212,776,173 


24-4336 


8.4202 


1.67504 


1875-5 


279.923 


666 


443.556 


295,408,296 


25.8070 


8 


7329 


1.S0150 


2092.3 


348.368 


598 


357.604 


213,847,192 


24-4540 


8.4249 


1.67224 


1878.7 


280,862 


667 


444.889 


296,740,963 


25.8263 


8 


7373 


1.49925 


2095.4 


349.415 


599 


358.801 


214,921,799 


24-4745 


8.4296 


1.6694s 


1881.8 


281,802 


668 


446,224 


298,077,632 


25.8457 


8 


7416 


1.49701 


2098.6 


350.464 


600 


360,000 


216,000,000 


24.4949 


8.4343 


1.66667 


1885.0 


282,743 


669 


447.561 


299,418,309 


25.8650 


8 


7460 


1.49477 


2101.7 


3SI.514 


601 


361,201 


217,081,801 


24.5153 


8.4390 


1.66389 


1888.1 


283,687 


670 


448,900 


300,763,000 


25.8844 


8 


7503 


1.49254 


2104.9 


352.565 


602 


362,404 


218,167,208 


24.5357 


8.4437 


1.66113 


1891.2 


284,631 


671 


450,241 


302,111,711 


25.9037 


8 


7547 


1.49031 


2108.0 


353.618 


603 


363,609 


219,256,227 


24.5561 


8.4484 


1.65837 


1894-4 


285,578 


672 


451.584 


303,464.448 


25.9230 


8 


7590 


1.48810 


2111.2 


354.673 


604 


364,816 


220,348,864 


24-5764 


8.4530 


1-65563 


1897-5 


286,526 


673 


452.929 


304,821,217 


25.9422 


8 


7634 


1.48588 


2114.3 


355.730 


605 


366,025 


221,445,125 


24.5967 


8.4577 


1.65289 


1900.7 


287,475 


674 


454.276 


306,182,024 


25.9615 


8 


7677 


1.48368 


2117.4 


356,788 


606 


367,236 


222,545,016 


24.6171 


8.4623 


I. 65017 


1903.8 


288,426 


675 


455.625 


307,546,87s 


25.9808 


8 


7721 


1.48148 


2120.6 


357.847 


607 


368,449 


223,648,543 


24.6374 


8.4670 


1-64745 


1907.0 


289,379 


676 


456.976 


308,915.776 


26.0000 


8 


7764 


1.47929 


2123.7 


358.908 


608 


369,664 


224,755.712 


24.6577 


8.4716 


1.64474 


1910.I 


290,333 


677 


458,329 


310,288,733 


26.0192 


8 


7807 


1.47711 


2126.9 


359.971 


609 


370,881 


225,866,529 


24.6779 


8.4763 


1.64204 


1913-2 


291,289 


678 


459.684 


311,665,752 


26.0384 


8 


7850 


1-47493 


2130.0 


361.03s 


610 


372.100 


226,981,000 


24.6982 


8.4809 


1-63934 


1916.4 


292,247 


679 


461,041 


313,046,839 


26.0576 


8 


7893 


1.47275 


2133. I 


362,101 


611 


373.321 


228,099.131 


24.7184 


8.4856 


1.63666 


1919.5 


293.206 


680 


462,400 


314,432,000 


26.0768 


8 


7937 


1.47059 


2136.3 


363.168 


612 


374,544 


229,220,928 


24.7386 


8.4902 


1-63399 


1922.7 


294,166 


681 


463.761 


315,821,241 


26.0960 


8 


7980 


1.46843 


2139.4 


364.237 


613 


375.769 


230,346,397 


24.7588 


8.4948 


I. 63132 


1925.8 


295,128 


682 


465.124 


317.214.568 


26.1151 


8 


8023 


1.46628 


2142.6 


365.308 


614 


376,996 


231.475.544 


24.7790 


8.4994 


1.62866 


1928.9 


296,092 


683 


466,489 


318,611,987 


26.1343 


8 


8066 


I .46413 


2145.7 


366,380 


615 


378,225 


232,608,375 


24.7992 


8.5040 


1.62602 


1932.1 


297.057 


684 


467.856 


320,013,504 


26.1534 


8 


8109 


1.46199 


2148.9 


367,453 


616 


379,456 


233.744.896 


24.8193 


8.5086 


1.62338 


1935-2 


298,024 


685 


469.225 


321,419,125 


26.1725 


8 


8152 


1.4598s 


2152.0 


368,528 


617 


380,689 


234,885,113 


24-8395 


8.5132 


1.62075 


1938.4 


298,992 


686 


470,596 


322,828,856 


26. 1916 


8 


8194 


1.45773 


2155-1 


369,60s 


618 


381.924 


236,029,032 


24.8596 


8.5178 


1.61812 


1941.5 


299,962 


687 


471.969 


324.242,703 26.2107 


8 


8237 


1.45560 


2158.3 


370,684 


619 


383,161 


237,176,659 


24-8797 


8.5224 


1.61551 


1944-7 


300,934 


688 


473.344 


325,660,672 26.2298 


8 


8280 


1.45349 


2161.4 


371.764 


620 


384,400 


238,328,000 


24.8998 


8.5270 


I. 61290 


1947-8 


301,907 


689 


474.721 


327,082,769 26.2488 


8 


8323 


1.45138 


2164.6 


372,845 


621 


385,641 


239,483.061 


24-9199 


8.5316 


I .61031 


1950.9 


302,882 


690I 476,100 


328,509,000126.2679 


8 


8366 


I .44928 


2167.7 


373.928 



542 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 





Table 


26. — Squares, Cubes, Square Roots, Cube Roots, Reciprocals, Circumferences and Circular Areas — {Contimied) 


No. 


Square 


Cube 


Sq. 


Cu. 


1000 X 


No 


= Dia 


No. 


Square 


Cube 


Sq. 


Cu. 


1000 X 


Nc 


. = Dia. 


root 


root 


recip. 


Circum. 


Area 


root 


root 


recip. 


Circum. 


Area 


691 


477.481 


329,939.371 


26.2869 


8.8408 


1.44718 


2170.8 


375,013 


760 


577.600 


438,976,000 


27.5681 


9-1258 


1-31579 


2387.6 


453,646 


692 


478,864 


331.373,888 


26.3059 


8-8451 


1-44509 


2174-0 


376,099 


761 


579,121 


440,711,081 


27.S862 


9- 1298 


I -31406 


2390.8 


454.841 


693 


480,249 


332,812,557 


26.3249 


8-8493 


1.44300 


2177.1 


377,187 


762 


580.644 


442,430,728 


27.6043 


9-1338 


1-31234 


2393-9 


456,037 


694 


481,636 


334.25S.384 


26.3439 


8.8536 


1.44092 


2180.3 


378,276 


763 


582,169 


444,194.947 


27.6225 


9-1378 


1.31062 


2397.0 


457.234 


69s 


483.02s 


335.702,375 


26.3629 


8.8578 


1.43885 


2183-4 


379,367 


764 


583,696 


445.943.744 


27.640s 


9.1418 


1.30890 


2400.2 


458,434 


696 


484,416 


337. 153. 536 


26.3818 


8.8621 


1.43678 


2186.6 


380,459 


76s 


585,225 


447,697,125 


27.6586 


9.1458 


1.30719 


2403.3 


459.63s 


697 


485,809 


338.608,873 


26.4008 


8.8663 


1.43472 


2189.7 


381,554 


766 


586,756 


449,455,096 


27.6767 


9 . 1498 


1.30548 


2406. 5 


460,837 


698 


487,204 


340,068,392 


26.4197 


8.8706 


1.43267 


2192.8 


382,649 


767 


588,289 


451,217,663 


27.6948 


9.1537 


1.30378 


2409.6 


462.042 


699 


488,601 


341.532.099 


26.4386 


8.8748 


1.43062 


2196.0 


383.746 


768 


589,824 


452,984.832 


27.7128 


9.1577 


1.30208 


2412.7 


463.247 


700 


490,000 


343.000,000 


26.4575 


8.8790 


1.42857 


2199. 1 


384.84s 


769 


591,361 


454.756,609 


27.7308 


9.1617 


1-30039 


241S-9 


464.454 1 


701 


491,401 


344,472,101 


26.4764 


8.8833 


1.42653 


2202.3 


385,945 


770 


592.900 


456,533.000 


27-7489 


9-1657 


1.29870 


2419.0 


465.663 


702 


492,804 


345,948,408 


26.4953 


8.8875 


1.42450 


2205.4 


387,047 


771 


594.441 


458,314.011 


27.7669 


9. 1696 


1.29702 


2422.2 


466,873 


703 


494.209 


347,428,927 


26.5141 


8-8917 


I .42248 


2208. 5 


388,151 


772 


595.984 


460.099,648 


27-7849 


9-1736 


1.29534 


2425-3 


468,085 


704 


495. 6t6 


348,913,664 


26.5330 


8.8959 


1 .42046 


2211.7 


389,256 


773 


597.529 


461,889.917 


27.8029 


9-1775 


1-29366 


2428.5 


469,298 


70s 


497,02s 


350,402,625 


26.5518 


8.9001 


1.41844 


2214.8 


390.363 


774 


599.076 


463,684,824 


27.8209 


9.1815 


1.29199 


2431-6 


470,513 


706 


498,436 


351,895,816 


26.5707 


8.9043 


1-41643 


2218.0 


391,471 


775 


600,62s 


465,484,375 


27-8388 


9-1855 


1.29032 


2434-7 


471,730 


707 


499,849 


353.393.243 


26.5895 


8.9085 


1-41443 


2221.1 


392.580 


776 


602,176 


467,288,576 


27.8568 


9-1894 


1.28866 


2437-9 


472,948 


708 


SOI, 264 


354,894.912 


26.6083 


8-9127 


1.41243 


2224.3 


393.692 


777 


603.729 


469,097,433 


27.8747 


9-1933 


1.28700 


2441-0 


474.168 


709 


S02,68l 


356,400,829 


26.6271 


8.9169 


1.41044 


2227.4 


394.805 


778 


605,284 


470,910,952 


27.8927 


9-1973 


1.2853s 


2444-2 


475.389 


710 


504,100 


357,911,000 


26.6458 


8.9211 


1.40845 


2230.5 


395.919 


779 


606,841 


472,729,139 


27.9106 


9.2012 


1.28370 


2447-3 


476,612 


711 


505,521 


359,425,431 


26.6646 


8.9253 


1.40647 


2233.7 


397,035 


780 


608,400 


474.552.000 


27.9285 


9.2052 


1.28205 


2450.4 


477,836 


712 


506,944 


360,944,128 


26.6833 


8.9295 


1.40449 


2236.8 


398.153 


781 


609,961 


476,379,541 


27-9464 


9.2091 


1.28041 


2453-6 


479,062 


713 


508,369 


362,467,097 


26.7021 


8.9337 


1.40253 


2240.0 


399.272 


782 


611.524 


478.211,768 


27.9643 


9.2130 


1.27877 


2456.7 


480,290 


714 


509,796 


363,994,344 


26.7208 


8.9378 


1.40056 


2243.1 


400.393 


783 


613.089 


480.048,687 


27.9821 


9.2170 


1-27714 


2459-9 


481,519 


71S 


511.225 


365,525,875 


26.7395 


8.9420 


1.39860 


2246.2 


401.SIS 


784 


614,656 


481,890,304 


2 8. 0000 


9.2209 


I-2755I 


2463.0 


482,750 


716 


512,636 


367.061,696 


26.7582 


8.9462 


1-39665 


2249-4 


402.639 


785 


616,22s 


483,736,62s 


28.0179 


9.2248 


1-27389 


2466.2 


483,982 


717 


514.089 


368,601,813 


26.7769 


8.9503 


1-39470 


2252.5 


403.765 


786 


617,796 


485.587,656 


28.0357 


9.2287 


1.27226 


2469-3 


485,216 


718 


515 524 


370.146,232 


26.7955 


8.9545 


1-39276 


2255-7 


404,892 


787 


619,369 


487,443.403 


28.0535 


9-2326 


1.27065 


2472.4 


486,451 


719 


S16.961 


371,694.959 


28.8142 


8-9587 


1.39082 


2258.8 


406,020 


788 


620.944 


489.303.872 


28.0713 


9-2365 


1.26904 


2475-6 


487,688 


720 


518,400 


373,248,000 


26.8328 


8.9628 


1.38889 


2261.9 


407.150 


789 


622,521 


491,169.069 


28.0891 


9 - 2404 


1 - 26743 


2478 . 7 


488,927 


721 


519,841 


374. 80s. 361 


26.8514 


8.9670 


I .38696 


226S-I 


408,282 


790 


624,100 


493.039.000 


28. 1069 


9.2443 


1.26582 


2481.9 


490,167 


722 


521,284 


376,367.048 


26.8701 


8-9711 


1-38504 


2268.2 


409,416 


791 


625,681 


494.913.671 


28.1247 


9.2482 


1.26422 


2485.0 


491,409 


723 


522,729 


377,933.067 


26.8887 


8.9752 


1-38313 


2271-4 


410.SSO 


792 


627,264 


496.793.088 


28. 1425 


9.2521 


1.26263 


2488.1 


492,652 


724 


524.176 


379.503,424 


26.9072 


8.9794 


I .38122 


2274.5 


411,687 


793 


628,849 


498,677.257 


28.1603 


9.2560 


I. 26103 


2491.3 


493.897 


72s 


525,625 


381,078,125 


26.9258 


8.9835 


1-37931 


2277.7 


412,825 


794 


630,436 


500,566,184 


28.1780 


9.2599 


1.25945 


2494-4 


495.143 


726 


527,076 


382.657,176 


26.9444 


8.9876 


1-37741 


2280.8 


413.96s 


795 


632,025 


502.459,875 


28.1957 


9.2638 


1.25786 


2497.6 


496,391 


727 


528,529 


384,240,583 


26.9629 


8.9918 


1-37552 


2283.9 


41s. 106 


796 


633.616 


504.358.336 


28.2135 


9.2677 


1.25628 


2500.7 


497.641 


728 


529,984 


385.828,352 


26.9815 


8.9959 


1-37363 


2287-1 


416.248 


797 


635.209 


506,261,573 


28.2312 


9.2716 


1.25471 


2503.8 


498.892 


729 


531,441 


387,420,489 


27.0000 


9.0000 


1-37174 


2290. 2 


417.393 


798 


636,804 


508,169,592 


28.2489 


9-2754 


1.25313 


2507.0 


500,145 


730 


532,900 


389,017,000 


27.0185 


9.0041 


1.36986 


2293.4 


418,539 


799 


638,401 


510,082,399 


28.2666 


9-2793 


1.25156 


2510.1 


501,399 


731 


534.361 


390,617,891 


27.0370 


9.0082 


1.36799 


2296. S 


419,686 


800 


640,000 


512,000.000 


28.2843 


9.2832 


1.25000 


2513-3 


502,655 


732 


535.824 


392,223,168 


27-0555 


9.0123 


1.36612 


2299.7 


420,83s 


801 


641,601 


513.922.401 


28.3019 


9.2870 


I .24844 


2516-4 


503,912 


733 


537.289 


393,832,837 


27.0740 


9.0164 


1.36426 


2302.8 


421,986 


802 


643.204 


515.849.608 


28.3196 


9.2909 


1.24688 


2519-6 


505,171 


734 


538,756 


395.446,904 


27.0924 


9.020s 


1.36240 


2305.9 


423,138 


803 


644,809 


517.781,627 


28.3373 


9.2948 


1-24533 


2522.7 


506,432 


735 


540.225 


397,065,375 


27.1109 


9.0246 


1-36054 


2309.1 


424.293 


804 


646,416 


519,718,464 


28.3549 


9.2986 


1-24378 


2525.8 


507,694 


736 


541.696 


398,688,256 


27-1293 


9.0287 


1.35870 


2312.2 


425,448 


805 


648,025 


521,660.123 


28.3725 


9-3025 


1.24224 


2529.0 


508,958 


737 


543,169 


400,315,553 


27-1477 


9-0328 


1-35685 


231S-4 


426,604 


806 


649,636 


523,606,616 


28.3901 


9-3063 


I . 24069 


2532.1 


510,223 


738 


544.644 


401,947,272 


27. 1662 


9-0369 


I -35501 


2318-5 


427,762 


807 


651.249 


525.557,943 


28.4077 


9-3102 


I. 23916 


2535-3 


511. 490 


739 


546,121 


403,583,419 


27-1846 


9.0410 


1-35318 


2321.6 


428,922 


808 


652,864 


527.514.112 


28.4253 


9-3149 


1.23762 


2538.4 


512,7S8 


740 


547,600 


405,224,000 


27.2029 


9-0450 


1.3513s 


2324.8 


430,084 


809 


654.481 


529.475,129 


28.4429 


9-3179 


I . 23609 


2541 -S 


514.028 


741 


549,081 


406,869,021 


27.2213 


9.0491 


1-34953 


2327 9 


431,247 


! 810 


656,100 


531,441,000 


28.460s 


9-3217 


1-23457 


2544-7 


515.300 


742 


550,564 


408,518,488 


27.2397 


9-0532 


1-34771 


2331-1 


432,412 


811 


657.721 


533,411.731 


28.4781 


9-3255 


1-23305 


2547-8 


516.573 


743 


552,049 


410,172,407 


27.2580 


9-0572 


1-34590 


2334-2 


433.578 


812 


659.344 


535.387.328 


28.4956 


9-3294 


1-23153 


2551-0 


517.848 


744 


553,536 


411.830,784 


27.2764 


9-0613 


1-34409 


2337-3 


434,746 


813 


660,969 


537.367.797 


28.5132 


9-3332 


I . 23001 


2554-1 


519.124 


745 


555,025 


413,493,625 


27.2947 


9-0654 


1.34228 


2340.5 


435,916 


814 


662,596 


539.353. 144 


28.5307 


9-3370 


1 . 22850 


2557-3 


520.402 


746 


S56,Sl6 


415,160,936 


27-3130 


9 - 0694 


1.34048 


2343-6 


437.087 


815 


664,225 


541,343,375 


28.5482 


9-3408 


1.22699 


2560.4 


321,681 


747 


558.009 


416,832,723 


27-3313 


9 -073s 


1.33869 


2346-8 


438.259 


816 


665,856 


543.338,496 


28.5657 


9-3447 


1.22549 


2563 -S 


522,962 


748 


559.504 


418,508,992 


27-3496 


9-0775 


1-33690 


2349.9 


439.423 


817 


667.489 


545.338. 513 


28.5832 


9.3485 


1-22399 


2566.7 


524.245 


749 


561,001 


420,189,749 


27-3679 


9.0816 


1.33511 


2353-1 


440,609 


818 


669,124 


547.343.432 


28.6007 


9-3523 


1.22249 


2569.8 


525.529 ' 


750 


562,500 


421,875.000 


27-3861 


9-0856 


1-33333 


2356.2 


441,786 


819 


670,761 


549.353.259 


28.6182 


9-3S6i 


I .22100 


2573-0 


526,814 i 


751 


564,001 


423.564.751 


27.4044 


9.0896 


1.33156 


2359.3 


442,965 


820 


672,400 


551.368. 000 


28.6356 


9-3599 


1 -21951 


2576-1 


528,102 


752 


565,504 


425,259,008 


27.4226 


9-0937 


1-32979 


2362.5 


444,146 


821 


674,041 


553.387.661 


28-6531 


9-3637 


1.21803 


2579-2 


529,391 


753 


567,009 


426.957.777 


27.4408 


9-0977 


1.32802 


2365.6 


445,328 


822 


675.684 


555.412,248 


28.670s 


9-3675 


1-21655 


2582-4 


530.681 


754 


568,516 


428,661,064 


27.4591 


9-IO17 


1.32626 


2368.8 


446,511 


823 


677.329 


557,441.767 


28-6880 


9-3713 


I-2I507 


2585-5 


531.973 


755 


570,025 


430,368,875 


27-4773 


9 -1057 


1.32450 


2371-9 


447.697 


824 


678,976 


559.476,224 


28.7054 


9-3751 


1.2I3S9 


2588.7 


533.267 


756 


571,536 


432,081,216 


27-4955 


9.1098 


1-32275 


2375-0 


448.883 


82s 


680.625 


561,515,625 


28.7228 


9-3789 


1.21212 


2591-8 


534.562 i 


)757 


573.049 


433.798,093 


27-5136 


9-1138 


1 .32100 


2378-2 


450.072 


826 


682,276 


563.559.976 


28-7402 


9-3827 


1.2106s 


2595-0 


535,858 


758 


574.564 


435.519,512 


27-5318 


9.I178 


1.31926 


2381-3 


451.262 


827 


683,929 


565.609.283 


28.7576 


9-3865 


I . 20919 


2598.1 


337,157 


759 


576,081 


437,245.479 


27-5500 


9.1218 


1.31752 


2384-5 


452,453 


828 


685.584 


567.663,552 


28.7750 


9-3902 


1.20773 


2601 .2 


538,456 



































MATHEMATICAL TABLES 



543 



p 


Table 


26. — Squares, Cubes, Square Roots, Cube Roots, Reciprocals Circumferences and 


Circular Areas — {Continued) 


No. 


Square 


Cube 


Sq. 


Cu. 


1000 X 


No. 


= Dia. 


No. 


Square 


Cube 


Sq. 


Cu. 


1000 X 


No 


= Dia. 


root 


root 


recip. 


Circum. 


Area 


root 


root 


recip. 


Circum. 


Area 


829 


687,241 


569,722,789 


28.7924 


9.3940 


1.20627 


2604.4 


539.758 


898 


806,404 


724,150,792 


29.9666 


9.6477 


I.II3S9 


2821.2 


633,348 


830 


688,900 


571,787,000 


28.8097 


9.3978 


1.20482 


2607. 5 


541.061 i 


899 


808,201 


726,572,699 


29-9833 


9.6513 


1.1123s 


2824.3 


634,760 


831 


690,561 


573.856,191 


28.8271 


9.4016 


1.20337 


2610.7 


542.36s 


900 


810,000 


729,000,000 


30-0000 


9.6549 


I. mil 


2827.4 


636,173 


832 


692,224 


575.930,368 


28.8444 


9.4053 


1.20192 


2613.8 


543.671 


901 


811,801 


731,432,701 


30.0167 


9.6585 


I. 10988 


2830.6 


637.587 


833 


693.889 


578,009,537 


28.8617 


9.4091 


I . 20048 


2616.9 


544.979 


902 


813,604 


733,870,808 


30.0333 


9.6620 


I. 10865 


2833.7 


639,003 


834 


695.536 


580,093.704 


28.8791 


9.4129 


1 . 19904 


2620.1 


546.288 


903 


815,409 


736,314,327 


30.0500 


9.6656 


I. 10742 


2836.9 


640,421 


835 


697.22s 


582,182,875 


28.8964 


9.4166 


1.19760 


2623.2 


547.599 


904 


817.216 


738,763,264 


30.0666 


9.6692 


I. 10619 


2840.0 


641,840 


836 


698,896 


584,277,056 


28.9137 


9.4204 


I. 19617 


2626.4 


548.912 


905 


819,025 


741.217.623 


30.0832 


9.6727 


1.10497 


2843.1 


643,261 


837 


700,569 


586,376,253 


28.9310 


9.4241 


1-19474 


2629. s 


550,226 


go6 


820,836 


743.677.416 


30.0998 


9.6763 


1.10375 


2846.3 


644,683 


838 


702,244 


588,480,472 


28.9482 


9.4279 


1.19332 


2632.7 


551.541 


907 


822,649 


746,142,643 


30.1164 


9.6799 


I. 10254 


2849.4 


646,107 


839 


703.921 


590,589.719 


28.9655 


9.4316 


I -19189 


2635.8 


552,838 


908 


824,464 


748.613.312 


30.1330 


9.6834 


1.10132 


2852.6 


647,533 


840 


70S.600 


592,704,000 


28.9828 


9-4354 


I .19048 


2638.9 


554.177 


909 


826,281 


751.089.429 


30. 1496 


9.6870 


1.10011 


2855.7 


648,960 


841 


707,281 


s94.823.321 


29.0000 


9.4391 


1. 18906 


2642. I 


355.497 


910 


828,100 


733.571.000 


30.1662 


9.6905 


I .09890 


2858.8 


650,388 


842 


708,964 


596,947.688 


29.0172 


9.4429 


1.1876s 


2645.2 


556,819 


911 


829,921 


736,038,031 


30.1828 


9.6941 


1.09769 


2862.0 


651.818 


843 


710,649 


599,077.107 


29.0345 


9.4466 


1.18624 


2648.4 


558,142 


912 


831,744 


758,550,528 


30.1993 


9.6976 


1.09649 


286s. 1 


653.250 


844 


712,336 


601,211,584 


29.0517 


9.4503 


1.18483 


26SI.5 


559.467 


913 


833,569 


761,048,497 


30.2159 


9.7012 


1.09529 


2868.3 


654.684 


84s 


714.02s 


603,331,123 


29.0689 


9.4541 


1.18343 


2654.6 


560,794 


914 


835,396 


763,551.944 


30.2324 


9.7047 


1.09409 


2871.4 


656,118 


846 


715.716 


603,495.736 


29.0861 


9-4578 


1.18203 


2657.8 


562,122 


915 


837.225 


766,060,873 


30.2490 


9.7082 


1.09290 


2874.6 


657. 555 


847 


717.409 


607,645,423 


29.1033 


9.4615 


I. 18064 


2660.9 


563.452 


916 


839.056 


768,575.296 


30.2655 


9.7118 


1.09170 


2877.7 


658,993 


848 


719.104 


609,800,192 


29. 1204 


9.4652 


1.17925 


2664.1 


564.783 


917 


840,889 


771.095,213 


30.2820 


9.7153 


1.09031 


2880.8 


660,433 


849 


720,801 


611,960,049 


29-1376 


9.4690 


1.17786 


2667.2 


566,116 


918 


842,724 


773,620,632 


30.2985 


9.7188 


1.08932 


2884.0 


661,874 


850 


722,500 


614,125,000 


29.1548 


9.4727 


1.17647 


2670.4 


567.450 


919 


844,561 


776,151,559 


30.3150 


9.7224 


1.08814 


2887.1 


663,317 


851 


724,201 


616,295,051 


29.1719 


9.4764 


1-17509 


2673-5 


568,786 


920 


846,400 


778,688,000 


30.3315 


9.7259 


1.08696 


2890.3 


664,761 


8S2 


725.904 


618,470,208 


29. 1890 


9.4801 


1. 17371 


2676.6 


570,124 


921 


848,241 


781,229,961 


30.3480 


9.7294 


1.08378 


2893.4 


666,207 


853 


727,609 


620,650,477 


29.2062 


9.4838 


I -17233 


2679.8 


571.463 


922 


850,084 


783,777,448 


30.3645 


9.7329 


1.08460 


2896. 5 


667,634 


854 


729.316 


622,835.864 


29.2233 


9.4873 


1.17096 


2682.9 


572.803 


923 


851.929 


786,330,467 


30.3809 


9-7364 


1.08342 


2899.7 


669,103 


8SS 


731.025 


625,026,373 


29.2404 


9.4912 


1.16959 


2686.1 


574.146 


92.1 


853.776 


788,889,024 


30.3974 


9-7400 


I .08225 


2902 .8 


670,554 


856 


732,736 


627,222,016 


29.2575 


9.4949 


1.16822 


2689.2 


575.490 


925 


855,623 


791,453,125 


30.4138 


9-7435 


I. 08108 


2906.0 


672,006 


857 


734.449 


629,422,793 


29.2746 


9.4986 


1.16686 


2692.3 


576,835 


926 


857,476 


794,022,776 


30.4302 


9-7470 


1.07991 


2909. 1 


673,460 


858 


736,164 


631,628,712 


29.2916 


9.5023 


1.16550 


2695 - 5 


578,182 


927 


859,329 


796,597,983 


30.4467 


9.750S 


1.0787s 


2912.3 


674,91s 


859 


737.881 


633.839.779 


29.3087 


9.5060 


I. 16414 


2698.6 


579.530 


928 


861,184 


799,178,752 


30.4631 


9.7540 


1.07759 


291s. 4 


676,372 


860 


739,600 


636,036,000 


29.3258 


9-5097 


I . 16279 


2701 .8 


380,880 


929 


863,041 


801,765,089 


30.4795 


9.7575 


1.07643 


2918.5 


677.831 


861 


741.321 


638,277.381 


29.3428 


9-5134 


1.16144 


2704.9 


382,232 


930 


864,900 


804,357,000 


30.4959 


9.7610 


1.07527 


2921.7 


679,291 


862 


743.044 


640,503,928 


29.3598 


9.5171 


1 . 16009 


2708.1 


583,585 


931 


866,761 


806,954,491 


30.5123 


9.7645 


1.07411 


2924.9 


680,732 


863 


744.769 


642,735.647 


29.3769 


9.5207 


1.15875 


2711 .2 


584,940 


932 


868,624 


809,557,568 


30.5287 


9.7680 


1.07296 


2928.0 


682,216 


864 


746.496 


644,972,544 


29.3939 


9.5244 


I. 15741 


2714-3 


586,297 


933 


870,489 


812,166,237 


30.5450 


9-7715 


1.07181 


2931. 1 


683,680 


865 


748.225 


647,214,625 


29.4109 


9.5281 


1-15607 


2717. S 


587.635 


934 


872,356 


814,780,504 


30.5614 


9-7750 


1.07066 


2934-2 


685,147 


866 


749.956 


649,461,896 


29.4279 


9-5317 


1-15473 


2720.6 


589,014 


935 


874.223 


817,400,375 


30.5778 


9-7783 


1.06952 


2937-4 


686,613 


867 


751.689 


651,714.363 


29.4449 


9-5354 


I. 15340 


2723.8 


590,375 


936 


876,096 


820,025,856 


30.5941 


9.7819 


1.06838 


2940. 5 


688,084 


868 


753.424 


633.972.032 


29.4618 


9.5391 


1.15207 


2726.9 


591.738 


937 


877,969 


822,636,953 


30.6105 


9-7854 


1.06724 


2943.7 


689,555 


869 


755.161 


636,234,909 


29.4788 


9-5427 


1-15075 


2730.0 


593.102 


938 


879,844 


825.293,672 


30.6268 


9-7889 


1.06610 


2946.8 


691,028 


870 


756,900 


638,303.000 


29.4958 


9-5464 


1 . 14943 


2733-2 


594.468 


939 


881,721 


827,936,019 


30.6431 


9-7924 


1.06496 


2930.0 


692,502 


871 


758,641 


660,776,311 


29.5127 


9.5501 


I .14811 


2736-3 


595.835 


940 


883,600 


830,584,000 


30.6594 


9-7939 


1.06383 


2933.1 


693.978 


872 


760,384 


663,054.848 


29.5296 


9-5537 


1.14679 


2739-5 


597.204 


941 


885,481 


833.237,621 


30.6757 


9.7993 


1.06270 


2936.2 


69S.4SS 


873 


762,129 


665,338,617 


29.5466 


9-5574 


1.14548 


2742.6 


598.575 


942 


887,364 


835,896,888 


30.6920 


9.8028 


1.06157 


2959.4 


696,934 


874 


763,876 


667,627,624 


29.5635 


9.5610 


1.14416 


2745.8 


399.947 


943 


889,249 


838,561,807 


30.7083 


9.8063 


1.06045 


2962.5 


698,415 


875 


765,625 


669,921,875 


29.5804 


9.5647 


1 .14286 


2748.9 


601,320 


944 


891,136 


841,232,384 


30.7246 


9.8097 


1.05932 


2965.7 


699.897 


876 


767.376 


672,221,376 


29.5973 


9.5683 


1.1415s 


2752.0 


602,696 


945 


893,025 


843,908,625 


30.7409 


9.8132 


1.05820 


2968.8 


701,380 


877 


769.129 


674,526,133 


29.6142 


9.5719 


I. 14025 


2755-2 


604,073 


946 


894,916 


846.590,536 


30.7571 


9.8167 


1.05708 


2971.9 


702,865 


878 


770.884 


676,836,152 


29.6311 


9.5756 


1- 1389s 


2758.3 


603,451 


947 


896,809 


849,278,123 


30.7734 


9.8201 


1.05597 


2975.1 


704.352 


879 


772,641 


679,151.439 


29.6479 


9.5792 


1.13766 


2761.5 


606,831 


948 


898,704 


851,971,392 


30.7896 


9.8236 


1.05485 


2978.2 


705,840 


880 


774.400 


681,472,000 


29.6648 


9-5828 


1.13636 


2764.6 


608,212 


949 


900,601 


854.670.349 


30.8058 


9.8270 


1.05374 


2981.4 


707,330 


881 


776,161 


683,797.841 


29.6816 


9.586s 


I. 13507 


2767.7 


609.395 


950 


902,500 


857. 375. 000 


30.8221 


9.8303 


1.05263 


2984.5 


708,822 


882 


777.924 


686,128,968 


29.6985 


9.5901 


I .13379 


2770.9 


610,980 


951 


904,401 


860,085,351 


30.8383 


9.8339 


1.05152 


2987.7 


710.313 


883 


779.689 


688,465,387 


29.7153 


9-5937 


1.13250 


2774-0 


612,366 


952 


906,304 


862,801,408 


30.8543 


9.8374 


1.05042 


2990.8 


711,809 


884 


781,456 


690,807,104 


29.7321 


9-5973 


1.13122 


2777-2 


613. 7S4 


953 


908,209 


865,523,177 


30.8707 


9 . 8408 


1.04932 


2993.9 


713.306 


88s 


783,225 


693. 154.12s 


29.7489 


9.6010 


1.12994 


2780.3 


615.143 


954 


910,116 


868,250,664 


30.8869 


9 . 8443 


1.04822 


2997.1 


714.803 


886 


784,996 


695.506,456 


29.7658 


9.6046 


1.12867 


2783-3 


616,534 


955 


912,025 


870,983,875 


30.9031 


9.8477 


1.04712 


3000.2 


716.303 


887 


786,769 


697,864,103 


29.7825 


9.6082 


1.12740 


2786-6 


617.927 


956 


913.936 


873,722,816 


30.9192 


9.8311 


1.04603 


3003.4 


717.804 


888 


788,544 


700,227,072 


29.7993 


9.6118 


I. 12613 


2789.7 


619,321 


957 


91S.849 


876,467,493 


30.9354 


9.8346 


I . 04493 


3006.5 


719,306 


889 


790,321 


702,595,369 


29.8161 


9.6154 


I. 12486 


2792.9 


620,717 


958 


917.764 


879,217,912 


30.9516 


9.8380 


1.04384 


3009.6 


720,810 


890 


792.100 


704,969,000 


29.8329 


9.6190 


1.12360 


2796.0 


622,114 


959 


919,681 


881,974,079 


30.9677 


9.8614 


1.04275 


3012.8 


722,316 


891 


793.881 


707.347,971 


29.8496 


9.6226 


1.12233 


2799.2 


623,513 


960 


921,600 


884,736,000 


30.9839 


9.8648 


1.04167 


3015.9 


723,823 


892 


795.664 


709.732.288 


29.8664 


9.6262 


1.12108 


2802.3 


624,913 


961 


923.521 


887,503.681 


31.0000 


9.8683 


1.04058 


3019. I 


725,332 


893 


797.449 


712,121,957 


29.8831 


9.6298 


I. 11982 


2805.4 


626,315 


962 


925.444 


890,277,128 


31.0161 


9.8717 


1.03950 


3022.2 


726,842 


894 


799.236 


714.516,984 


29.8998 


9.6334 


1.11857 


2808.6 


627,718 


963 


927.369 


893,056,347 


31.0322 


9.8751 


1.03842 


3025.4 


728,334 


89s 


801,025 


716,917.375 


29.9166 


9.6370 


1.11732 


2811.7 


629,124 


964 


929,296 


893,041,344 


31.0483 


9.8785 


1.03734 


3028. 5 


729,867 


896 


802,816 


719.323.136 


29.9333 


9.6406 


1 . 11607 


2814.9 


630,530 


965 


931.225 


898,632,125 


31.0644 


9.8819 


1.03627 


3031.6 


731,382 


897 


804,609 


721,734.273 


29.9500 


9.6442 


1. 1 1483 


2818.0 


631.938 


966 


933.156 


901,428,696 


31.0805 


9.8854 


1.03320 


3034 -8 


732,899 



544 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 





Table 


26. — Squares, Cubes, Square Roots, Cube Roots, Reciprocals, Circumeerences and Circular Areas — {Continued) 


No. 


Square 


Cube 


Sq. 
root 


Cu. 
root 


icooX 
recip. 


No 


= Dia. 


No. 


Square 


Cube 


Sq. 
root 


Cu. 
root 


1000 X 
recip. 


No. = Dia. 


Circum. 


Area 


Circum. 


Area 


967 


935,089 


904,231,063 


31.0966 


9.8888 


1.03413 


3037-9 


734,417 


984 


968,256 


952,763,904 


31.3688 


9.9464 


I .01626 


3091.3 


760,466 


968 


937.024 


907,039,232 


31.1127 


9.8922 


1.03306 


3041. I 


735,937 


985 


970,225 


955,671,625 


31.3847 


9 . 9497 


I. 01523 


3094.5 


762,013 


969 


938,961 


909,853,209 


31.1288 


9.8956 


I. 03199 


3044.2 


737,458 


986 


972,196 


958,585,256 


3 1 . 4006 


9.9531 


I .01420 


3097.6 


763,561 


970 


940,900 


912,673,000 


31.1448 


9.8990 


1.03093 


3047.3 


738,981 


987 


974,169 


961,504,803 


31.4166 


9.9565 


1.01317 


3100.8 


765,111 


971 


942,841 


915,498,611 


31. 1609 


9.9024 


1.02987 


3050. S 


740,506 


988 


976,144 


964,430,272 


31.4325 


9.9598 


1.0121S 


3103.9 


766,662 


972 


944,784 


918,330,048 


31.1769 


9.9058 


I. 02881 


3053.6 


742,032 


989 


978,121 


967,361,669 


31.4484 


9.9632 


1.01112 


3107.0 


768,214 


973 


946,729 


921,167,317 


31.1929 


9.9092 


1.02775 


3056.8 


743,559 


990 


980,100 


970,299,000 


31.4643 


9.9666 


I.OIOIO 


3110.2 


769,769 


974 


948,676 


924,010,424 


31.2090 


9.9126 


1.02669 


3059.9 


745,088 


991 


982,081 


973,242,271 


31.4802 


9 • 9699 


1.00908 


3113.3 


771,32s 


975 


950,625 


926,859,375 


31.2250 


9.9160 


1.02564 


3063.1 


746,619 


992 


984,064 


976,191,488 


31.4960 


9.9733 


1.00806 


3116.S 


772,882 


976 


952,576 


929,714,176 


31.2410 


9.9194 


1.02459 


3066.2 


748,151 


993 


986,049 


979,146,657 


31.S119 


9.9766 


I. 0070s 


3119.6 


774,441 


977 


954,529 


932,574,833 


31.2570 


9.9227 


I. 02354 


3069.3 


749,68s 


994 


988,036 


982,107,784 


31.5278 


9.9800 


1.00604 


3122.7 


776,002 


978 


956,484 


935,441,352 


31.2730 


9.9261 


I .02249 


3072. s 


751,221 


995 


990,025 


985,074,875 


31.5436 


9.9833 


1.00503 


3125.9 


777,564 


979 


958,441 


938,313,739 


31.2890 


9.9295 


I. 02145 


3075.6 


752,758 


996 


992,016 


988,047,936 


31.5595 


9.9866 


1.00402 


3129.0 


779,128 


980 


960,400 


941,192,000 


31.3050 


9.9329 


I. 02041 


3078.8 


754,296 


997 


994,009 


991,026,973 


31.5753 


9.9900 


I. 00301 


3132.2 


780,693 


981 


962,361 


944,076,141 


31.3209 


9.9363 


I. 01937 


3081.9 


755,837 


998 


996,004 


994,011,992 


31.5911 


9.9933 


1.00200 


3135.3 


782,260 


982 


964.324 


946,966,168 


31.3369 


9.9396 


I. 01833 


3085.0 


757,378 


999 


998,001 


997,002,999 


31.6070 


9.9967 


I .00100 


3138.5 


783.828 


983 


966,289 


949,862,087 


31.3528 


9.9430 


I. 01729 


3088.2 


758,922 























Table 27. — Areas and 


CiRCUMEERENCES OF CIRCLES — DECIMAL DIVISIONS 






Diam- 


Area 


Circum- 


Diam- 


Area 


Circum- 


Diam- 


Area 


Circum- 


Dia- 


Area 


Circum- 


eter 


ference 


eter 


ference 


eter 


ference 


eter 


ference 


0.0 






.5 


15.9043 


14.1372 ; 


9.0 


63.6173 


28.2743 


.5 


143.1388 


42.411S 


". I 


.007854 


.31416 


.6 


16.6190 


14.4513 


. 1 


65.0388 


28.5885 


.6 


145.2672 


42.7257 


.2 


.031416 


.62832 


■ 7 


17.3494 


14.7655 


. 2 


66.4761 


28.9027 


.7 


147.4114 


43.0398 


.3 


.070686 


.94248 


.8 


18.0956 


15.0796 


.3 


67.9291 


29. 2168 


.8 


149.5712 


43.3540 


■4 


.12566 


1.2566 


■ 9 


18.8574 


15.3938 


■4 


69.3978 


29.5310 


.9 


151.7468 


43.6681 


.5 


.19635 


1.5708 


5-0 


19.6350 


15.7080 


.5 


70.8822 


29.8451 


14.0 


153.9380 


43.9823 


-6 


.28274 


1.8850 


. I 


20.4282 


16.0221 


.6 


72.3823 


30.1593 


. I 


156. 1450 


44.296s 


.7 


■38485 


2.1991 


.2 


21.2372 


16.3363 


.7 


73.8981 


30.4734 


. 2 


158.3677 


44.6106 


.8 


.50265 


2.5133 


.3 


22.0618 


16.6504 


.8 


75.4296 


30.7876 


.3 


160.6061 


44.9248 


.9 


.63617 


2.8274 


■4 


22.9022 


16.9646 


.9 


76.9769 


31. 1018 


.4 


162.8602 


45.2389 


I.O 


.7854 


3.1416 


.5 


23.7583 


17.2788 


10. 


78.5398 


31.4159 


.5 


16s. 1300 


45.5531 


. I 


.9503 


3.4558 


.6 


24.6301 


17.5929 


. I 


80.1x85 


31.7301 


.6 


167.4155 


45.8673 


.2 


1. 1310 


3.7699 


.7 


25.5176 


17.9071 


.2 


81.7128 


32.0442 


.7 


169.7167 


46. 1814 


.3 


1.3273 


4.0841 


.8 


26.4208 


18.2212 


.3 


83.3229 


32.3584 


.8 


172.0336 


46.4956 


■4 


1.5394 


4.3982 


.9 


27.3397 


18.5354 


•4 


84.9487 


32.6726 


.9 


174.3662 


46.8097 


.5 


1.7671 


4.7124 


6.0 


28.2743 


18.8496 


•5 


86.5901 


32.9867 


15.0 


176.7146 


47.1239 


.6 


2.0106 


5.026s 


. I 


29.2247 


19.1637 


.6 


88.2473 


33 1009 


. I 


179.0786 


47.4380 


.7 


2.2698 


5.3407 


. 2 


30.1907 


19.4779 


.7 


89.9202 


33.0150 


. 2 


181.4584 


47.7S22 


.8 


2 . 5447 


5.6S49 


.3 


31.172s 


19.7920 


.8 


91.6088 


33.9292 


.3 


183.8539 


48 . 0664 


.9 


2.8353 


5 . 9690 


.4 


32.1699 


20. 1062 


.9 


93.3132 


34.2434 


.4 


186.2650 


48.380s 


2.0 


3.1416 


6.2832 


.5 


33.1831 


20.4204 


11. 


95.0332 


34.5575 


.5 


188.6919 


48 . 6947 


. I 


3.4636 


6.5973 


.6 


34.2119 


20.7345 


. I 


96.7689 


34.8717 


6 


191.1345 


49.0088 


.2 


3.8013 


6.911S 


•7 


35.2565 


21.0487 


. 2 


98.5203 


35.1858 


7 


193.5928 


49.3230 


.3 


4.1548 


7.2257 


.8 


36.3168 


21.3628 


.3 


100.2875 


35.5000 


8 


196.0668 


49.6372 


.4 


4.5239 


7.5398 


• 9 


37.3928 


21.6770 


•4 


102.0703 


35.8142 


.9 


198.556s 


49.9513 


.5 


4.9087 


7.8540 


7.0 


38.4845 


21.9911 


.5 


103.8689 


36.1283 


16.0 


201 .0619 


50.26SS 


.6 


5.3093 


8.1681 


. I 


39.5919 


22.3053 


.6 


105.6832 


36.442s 


. I 


203.5831 


50.5796 


.7 


5.7256 


8.4823 


.2 


40.71SO 


22.619s 


.7 


107.5132 


36.7566 


. 2 


206. 1199 


50.8938 


.8 


6.1575 


8.7965 


.3 


41.8539 


22.9336 


.8 


109.3588 


37.0708 


.3 


208.6724 


SI. 2080 


.9 


6.6052 


9. I106 


•4 


43.0084 


23.2478 


.9 


III . 2202 


37.3850 


.4 


211.2407 


51.5221 


3.0 


7.0686 


9.4248 


.5 


44. 1786 


23.5619 


12.0 


113.0973 


37.6991 


.5 


213.8246 


51.8386 


. I 


7.5477 


9.7389 


.6 


45.3646 


23.8761 


. I 


114.9901 


38.0133 


.6 


216.4243 


52.1S04 


. 2 


8.042s 


10.0531 


•7 


46 . 5663 


24.1903 


.2 


116.8987 


38.3274 


.7 


2190397 


52.4646 


.3 


8.5530 


10.3673 


.8 


47.7836 


24.5044 


.3 


118.8229 


38.6416 


.8 


221.6708 


52.7788 


• 4 


9.0792 


10.6814 


.9 


49. 1067 


24.8186 


.4 


120.7628 


38.9557 


.9 


224.3176 


53.0929 


.5 


9.6211 


10.9956 


8.0 


50.2655 


25.1327 


.5 


122.7185 


39.2699 


17.0 


226.9801 


53.4071 


.6 


10. 1788 


11.3097 


. I 


51.5300 


25.4469 


.6 


124.6898 


39.5841 


. 1 


229.6583 


53.7212 


.7 


10.7521 


11.6239 


. 2 


52.8102 


25.7611 


.7 


126.6769 


39.8982 


.2 


232.3522 


54.0354 


8 


II.3411 


II. 9381 


.3 


54. 1061 


26.0752 


.8 


128.6796 


40. 2124 


.3 


2350618 


54.3496 


.9 


11.9459 


12.2522 


■4 


55.4177 


26.3894 


.9 


130.6981 


40.526s 


■ 4 


237.7871 


54.6637 


4.0 


12.5664 


12.5664 


.5 


56.7450 


26.703s 


13.0 


132.7323 


40.8407 


.5 


240.5282 


54.9779 


. I 


13.202s 


12. 880s 


.6 


58.0880 


27.0177 


. 1 


134.7822 


41.1549 


.6 


243.2849 


55.2920 


.2 


13.8544 


13.1947 


.7 


59.4468 


27.3319 


.2 


136.8478 


41.4690 


.7 


246.0574 


55.6062 


.3 


14.5220 


13.5088 


.8 


60.8212 


27 .6460' 


.3 


138.9291 


41.7832 


.8 


248.8456 


55.9203 


.4 


15.2053 


13.8230 


.9 


62.2114 


27.9602 


.4 


141. 0261 


42.0973 


.9 


251.6494 


56.2345 



MATHEMATICAL TABLES 



545 



Table 27. — Areas and Circumferences of Circles — Decimal Divisions — (Conlinued) 



Diam- 
eter 


Area 


Circum- 
ference 


. Diam- 
eter 


Area 


Circum- 
ference 


Diam- 
eter 


Area 


Circum- 
ference 


Diam- 
eter 


Area 


Circum- 
ference 


18.0 


254.4690 


36.5487 


.5 


471-4352 


76.9690 


310 


754-7676 


97.3894 


-3 


I 104. 4662 


117 . 8097 


.1 


257.3043 


56.8628 


.6 


473.2916 


77.2832 


. I 


759-6450 


97.7035 


.6 


mo. 3643 


118. 1239 


. 2 


260. 1553 


37. 1770 


.7 


479- 1636 


77-5973 


. 2 


764.5380 


98.0177 


.7 


1116. 2786 


118.4380 


• 3 


263.0220 


57-4911 


.8 


483-0313 


77.9113 


■ 3 


769.4467 


98.3319 


.8 


1122.2083 


118.7322 


• 4 


265.9044 


57-8053 


.9 


486.9547 


78.2257 


.4 


774-3712 


98.6460 


■9 


1128.1S38 


119.0664 


• S 


268.802s 


58.1193 


25.0 


490.8739 


78.5398 


.5 


779.3113 


98.9602 


38.0 


1134.1149 


119.3805 


.6 


271.7163 


58.4336 


. I 


494. 8087 


78.8540 


.6 


784. 2672 


99.2743 


. I 


1 1 40. 09 1 8 


119.6947 


.7 


274.6459 


58.7478 


. 2 


498.7592 


79. 1681 


.7 


789.2388 


99.588s 


. 2 


I146.0844 


120.0088 


.8 


277.5911 


59.0619 


.3 


502.725s 


79.4823 


.8 


794. 2260 


99.9026 


.3 


1132.0927 


120.3230 


■ 9 


280.5521 


39.3761 


-4 


506.707s 


79.7963 


■ 9 


799.2290 


100.2168 


-4 


1138. I167 


120.6372 


19.0 


283.5287 


59.6903 


.5 


510.7052 


80. I106 


32.0 


804.2477 


100.5310 


-5 


1164.1564 


120.9513 


. I 


286.5211 


60.0044 


.6 


S14-7183 


80.4248 


. I 


809. 2821 


100.8451 


.6 


1170.2118 


121.265s 


.2 


289.5292 


60.3186 


-7 


318.7476 


80.7389 


. 2 


814.3322 


I0I.IS93 


.7 


1176. 2830 


121.5796 


• 3 


292.5530 


60.6327 


.8 


522.7924 


81.0531 


.3 


819.3980 


101.4734 


.8 


I182.3698 


121.8938 


• 4 


29s. 592s 


60.9469 


-9 


526.8529 


81.3672 


-4 


824.4796 


loi .7876 


.9 


I188.4723 


122.2080 


■ 5 


298.6477 


61.2611 


26.0 


530.9292 


81.6814 


.5 


829.5768 


102. 1018 


39.0 


1194.3906 


I22.S22I 


.6 


301.7186 


61.3732 


. I 


533.0211 


81.9936 


.6 


834-6897 


102.4159 


. 1 


1200.7246 


122.8363 


■ 7 


304.8052 


61.8894 


. 2 


339-1287 


82.309T 


.7 


839-8184 


102.7301 


.2 


1206.8742 


123.1504 


.8 


307.907s 


62.203s 


.3 


343-2521 


82.6239 


.8 


844.9628 


103.0442 


.3 


1213.0396 


123.4646 


■ 9 


311.0255 


62.S177 


-4 


347.3911 


82.9380 


■ 9 


850. 1229 


103.3384 


.4 


1219.2207 


123.7788 


20.0 


314-1593 


62.8319 


-3 


351.5459 


83.2522 


33.0 


855.2986 


103.6726 


.3 


1225.417s 


124.0929 


. I 


317.3087 


63. 1460 


.6 


533-7163 


83.5664 


. I 


860.4902 


103.9867 


.6 


1231 .6300 


124.4071 


.2 


320.4739 


63.4602 


• 7 


559.9025 


83.8805 


. 2 


865.6973 


104.3009 


.7 


1237.8582 


124. 7212 


• 3 


323.6547 


63.7743 


.8 


564.1044 


84.1947 


.3 


870.9202 


104.6150 


.8 


1244. I02I 


123.0334 


• 4 


326.8513 


64.0885 


.9 


368.3220 


84.5088 


■4 


876.1588 


104.9292 


.9 


1250. 3617 


125-3493 


• S 


330.0636 


64.4026 


27.0 


372.5553 


84.8230 


.5 


881.4131 


105. 2434 


40.0 


1256. 6371 


125.6637 


.6 


333.2916 


64.7168 


. I 


576.8043 


85.1372 


.6 


886.6831 


105.5573 


. I 


1262. 9281 


125.9779 


.7 


336.5353 


65.0310 


. 2 


581.0690 


85.4513 


.7 


891.9688 


103.8717 


.2 


1269.2348 


126. 2920 


.8 


339.7947 


6S.3451 


.3 


585.3494 


85.7655 


.8 


897.2703 


106. 1858 


.3 


1273.3373 


126.6062 


.9 


343.0698 


63-6593 


•4 


589.6455 


86.0796 


.9 


902.5874 


106.5000 


.4 


1281.8955 


126.9203 


21.0 


346.3606 


65-9734 


-3 


393-9574 


86.3938 


34.0 


907.9203 


106.8142 


.5 


1288. 2493 


127.234s 


.1 


349.6671 


66.2876 


.6 


398.2849 


86.7080 


. I 


913.2688 


107.1283 


.6 


1294. 6189 


127.3487 


.2 


332.9893 


66.6018 


.7 


602.6282 


87.0221 


. 2 


918.6331 


107.4423 


.7 


1301.0042 


127.8628 


.3 


356.3273 


66. 9 1 59 


.8 


606.9871 


87.3363 


.3 


924.0131 


107.7366 


.8 


1307.4052 


128. 1770 


.4 


3S9.6809 


67.2301 


.9 


611. 3618 


87.6504 


-4 


929.4088 


108.0708 


.9 


I3I3.82I9 


128.4911 


• 5 


363.0503 


67-5442 


28.0 


615.7522 


87.9646 


.3 


934.8202 


108.3846 


41.0 


1320.2543 


128.8053 


.6 


366.4354 


67-8584 


. I 


620. 1582 


88.2788 


.6 


940.2473 


108. 6991 


. I 


1326.7024 


129.1193 


•7 


369.8361 


68. 1726 


, 2 


624.5800 


88.5929 


.7 


945 6901 


109.0133 


.2 


1333. 1663 


129.4336 


.8 


373.2526 


68.4867 


.3 


629.017s 


88.9071 


.8 


951 . i486 


109.3274 


.3 


1339.6458 


129.7478 


■9 


376.6848 


68.8009 


■4 


633.4707 


89.2212 


.9 


956.6228 


109.6416 


.4 


1346. I4IO 


130.0619 


22.0 


380. 1327 


69. 1150 


.5 


637.9397 


89.5354 


35.0 


962. I127 


109.9357 


.3 


1332.6520 


130.3761 


. I 


383.5963 


69.4292 


.6 


642.4243 


89.8495 


. I 


967.6184 


110.2699 


.6 


1359.1786 


130.6903 


.2 


387.0756 


69.7434 


-7 


646.9246 


90.1637 


.2 


973-1397 


110. 5841 


.7 


1365. 7210 


131 .0044 


• 3 


390.5707 


70.037s 


.8 


651.4406 


90.4779 


.3 


978.6768 


110.8982 


.8 


1372.2791 


131. 3186 


• 4 


394-0814 


70.3717 


.9 


655-9724 


90.7920 


.4 


984.2296 


1 1 1 . 2 1 24 


.9 


1378.8529 


131.6327 


.5 


397.6078 


70.6858 


29.0 


660.5199 


91. 1062 


.5 


989.7980 


III .5265 


42. 


1383.4424 


131.9469 


.6 


401 . 1500 


71.0000 


. I 


665.0830 


91.4203 


.6 


995.3822 


III .8407 


. 1 


1392.0476 


132.2611 


.7 


404.7078 


71.3142 


. 2 


669.6619 


91.7343 


.7 


1000. 9821 


112. 1549 


. 2 


1398.6685 


132.3752 


.8 


408. 2814 


71-6283 


-3 


674-2565 


92.0487 


.8 


1006.5977 


112.4690 


.3 


1405.3051 


132.8894 


.9 


411.8706 


71-9423 


-4 


678.8668 


92.3628 


.9 


1012. 2290 


112.7832 


.4 


I4II.9574 


133.203s 


23.0 


415.4756 


72.2566 


-3 


683.4927 


92.6770 


36.0 


1017. 8760 


113.0973 


-5 


1418.6254 


133.3177 


. I 


419.0963 


72.5708 


.6 


688.134s 


92.9911 


. 1 


1023.5387 


113-4113 


.6 


1425.3092 


133.831S 


. 2 


422.7327 


72.8849 


-7 


692.7919 


93.3053 


.2 


1029. 2172 


113.7257 


•7 


1432.0086 


134. 1460 


.3 


426.3848 


73.1991 


.8 


697.4650 


93.6193 


.3 


1034-9113 


114.0398 


.8 


1438.7238 


134.4602 


•4 


430.0526 


73.5133 


.9 


702.1538 


93.9336 


.4 


1040. 6211 


114.3540 


.9 


1443.4546 


134.7743 


• S 


433.7361 


73-8274 


30.0 


706.8583 


94.2478 


• 3 


1046.3467 


114. 6681 


A3-0 


1452 .2012 


133.0885 


.6 


437.4354 


74-1416 


. I 


711.3786 


94.5619 


.6 


1052.0880 


114.9823 


. I 


1458.9635 


133.4026 


■ 7 


441.IS03 


74-4557 


. 2 


716.3145 


94.8761 


. 7 


1057.8449 


115.2963 


. 2 


1465.7413 


133.7168 


.8 


444.8809, 


74-7699 


• 3 


721.0662 


95.1903 


.8 


1063. 6176 


115-6106 


.3 


1472.5352 


136.0310 


• 9 


448.6273 


75.0841 


■4 


723.8336. 


95.5044 


.9 


1069.4060 


115.9248 


•4 


1479.3446 


136.3431 


24.0 


432.3893 


75.3982 


■5 


730.6167 


93.8186 


37.0 


1075.2101 


116.2389 


.3 


i486. 1697 


136.6593 


. I 


456. 1671 


73.7124 


.6 


733.4154 


96.1327 


. I 


1081.0299 


116. 5531 


.6 


1493.0105 


136.9734 


. 2 


459.9606 


76.026s 


. 7 


740.2299 


96.4469 


. 2 


1086.8654 


116.8672 


.7 


1499.8670 


137.2876 


■ 3 


463.7698 


76.3407 


.8 


745.0601 


96.7611 


.3 


1092.7166 


117. 1814 


.8 


1306.7393 


137.6018 


•4 


467 . 5046 


76,6549 


.9 


749.9060 


97.0732 


.4 


1098.5835 


117.4956 


■ 9 


1513.6272 


137.9139 



35 



646 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 







Table 27. 


—Areas and Circumterences of Circles- 


-Decimal Divisions — {Continued) 






Diam- 
eter 


Area 


Circum- 
ference- 


Diam- 
eter 


Area 


Circum- 
ference 


Diam- 
eter 


Area 


Circum- . 
ference 


Diam- 
eter 


Area 


Circum- 
ference 


44 -o 


1520.5308 


138. 2301 


.5 


2002.9617 


158.6504 


S7.0 


2551. 7S86 


179.0708 


.5 


3166.9217 


199.4911 


. I 


1527.4502 


138.5442 


.6 


2010.9020 


158.9646 


.1 


2560.7200 


179.3849 


.6 


3176. 9041 


199.8053 


. 2 


1534.3853 


138.8584 


.7 


2018.8581 


159.2787 


.2 


2569.6971 


179.6991 


.7 


3186.9023 


200.1195 


• 3 


1541.3360 


139.1726 


.8 


2026.8299 


159.5929 


.3 


2578.6899 


180.0133 


.8 


3196. 9161 


200.4336 


•4 


1548.3025 


139.4867 


.9 


2034.8174 


159.9071 


■4 


2587.6984 


180.3274 


.9 


3206.9456 


200.7478 


• 5 


1555.2847 


139.8009 


51.0 


2042.8206 


160. 2212 


.5 


2596.7227 


180.6416 


64.0 


3216.9909 


201 .0620 


.6 


1562. 2826 


140. 1150 


. I 


2050.839s 


160.5354 


.6 


2605.7626 


180.9557 


. I 


3227.0518 


201.3761 


• 7 


1569. 2962 


140.4292 


. 2 


2058.8742 


160.849S 


.7 


2614.8182 


181.2699 


.2 


3237.1285 


201.6902 


.8 


1576.325s 


140.7434 


■ 3 


2066.924s 


161. 1637 


.8 


2623.8896 


181.5841 


.3 


3247.2209 


202.0044 


• 9 


1583. 370s 


141.0575 


• 4 


2074.990s 


161.4779 


.9 


2632.9766 


181.8982 


.4 


3257.3289 


202.3186 


43. o 


1590.4313 


141. 3717 


.5 


2083.0723 


161.7920 


58.0 


2642 .0794 


182.2124 


.s 


3267.4527 


202.6327 


. I 


1S97.5077 


141 .6858 


.6 


2091 . 1697 


162. 1062 


. I 


2651. 1979 


182.5265 


.6 


3277.5922 


202.9469 


.2 


1604.5999 


142.0000 


.7 


2099.2829 


162.4203 


.2 


2660.3321 


182.8407 


.7 


3287.7474 


203.2610 


■ 3 


1611.7077 


142.3141 


.8 


2107.4118 


162.734s 


3 


2669.4820 


183.1549 


.8 


3297.9183 


203.3732 


• 4 


1618.8313 


142.6283 


.9 


2115.5563 


163.0487 


4 


2678.647s 


183.4690 


.9 


3308.1049 


203.8894 


.5 


1625.9705 


142.9425 


52.0 


2123. 7166 


163.3628 


S 


2687.8289 


183.7832 


65.0 


3318.3072 


204.203s 


.6 


1633.1255 


143.2566 


. I 


2131.8926 


163.6770 


.6 


2697.0259 


184.0973 


. I 


3328.5253 


204.5177 


.7 


1640. 2962 


143.5708 


. 2 


2140.0843 


163.9911 


* .7 


2706.2386 


184.4115 


. 2 


3338.7590 


.204.8318 


.8 


1647.4826 


143.8849 


.3 


2148.2917 


164.3053 


.8 


2715.4670 


184.7256 


.3 


3349.008s 


20s. 1460 


• 9 


1654.6847 


144.1991 


.4 


2156.5149 


164.619s 


.9 


2724.7112 


185.0398 


.4 


3359.2736 


205.4602 


46.0 


1661 .9025 


144. 5 133 


.5 


2164.7537 


164.9336 


59.0 


2733.9710 


185.3540 


.5 


3369.554s 


205.7743 


. I 


1669. 1360 


144.8274 


.6 


2173.0082 


165. 2478 


.1 


2743.246s 


185.6681 


.6 


3379.8510 


206.0885 


. 2 


1676.3852 


145. 1416 


.7 


2181.278s 


165.5619 


. 2 


2752.5378 


185.9823 


.7 


3390.1633 


206.4026 


.3 


1683.6502 


145.4557 


.8 


2189.5644 


165.8761 


■ 3 


2761.8448 


186.2964 


.8 


3400.4913 


206.7168 


•4 


1690.9308 


145.7699 


.9 


2197.8661 


166. 1903 


.4 


2771.167s 


186.6106 


.9 


3410.8350 


207.0310 


.5 


1698 .2272 


146.0841 


53.0 


2206. 1834 


166.5044 


■ S 


2780.5058 


186.9248 


66.0 


3421.1944 


207.3451 


.6 


1705.5392 


146.3982 


. I 


2214.5165 


166.8186 


.6 


2789.8599 


187.2389 


. 1 


3431.569s 


207.6593 


•7 


1712.8670 


146.7124 


.2 


2222.8653 


167.1327 


.7 


2799.2297 


187.5531 


. 2 


3441.9603 


207.9734 


.8 


1720. 21OS 


147.026s 


.3 


2231 . 2298 


167.4469 


.8 


2808.6152 


187.8672 


.3 


3452.3669 


208.2876 


■ 9 


1727.5696 


147.3407 


.4 


• 2239.6100 


167.7610 


.9 


2818.0165 


188.1814 


.4 


3462.7891 


208.6017 


47.0 


1734.944s 


147.6549 


.5 


2248.0059 


168.1752 


60.0 


2827.4334 


188.4956 


.5 


3473.2270 


208.9159 


. I 


1742. 3351 


147.9690 


.6 


2256.4175 


168.3894 


. I 


2836.8660 


188.8097 


.6 


3483.6807 


209.2301 


.2 


1749.7414 


148.2832 


.7 


2264.8448 


168.7035 


.2 


2846.3143 


189. 1239 


.7 


3494. ISOO 


209.5442 


.3 


1757. 1634 


148.5973 


.8 


2273.2879 


169.0177 


•3 


2855.7784 


189.4380 


.8 


3504.6351 


209.8584 


.4 


1764. 6012 


148.9115 


.9 


2281.7466 


169.3318 


•4 


2865.2582 


189.7522 


.9 


3515.1359 


210. 1725 


• S 


1772. OS46 


149.2257 


54.0 


2290.2210 


169.6460 


.5 


2874.7536 


190.0664 


67.0 


3523.6524 


210.4867 


.6 


1779.5237 


149.5398 


. I 


2298. 7112 


169.9602 


.6 


2884.2648 


190.380s 


. I 


3336.1845 


210.8009 


■7 


1787.0086 


149.8540 


. 2 


2307 .2X71 


170.2743 


.7 


2893.7917 


190.6947 


. 2 


3546.7324 


211. 1150 


.8 


1794.5091 


150. 1681 


•3 


2315.7386 


170.588s 


.8 


2903.3343 


191 .0088 


.3 


3557.2960 


211.4292 


•9 


1802.0254 


150.4823 


■ 4 


2324.2759 


170.9026 


.9 


2912.8925 


191.3230 


.4 


3567.8754 


211.7433 


48.0 


1809.5574 


150.7964 


• 5 


2332.8289 


171.2168 


61.0 


2922.4666 


191.6372 


.5 


3378.4704 


2I2.0S7S 


. I 


1817. 1050 


151. I106 


.6 


2341.3976 


171.5310 


. I 


2932.0563 


191.9513 


6 


3589. 0811 


212.3717 


. 2 


1824.6684 


ISI.4248 


.7 


2349.9820 


171. 8451 


.2 


2941.6617 


192.265s 


7 


3399.7075 


212.6858 


• 3 


1832.2475 


151.7389 


.8 


2358.5821 


172.1593 


.3 


2951.2828 


192.5796 


.8 


3610.3497 


213.0000 


■4 


1839.8423 


152.0531 


.9 


2367.1979 


172.4734 


.4 


2960.9196 


192.8938 


.9 


3621.007s 


213.3141 


• 5 


1847.4528 


152.3672 


55.0 


2375.8294 


172.7876 


.5 


2970.5722 


193.2079 


68.0 


3631. 6811 


213.6283 


.6 


1855.0790 


IS2.6814 


.1 


2384.4767 


173. 1018 


.6 


2980. 2404 


193.5221 


. I 


3642.3704 


213.9425 


.7 


1862. 7210 


152.9956 


.2 


2393 1396 


173.41S9 


.7 


2989.9244 


193.8363 


. 2 


3633.0754 


214.2566 


8 


1870.3786 


153.3097 


.3 


2401.8183 


173.7301 


.8 


2999.6241 


194.1504 


.3 


3663.7960 


214.5708 


9 


1878. 0519 


153.6239 


■ 4 


2410.5126 


174.0442 


.9 


3009.3394 


194.4646 


• 4 


3674.5324 


214.8849 


49.0 


1885.7410 


153.9380 


.5 


2419.2227 


174.3584 


62.0 


3019.0705 


194.7787 


.5 


3685.284s 


21s. 1991 


. I 


1893.4457 


154.2522 


.6 


2427.9485 


174.6726 


. 1 


3028.8173 


195.0929 


.6 


3696.0523 


215. 5133 


. 2 


1901 . 1662 


154.5664 


.7 


2436.6899 


174.9867 


. 2 


3038.5798 


195.4071 


.7 


3706.8359 


215.8274 


• 3 


1908.9024 


154. 880s 


.8 


2445.4417 


175.3009 


.3 


3048.3580 


195.7212 


.8 


3717.6351 


216. 1416 


•4 


1916.6543 


155.1947 


.9 


2454.2200 


175.6150 


.4 


3058.1519 


196.0354 


.9 


3728.4500 


216.4556 


• S 


1924. 4218 


155.S088 


56.0 


2463.0086 


175.9292 


.5 


3067.9616 


196.349s 


69.0 


3739.2807 


216.7699 


.6 


1932. 2051 


155.8230 


. I 


2471.8129 


176.2433 


.6 


3077.7869 


196.6637 


. 1 


3730.1270 


217.0841 


.7 


1940. 0041 


156.1372 


.2 


2480.6330 


176.5575 


.7 


3087.6279 


196.9779 


. 2 


3760.9891 


217.3982 


.8 


1947. 8189 


156.4S13 


.3 


2489.4687 


176.8717 


.8 


3097.4847 


197. 2920 


.3 


3771.8668 


217.7124 


• 9 


1955.6493 


156.765s 


• 4 


2498.3201 


177.1858 


.9 


3107.3571 


197.6062 


.4 


3782.7603 


218.0265 


50.0 


1963.4954 


157.0796 


.5 


2507.1873 


177.5000 


63.0 


3117.2453 


197.9203 


.5 


3793.6693 


218.3407 


. I 


1971.3572 


157.3938 


.6 


2516. 0701 


177.8141 


. 1 


3127.1492 


198.2345 


.6 


3804.5944 


218.6548 


.2 


1979.2348 


157 .7080 


.7 


2524.9687 


178.1283 


.2 


3137.0687 


198.5487 


. 7 


3813.5330 


218.9690 


• 3 


1987. 1280 


158 .0221 


.8 


2533.8830 


178.4425 


.3 


3147.0040 


198.8628 


.8 


3826.4913 


219.2832 


■4 


1995.0370 


158.3363 


■9 


2542.8129 


178.7566 


.4 


3156.9550 


199.1770 


■9 


3837.4633 


219.3973 



MATHEMATICAL TABLES 



547 







Table 27 


. — Areas 


AND Circumferences of Circles- 


-Decimal Divisions — ( 


Continued) 






Diam- 
eter 


Area 


Circum- 
ference 


Diam- 
eter 


Area 


Circum- 
ference 


Diam- 
eter 


Area 


Circum- 
ference 


Diam- 
eter 


Area 


Circum- 
ference 


70.0 
. I 
.2 
.3 

.4 


3848. 4SI0 
3859.4544 
3870.4735 
3881. S084 
3892.5589 


219.911S 
220. 2256 
220.5398 
220.8540 
221. 1681 


.5 
.6 
.7 
.8 
.9 


4596.3464 
4608.3708 
4620. 4110 
4632.4668 
4644.5384 


240.3318 
240.6460 
240.9602 
241.2743 
241.5885 


83.0 

. I 
.2 
.3 
-4 


S410-6079 
S423-6534 
5436.7146 
5449.7914 
5462.8840 


260.7522 
261.0663 
261.380s 
261.6947 
262.0088 


.5 
.6 
■ 7 
.8 
• 9 


6291.2356 
6305.3021 
6319-3843 
6333.4822 
6347.5958 


281. 1725 
281.4867 
281.8009 
282. iiso 
282.4292 


.5 

.6 

•7 
.8 
-9 


3903.6252 

3914-7072 
3925.8048 
3936.9182 
3948.0473 


221.4823 
221.7964 
222. 1106 
222.4248 
222.7389 


77.0 
. 1 
.2 
.3 
.4 


4656.6257 
4668.7287 
4680. 8474 
4692.9818 
4705. 1319 


241.9026 
242. 2168 
242.5310 
242.84S1 
243.1592 


.5 
.6 

-7 
.8 
-9 


5475.9923 
5489. 1163 
5502.2560 
551S.4115 
5528.5826 


262.3230 
262.6371 
262.9513 
263.265s 
263.5796 


90.0 
. I 

.2 

-3 

-4 


6361.7251 
6375.8701 
6390.0308 
6404.2073 
6418.3994 


282.7433 
283.0575 ■ 
283.3717 
283.6858 
284.0000 


71.0 
. I 

.2 

• 3 

•4 


3959.1921 
3970.3526 
3981.5288 
3992.7208 
4003.9284 


223.0531 
223.3672 
223.6814 
223.9956 
224.3097 


-5 
.6 

.7 
.8 
.9 


4717.2977 
4729.4792 
4741-6765 
4753-8894 
4766. 1180 


243.4734 
243.7876 
244. 1017 
244.4159 
244.7301 


84.0 
. I 
. 2 
.3 
-4 


5541.7694 
5554-9720 
5568.1902 
5581.4242 
5594.6738 


263.8938 
264.2079 
264.5221 
264.8363 
26s. 1504 


5 
6 

7 
.8 
■ 9 


6432.6073 
6446.8308 
6461. 0701 
6475.3251 
6489.5958 


284.3141 
284.6283 
284.942s 
285-2566 
285.5708 


■ S 
.6 

■ 7 
.8 
.9 


4015. 1517 
4026.3908 
4037.6455 
4048.9160 
4060.2022 


224.6239 
224.9380 
225. 2522 
225.5664 
225.8805 


78.0 
. I 
. 2 
■ 3 

-4 


4778.3624 
4790.622s 
4802.8982 
4815. 1897 
4827.4969 


245.0442 
245.3584 
245-6725 
245-9867 
246.3009 


-5 
.6 
.7 
.8 
■ 9 


5607.9392 
5621.2203 ■ 
5634-5171 
5647.8296 
S66i. 1578 


265.4646 
265.7787 
266.0929 
266.4071 
266. 7212 


91.0 
. 1 

.2 
.3 

■ 4 


6503.8822 
6518.1843 
6532. S021 
6546.8356 
6561.1848 


28S-8849 
286. 1991 
286.5132 
286.8274 
287. 1416 


72.0 
. I 
. 2 
.3 

• 4 


4071. 5041 
4082.8216 
4094-1549 
4105. SO39 
4116.8687 


226. 1947 
226.S088 
226. 8230 
227.1371 
227.4513 


-5 
.6 

.7 
.8 
■ 9 


4839.8198 
4852.1584 
4864.5127 
4876.8828 
4889.2685 


246.6150 
246.9292 

247.2433 
247.5575 
247.8717 


8s.o 
. I 
. 2 
-3 

-4 


5674-5017 
5687.8613 
5701.2367 
5714.6277 
5728.0344 


267.0354 
267.3495 
267.6637 
267.9779 
268 .2920 


-5 
.6 

.7 
.8 
• 9 


6575.5497 
6S89.9304 
6604.3267 
6618.7388 
6633. 1666 


287.4557 
287.7699 
288. 0840 
288.3982 
288.7124 


.5 
.6 

■7 
.8 
.9 


4128. 2491 
4139.6452 
4151.0570 
4162.4846 
4173.9278 


227.7655 
228.0796 
228.3938 
228.7079 
229.0221 


79-0 
. I 
.2 
-3 

.4 


4901.6699 
4914-0871 
4926.5199 
4938.9685 
4951.4328 


248.1858 
248.5000 
248.8141 
249.1283 
249.442s 


-5 
.6 

• 7 
.8 

• 9 


5741-4569 
5754-8951 
5768.3489 
5781.8185 
5795.3038 


268.6062 
268.9203 
269.234s 
269.5486 
269.8628 


92.0 
. I 
. 2 
.3 

-4 


6647.6100 
6662.0692 
6676.5441 
6691.0347 
6705.5410 


289.0265 
289.3407 
289.6548 
289.9690 
290. 2832 


730 
. I 

. 2 

■3 

• 4 


4185.3868 
4196. 8615 
4208.3518 
4219.8579 
4231.3797 


229.3363 
229.6504 
229.9646 
230.2787 
230.5929 


.5 
.6 

-7 
.8 
.9 


4963.9127 
4976.4084 
4988.9198 
5001.4469 
5013.9897 


249.7566 
250.0708 
250.3849 
250.6991 
251.0133 


86,0 
. I 
. 2 
-3 

•4 


S808.8048 
S822.321S 
5835-8539 
S849.4020 
5862.9659 


270. 1770 
270.4911 
270.8053 
271.1194 
271.4336 


-5 
.6 

.8 
-9 


6720.0630 
6734.6007 
6749.1542 
6763.7233 
6778.3081 


290.5973 
290.911S 
291 . 2256 
291-5398 
291.8S40 


.5 
.6 
.7 
.8 
.9 


4242.9172 
4254-4704 
4266.0393 
4277.6240 
4289.2243 


230.9071 
231.2212 
231-5354 
231-8495 
232.1637 


80.0 
. I 
. 2 
.3 

.4 


5026.5482 
5039.1224 
5051.7124 
5064.3180 
S076.9394 


251.3274 
251.6416 
251.9557 
252.2699 
252.5840 


-5 
.6 

.8 
-9 


5876.5454 
5890. 1406 
5903.7516 
5917.3782 
5931.0206 


271.7478 
272.0619 
272.3761 
272.6902 
273.0044 


93-0 
. 1 
. 2 
•3 

4 


6792.9087 
6807.5249 
6822 . 1569 
6836.8046 
6851.4680 


292. 1681 
292.4823 
292. 7964 
293. 1106 
293-4248 


74-0 
. I 
. 2 
■ 3 

• 4 


4300.8403 

4312. 4721 
4324-1195 
4335-7827 
4347.4616 


232.4779 
232.7920 
233. 1062 
233.4203 
233.7345 


-5 
..6 
-7 
.8 
.9 


5089.5764 
5102.2292 
5114.8977 
5127. 5818 
5140. 2817 


252.8982 
253.2124 
253.5265 
253.8407 
254.1548 


87.0 
. I 
. 2 
• 3 

.4 


5944.6787 
5958.352s 
5972.0419 
5985.7471 
5999.4680 


273-3186 
273-6327 
273.9469 
274. 2610 
274-5752 


5 
6 

7 
8 
9 


6866. 1471 
6880.8419 
6895.5524 
6910.2786 
692s. 0205 


293-7389 
294-0531 
294- 3672 
294.6814 
294-9956 


■ S 
.6 

•7 
.8 
.9 


4359.1562 
4370.8664 
4382.5924 
4394-3341 
4406.0915 


234.0487 
234-3628 
234-6770 
234-9911 
235-3053 


81.0 
. I 

.2 
.3 

.4 


5152.9973 
5165.7286 
S178.4756 
5191.2384 
5204.0168 


254.4690 
254.7832 
255.0973 
255.4115 
255.7256 


.5 
.6 
•7 
8 
.9 


6013.2047 
6026.9570 
6040.7250 
6054.5088 
6068.3082 


274.8894 
275-2035 
275.5177 
275-8318 
276. 1460 


940 
. I 

.2 

-3 

• 4 


6939-7781 
6954-5515 
6969 . 3405 
6984.1453 
6998.9657 


295 -3097 
295-6239 
295-9380 
296. 2522 
296. 5663 


75.0 
. I 

.2 
.3 

• 4 


4417.8647 
4429.6535 
4441 -4580 
4453-2783 
4465.1142 


23s. 6194 
235 9336 
236,2478 
236.5619 
236.8761 


.5 

.6 

.7 
.8 
-9 


5216. 8109 
5229.6208 
5242.4463 
5255.2876 
5268. 1446 


256.0398 
256.3540 
256.6681 
256.9823 
257.2964 


88.0 
. I 
.2 
.3 

•4 


6082. 1234 
6095.9542 
6109.8008 
6123. 6631 
6137. 5410 


276.4602 
276.7743 
277.0885 
277.4026 
277-7168 


.5 

.6 
.7 
8 
.9 


7013. 8019 
7028.6538 
7043.5214 
7058.4047 
7073-3037 


296.8805 
297.1947 
297.5088 
297 8230 
298.1371 


.5 
.6 
• 7 
.8 
.9 


4476.9659 
4488.8332 
4500.7163 
4512. 6151 
4524.5296 


237. 1902 
237.5044 
237.8186 
238.1327 
238.4469 


82.0 
. I 
. 2 
.3 

.4 


5281. 0172 
5293.9056 
5306.8097 
5319.729s 
5332.6650 


257.6106 
257.9248 
258.2389 
258.5531 
258.8672 


-5 
.6 

-7 
.8 
.9 


61S1.4347 

6165.3441 
6179. 2692 
6193. 2101 
6207. 1666 


278.0309 
278.3451 
278.6593 
278.9734 
279.2876 


95.0 
. I 
. 2 
■3 
-4 


7088.2184 
7103. 1488 
7118 .0949 
7133-0568 
7148-0343 


298.4513 
298.765s 
299. 0796 
299.3938 
299.7079 


76.0 

. I 
. 2 
• 3 
.4 


4536.4598 
4548.4057 
4560.3673 
4572.3446 
4584.3376 


238.7610 
239-0752 
239 3894 
239-7035 
240.0177 


.5 
.6 
.7 
.8 
.9 


5345.6162 
5358.5832 
5371.5658 
5384.5641 
5397.5782 


259. 1814 
259-4956 
259.8097 
260. 1239 
260.4380 


89.0 
. I 
.2 
-3 
•4 


6221. 1388 
6235. 1268 
6249.1304 
6263. 1498 
6277.1848 


279.6017 
279.9159 
280. 2301 
280.5442 
280.8584 


-5 
.6 
-7 
-8 
.9 


7163.0276 
7178.0365 
71930612 
7208. 1016 
7223.1577 


300.0221 
300.3363 
300.6SO4 
300. 9646 
301.2787 



548 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 







Table 27 


. — Areas 


4ND Circumferences of Circles- 


-Decimal Divisions — ( 


Continued) 






Diam- 




Circum- 


Diam- 


Area 


Circum- 


Diam- 


Area 


Circum- 


Diam- 


Area 


Circum- 


eter 




ference 


eter 




ference 


eter 




ference 


eter 




ference 


96.0 


7238.2294 


301.5929 


97.0 


7389. 8113 


304.7345 


98.0 


7542-9639 


307.8761 


99-0 


7697.6874 


311-0177 


. I 


7253.3169 


301.9071 


. I 


7405.0559 


305.0486 


. I 


7558.3656 


308. 1902 


. I 


7713-2461 


311-3318 


. 2 


7268.4201 


302.2212 


2 


7420.3162 


305.3628 


2 


7573-7830 


308.5044 


.2 


7728.820s 


311.6460 


.3 


7283.5391 


302.5354 


.3 


7435.5921 


305.6770 


3 


7589. 2161 


308.8186 


.3 


7744-4107 


311.9602 


■4 


7298.6737 


302.849s 


-4 


7450.8838 


30S.9911 


.4 


7604.6648 


309.1327 


4 


7760.0166 


312.2743 


.5 


7313.8240 


303.1637 


-5 


7466. 1913 


306.3053 


.5 


7620. 1293 


309-4469 


5 


7775.6381 


312.588s 


.6 


7328.9901 


303.4779 


.6 


7481. 5144 


306.6194 


.6 


7635.6095 


309.7610 


6 


7791.2754 


312.9026 


■7 


7344. 1718 


303.7920 


.7 


7496.8532 


306.9336 


-7 


7651. 1054 


310.0752 


7 


7806.9284 


313.2168 


.8 


7359.3693 


304. 1062 


.8 


7512.2077 


307.2478 


.8 


7666.6170 


310.3894 


.8 


7822.5971 


313.5309 


.9 


7374-5824 


304.4203 


.9 


7527.5780 


307.5619 


-9 


7682.1443 


310.703s 


.9 
100. 


7838. 281S 
7853-9816 


313.841S 
314-1593 



For larger integral circles see Table 20. 

Table 28. — Areas, Circumferences, Squares, Cubes and Fourth Powers of 

(F. E. Kelley, Amer. Mach., July 25, 1901) 



Binary Fractional Quantities 



1 


Area 


Cir. 1 


Square | 


Cube 1 


4th power 1 


1 


Area 


Cir. 


Square 


Cube 


4th power 


A 


.00019 


. 04909 


.000244 


.0000038 


. 00000006 


I 


.78540 


3-14159 


1.0 


1 .0 


1.0 


A 


.00077 


.0982 


.000977 


.0000305 


.0000009 


A 


.83525 


3.2398 


1.0635 


I .0967 


1.13 


A 


.00173 


.1473 


.002197 


.0001030 


.0000048 


is 


.88664 


3.3379 


1 .1289 


1-1995 


1.27 


A 


.0031 


.1963 


.003906 


.0002441 


.0000152 


A 


.93956 


3.4361 


I. 1963 


1.3084 


1.43 


A 


.0048 


.2454 


.006104 


.0004768 


.0000372 


i 


.99402 


3.5343 


1.2656 


1.4238 


1.60 


A 


.0069 


• 294s 


.008789 


.0008240 


.0000771 


A 


1.05 


3.632s 


1.3369 


1.5458 


1.78 


A . 


.0094 


.3436 


.011963 


.0013084 


.0001430 


A 


1.1075 


3.7306 


I. 4102 


1.6746 


1.98 


i 


.0123 


.3927 


.015625 


.0019531 


.000244 


A 


I. 1666 


3.8288 


1.4854 


1.8103 


2.20 


A 


.0155 


.4418 


.019775 


.002781 


.0003908 


i 


1.2272 


3.9270 


1.5625 


I.9S3I 


2.4s 


A 


.0192 


.4909 


.024414 


.003815 


.0005954 


A 


1.2893 


4.0252 


1.6416 


2.1033 


2 .69 


ik 


.0232 


.5400 


.029541 


.005077 


.0008702 


■h 


I. 3530 


4.1233 


1.7227 


2. 2610 


2.97 


* 


.0276 


.5890 


.035156 


.006592 


.001232 


^ 


1.4182 


4.221s 


1.8057 


2.4264 


3-26 


« 


.0324 


.6381 


.041260 


.008381 


.001697 


f 


1.4849 


4.3197 


1.8906 


2.5996 


3-57 


A 


.0376 


.6872 


.047852 


.010468 


.002284 


M 


I.SS32 


4.4179 


1.977s 


2.7809 


3.91 


M 


.0431 


.7363 


.054932 


.01287s 


.003014 


A 


1.6230 


4-Si6o 


2 . 0664 


2.970s 


4.27 


i 


.0491 


• 7854 


.0625 


.015625 


.003906 


M 


1.6943 


4.6142 


2.1572 


3.1684 


4.6s 


H 


.0554 


.8345 


.070SS7 


.018742 


.004970 


h 


1.7671 


4.7124 


2.2s 


3.375 


S.06 


A 


.062 1 


.8836 


.079102 


.022247 


.006256 


ii 


1.841S 


4.8106 


2.3447 


3.5904 


S.47 


H 


.0692 


.9327 


.088135 


.02616s 


.007762 


A 


1.917s 


4.9087 


2.4414 


3-8147 


S.95 


A 


.0767 


.9817 


.097656 


.030518 


.009530 


M 


I ■ 9949 


5 . 0069 


2 . S400 


4.0482 


6-45 


li 


.0846 


1.0308 


. 107666 


.035328 


.01158 


t 


2.0739 


S.iosi 


2.6406 


4.2910 


6.97 


il 


.0928 


1.0799 


.118164 


.040619 


.01395 


f^ 


2.1545 


S.2033 


2.7432 


4-5434 


7-50 


ii 


. 1014 


1. 1290 


.129150 


.046413 


.01668 


H 


2.2365 


5.3014 


2.8477 


4-8054 


8.07 


1 


. 1104 


1.1781 


. 140625 


.052734 


■01977 


M 


2.3201 


5.3996 


2.9541 


5-0774 


8.70 


n 


.1198 


1.2272 


.152588 


.05960s 


.02328 


1 


2.4053 


5-4978 


3.062s 


5-3594 


9.38 


if 


. 1296 


1.2763 


.165039 


.067047 


.02723 


4 


2.4920 


S.5960 


3.1729 


5.6516 


10. OS 


n 


.1398 


1.3254 


•177979 


.07508s 


.03165 


ii 


2.5802 


S.6941 


3-2852 


S-9543 


10.76 


A 


.1503 


1.3744 


. 191406 


.083740 


.03663 


M 


2 . 6699 


5.7923 


3-3994 


6.2677 


11.56 


H 


. 1613 


1.4235 


.205322 


.093037 


.0421s 


i 


2.7612 


5.8905 


3-5156 


6.5918 


12.4 


M 


.1726 


1.4726 


.219727 


. 102997 


.04837 


^ 


2.8540 


5. 9887 


3.6338 


6.9269 


13-2 


ii 


.1843 


I-5217 


.234619 


.113644 


■05504 


tt 


2-9483 


6.0868 


3-7539 


7-2732 


14.1 


h 


.1963 


1.5708 


.25 


.125 


.062 s 


fi 


3.0442 


6.1850 


3-8760 


7.6308 


14.9 


M 


.2088 


I. 6199 


.265869 


.137089 


.07071 


2 


3-1416 


6.2832 


4.0 


8.0 


16.0 


H 


.2217 


1.6690 


.282227 


. 149933 


.07963 


A 


3.3410 


6.^795 


4.2539 


8.7737 


18. 1 


M 


.2349 


1.7181 


.299072 


.163555 


.08946 


J 


3.5466 


6.6759 


4.5156 


9.5957 


20.4 


A 


.248s 


I. 7671 


.316406 


.177979 


. 10011 


_s. 


3.7583 


6.8722 


4.7852 


10.4675 


22.9 


e 


.2625 


I. 8162 


.334229 


. 193226 


. 11169 


i 


3.9761 


7.0686 


5.0625 


11.3906 


25.6 


^ 


.2769 


I. 8653 


.352559 


. 209320 


. I24I9I 


A 


4.2000 


7.2649 


5.3477 


12.3665 


28.6 


M 


.2916 


I. 9144 


.371338 


.226284 


.137901 


i 


4.4301 


7.4613 


5 . 6406 


13.3965 


31.8 


1 


.3068 


1-9635 


.390625 


.244141 


.152568 


A 


4.6664 


7.6576 


5.9414 


14.4822 


35. 3 


ii 


-3252 


2.0126 


.410400 


.262913 


. 168428 


i 


4.9087 


7.8540 


6.25 


15.625 


39.1 


B 


-3382 


2.0617 


-430664 


.282623 


.18546 


A 


5.1572 


8.0503 


6.5664 


16.8264 


43-0 


-Si 


.354s 


2. 1108 


-451416 


.303295 


.20376 


1 


5.4119 


8.2467 


6.8906 


18.0879 


47-5 


^ 


.3712 


2.1598 


.472656 


.324951 


.22340 


ii 


5.6727 


8.4430 


7.2227 


19-4109 


52.1 


M 


.3883 


2.2089 


■494385 


-347614 


• 24438 


a 


5.9396 


8.6394 


7.5625 


20. 7969 


57.2 


H 


.4057 


2.2580 


-516602 


.371307 


.26688 


11 


6.2126 


8.8357 


7.9102 


22.2473 


62.6 


fi 


.4236 


2.3071 


.539307 


•396053 


.29084 


5 


6.4918 


9.0321 


8.2656 


23-7637 


68.3 


a 


.4418 


2.3562 


• 5625 


.421875 


.31641 


li 


6.7771 


9.2284 


8.6289 


25-3474 


74.5 


M 


.4602 


2.4048 


.586182 


•448795 


.34357 


3 


7.0686 


9-4248 


9.0 


27-0 


81.0 


M 


.4794 


2-4544 


.610352 


.476837 


.3721 


1 


7.6699 


9.817s 


9.7656 


30-5176 


95-4 


fi 


.4987 


2.5030 


.635010 


.506023 


.4032 


i 


8.2958 


10.210 


10.5626 


34-3281 


112. 


if 


.5185 


2.5525 


.660156 


.536377 


.4356 


1 


8.9462 


10.603 


11-391 


38-443 


130 


« 


.5387 


2.6011 


.685791 


.567921 


.4704 


h 


9.6211 


10.996 


12.25 


42-875 


ISO 


!i 


-SS9I 


2.6507 


.711914 


.600677 


.5069 


5 


10.321 


11.388 


13.141 


47-635 


173 


M 


-5796 


2.7017 


-738525 


.634670 


• 5456 


\ 


11.045 


11.781 


14.062 


52-734 


198 


1 


.6013 


2-7499 


.765625 


.669922 


-5861 


1 


11-793 


12.174 


15.016 


58.186 


225 


H 


.6228 


2. 7960 


.793213 


.706455 


.6292 


4 


12. 566 


12.S66 


16.0 


64-0 


256 


f* 


.6450 


2.8471 


.821289 


-744293 


.6750 


J 


13.364 


12.959 


17.016 


70. 189 


290 


11 


-6675 


2.8965 


-849854 


-783459 


.722s 


i 


14.186 


13.352 


18.062 


76.766 


326 


H 


.6903 


2.9452 


.878906 


-82397s 


.7725 


i 


15-033 


13.744 


19-141 


83-740 


366 


»li 


• 7131 


2.9939 


.908447 


.865864 


.8248 


i 


15-904 


14-137 


20.25 


91.12s 


410 


*i 


.7371 


3-0434 


-938477 


.909149 


.8798 


5 


16.800 


14.530 


21.391 


98.932 


458 


S 


.7610 


3.0913 


-968994 


-953854 


.938s 


i 


17.721 


14-923 


22 .562 


107.172 


509 


I 


.7854 


3-1416 


I.O 


I . 


I . 0000 


5 


19.63s 


15.708 


25.0 


I2S.0 


62s 



MATHEMATICAL TABLES 



549 



Table 29. — Areas and Circumferences of Circles — Binary Divisions 
For more Minute Divisions of Small Circles see Table 22 



Diam. | 


Cir. Area | 


Diam. 


Cir. Area 


Diam. 


Cir. Area | 


Diam. Cir. | Area | 


Diam. | Cir. | Area | 


Diam. Cir. 


Area 


A 


.1963 


.00307 


9 ins. 


28.274 


63.617 


18 ins. 


56.549 


254-469 


27 ins. 


84.823 


572.556 


36 ins. 


113.097 


1017.87 


45 ins. 


141-372 


1590.43 


X 


.3927 


.0123 


1 


28.667 


65.397 


i 


56.941 


258.016 


i 


85.216 


577.870 


i 


113.490 


1024.95 


i 


141.764 


1599.28 


l 


.7854 


.0491 


i 


29.060 


67.201 


i 


57.334 


261.587 


i 


85.608 


583.208 


i 


113.883 


1032.96 


i 


142-157 


1608.1s 


i 


I. 1781 


. I104 


1 


29.452 


69.029 


i 


57.727 


265.182 


i 


86.001 


588.571 


i 


114.275 


1039.19 


8 


142.550 


1617.04 


1 


1.5708 


• 1963 


i 


29.845 


70.882 


\ 


58.119 


268.803 


i 


86.394 


593.958 


^ 


114.668 


1046.3s 


1 


142.942 


1625.97 


1 


1.9635 


.3068 


8 


30.238 


72.760 


1- 


58.512 


272.447 


i 


86.786 


599.370 


f 


115.061 


1053.52 


i 


143.335 


1634-92 


A 


2.3562 


.4418 


1 


30.631 


74.662 


I 


58.905 


276. 117 


i 


87.179 


604.807 


i 


115.453 


1060.73 


a 


143.728 


1643.89 


i 


2.7489 


.6013 


a 


31.023 


76.589 


1 


59.298 


279.811 


i 


87.572 


610.268 


1 


115.846 


1067.9s 


s 


144.121 


1652.88 


I in. 


3.1416 


.7854 


10 ins. 


31.416 


78.540 


19 ins. 


59.690 


283.529 


28 ins. 


87.965 


615.753 


37 ins. 


116.239 


1075.21 


46 ins. 


144-S13 


1661.90 


1 


3.5343 


.9940 


1 


31.809 


80.516 


-J 


60.083 


287.272 


i 


88.357 


621.263 


i 


116.631 


1082.48 


i 


144.906 


1670.95 


i 


3.9270 


1.2272 


I 


32.201 


82.516 


1 


60.476 


291.039 


i 


88.750 


626.798 


i 


117.024 


1089.79 


i 


145-299 


1680.01 


t 


4.3197 


1 . 4849 


i 


32.594 


84.541 


f 


60.868 


294-831 


i 


89.143 


632.357 


1 


117.417 


1097.11 


t 


145-691 


1689.10 


i 


4.7124 


I. 767 1 


5 


32.987 


86.590 


i 


61 .261 


298.648 


1 


89.535 


637.941 


i 


117.810 


1104.46 


i 


146.084 


1698.23 


1 


5.1051 


2.0739 


1 


33.379 


88.664 


1 


61 .664 


302.489 


1 


89.928 


643.554 


1 


118.202 


II II. 84 


f 


146.477 


1707.37 


i 


5.4978 


2.4053 


i 


33-772 


90.763 


a 


62.046 


306.355 


3 


90.321 


649. 182 


J 


118.596 


1119.24 


1 


146.869 


1716.54 


i 


5.8905 


2.7611 


i 


34.165 


92.886 


I 


62.439 


310.245 


i 


90.713 


654.839 


i 


118.988 


1126.66 


s 


147.262 


1725.73 


2 ins. 


6.2832 


3.1416 


II ins. 


34.558 


95-033 


20 ins. 


62.832 


314.. 160 


29 ins. 


91.106 


660.521 


38 ins. 


119. 381 


1134-11 


47 ins. 


147.655 


1734-94 


i 


6.6759 


3 . 5466 


i 


34-950 


97 - 205 


i 


63.225 


318.099 


i 


91.499 


666. 237 


i 


119.773 


I 141. 59 


1 


148.048 


1744-18 


i 


7.0686 


3.9761 


i 


35-343 


99-402 


i 


63.617 


322.063 


1 


91.892 


671 J>S8 


i 


120.166 


1149.08 


i 


148.440 


1753-45 


i 


7.4613 


4-4302 


i 


35-736 


101.623 


* 


64.010 


326.051 


i 


92.284 


677.714 


i 


120.559 


1156.61 


i 


148.833 


1762.73 


1 


7.8540 


4-9087 


i 


36.128 


103.869 


1 


64.403 


330.064 


1 


92.677 


683.494 


i 


120.951 


1164.15 


i 


149.226 


1772. OS 


i 


8.2467 


5.4119 


i 


36.521 


106.139 


f 


64.795 


334-101 


f 


93.070 


689.298 


1 


121.344 


1171.73 


i 


149.618 


1781.39 


i 


8.6394 


5.9396 


i 


36.914 


108.434 


a 


65.188 


338.163 


i 


93.462 


695.128 


a 


121.737 


1179.32 


a 


150.011 


1790.76 


i 


9.0321 


6 . 49 I 8 


i 


37.306 


110.753 


7 


65.581 


342.250 


i 


93.855 


700.981 


l 


122.129 


1186.94 


1 


150.404 


1800.14 


3 ins. 


9.4248 


7.0686 


12 ins. 


37.699 


113.097 


21 ins. 


65.973 


346.361 


30 ins. 


94.248 


706.860 


39 ins. 


122.522 


1194-59 


48 ins. 


150.796 


1809.56 


i 


9.8175 


7.6699 


i 


38.092 


115.466 


i 


66.366 


350.497 


-1 


94.640 


712.762 


i 


122.915 


1202. 26 


i 


151. 189 


1818.99 


1 


10. 210 


8.2958 


i 


38.485 


117.859 


i 


66.759 


354.657 


i 


95.033 


718.690 


i 


123.308 


1209.95 


i 


151. S82 


1828.46 


f 


10.603 


8.9462 


t 


38.877 


120.276 


1 


67.152 


358.841 


i 


95.426 


724.641 


t 


123.700 


1217.67 


f 


151. 975 


1837.93 


} 


10.996 


9.6211 


3 


39.270 


122.718 


X 


67.544 


363.051 


i 


95.819 


730.618 


i 


124.093 


1225.42 


4 


152.367 


1847.4s 


f 


11.388 


10.321 


1 


39.663 


125.184 


5 


67.937 


367.284 


5 


96.211 


736.619 


f 


124.486 


1233.18 


5 


152.760 


1856.99 


f 


II. 781 


11.045 


i 


40.055 


127.676 


■J 


68.330 


371.543 


I 


96.604 


742.644 


i 


124.878 


1240.98 


a 


153.153 


1866. ss 


i 


12.174 


11.793 


i 


40.448 


130. 192 


« 


68.722 


375.826 


' 


96.997 


748.694 


i 


125.271 


1248.79 


1 


153.54s 


1876.13 


4 ins. 


12.566 


12.566 


13 ins. 


40.841 


132.732 


22 ins. 


69. lis 


380.133 


31 ins. 


97.389 


754-769 


40 ins. 


125.664 


1256.64 


49 ins. 


153.938 


1885.74 


i 


12.959 


13.364 


i 


41.233 


135.297 


\ 


69.508 


384.465 


i 


97.782 


760.868 


i 


126.056 


1264.50 


i 


154.331 


1895-37 


i 


13.352 


14.186 


i 


41.626 


137.886 


\ 


69.900 


388.822 


i 


98.175 


766.992 


i 


126.449 


1272.39 


i 


154.723 


I 90s -03 


i 


13.744 


15.033 


i 


42.019 


140.500 


3 


70.293 


393.203 


i 


98.567 


773.140 


* 


126.842 


1280.31 


i 


155. 116 


1914-70 


i 


14.137 


15.904 


i 


42.412 


143.139 


i 


70.686 


397.608 


i 


98.960 


779.313 


i 


127.235 


1288.25 


i 


155. S09 


1924-42 


i 


14.530 


16. 800 


i 


42 . 804 


145.802 


! 


71.079 


402.038 


i 


99-353 


785-510 


1 


127.627 


1296.21 


i 


155.902 


1934- IS 


i 


14-923 


17.721 


a 


43.197 


148.489 


I 


71.471 


406.493 


i 


99-746 


791-732 


f 


128.020 


1304.20 


i 


156.294 


1943.91 


J 


IS. 315 


18.665 


i 


43 . 590 


151.201 


i 


71.864 


410.972 


i 


100.138 


797-978 


i 


128.413 


1312.21 


i 


156.687 


1953.69 


5 ins. 


15.708 


19.63s 


14 ins. 


43.982 


153.938 


23 ins. 


72.257 


4IS.476 


32 ins. 


100.531 


804.249 


41 ins. 


128.805 


1320.25 


50 ins. 


157.080 


1963.50 


i 


16. lOI 


20.629 


i 


44-375 


156.699 


\ 


72.649 


420.004 


1 


100.924 


810.545 


X. 


129.198 


1328.32 


i 


157.865 


1983.18 


i 


16.493 


21.648 


i 


44.768 


159.485 


i 


73.042 


424.557 


1 


101.316 


816.865 


I 


129.591 


1336.40 


i 


158.650 


2002.96 


i 


16.886 


22.691 


f 


45. 160 


162.295 


! 


73.435 


429.135 


I 


101.709 


823.209 


3 


129.983 


1344.51 


t 


159.436 


2022.84 


i 


17.279 


23.758 


"i 


45.553 


165.130 


i 


73.827 


433.731 


h 


102.102 


829. S78 


1 


130.376 


1352.65 








i 


17-671 


24.850 


1 


45 • 946 


167.989 


» 


74-220 


438.363 


i 


102.494 


835.972 


-I 


130.769 


1360.81 


SI ins. 


160.221 


2042.82 


i 


18.064 


25.967 


3 


46.338 


170.873 


J 


74-613 


443.014 


i 


102.887 


842.390 


4 


131.161 


I36SV.OO 


} 


161.007 


2062.90 


i 


18.457 


27.109 


1 


46.731 


173.782 


8 


75.006 


447 . 699 


1 


103.280 


848.833 


5 


131-554 


1377.21 


i 


161.792 


2083.07 


6 ins. 


18.850 


28.274 


15 ins. 


47.124 


176. 71S 


24 ins. 


75-398 


452.390 


33 ins. 


103.673 


855.30 


42 ins. 


131-947 


138s. 44 


i 


162.577 


2103.3s 


i 
i 


19.242 

19.635 
20.028 


29.46s 
30.680 
31.919 


i 

i 
1 


47.517 
47.909 
48.302 


179.672 
182.654 
185.661 


i 

i 


75-791 
76. 184 
76.576 


457.115 
461.864 
466.638 


i 

i 


104.065 
104.458 
104.851 


861.79 
868.30 
874.85 


i 
I 

i 


132.340 
132.732 
133.125 


1393.70 
1401.98 
1410. 29 


52 ins. 

i 

i 

i 


163.363 

164. 148 
164.934 
165.719 


2123.72 
2144.19 
2164.75 
2185.42 


i 

1 


20.420 
20.813 


33.183 

34.472 


i 

i 


48.694 
49.087 


188.692 
191.748 




76.969 
77.362 


471.436 
476.259 




105.243 
105.636 


881.41 
888.00 


2" 


133.518 
133.910 


1418.62 
1426.98 


i 
i 


21.205 

21.598 


35.784 
37.122 


i 
-I 


49.480 
49.873 


194.828 
197.933 


i 

i 


77.754 
78.147 


481 . 106 
485.978 


i 


106.029 
106.421 


894.62 
901.26 


a 

1 


134.303 
134.696 


1435.36 
1443.77 


53 ins. 
i 


166.504 
167.290 


2206.18 
2227.0s 


7 ins. 


21.991 


38.48s 


16 ins. 


50.265 


201.062 


25 ins. 


78.540 


490.875 


34 ins. 


106.814 


907.92 


43 ins. 


135.088 


1452.20 


§ 


168.07s 


2248.01 


i 


22.384 


39.871 


i 


50.658 


204.216 


i 


78.933 


495.796 


i 


107.207 


914.61 


i 


135.481 


1460.65 


i 


168.861 


2269.06 


i 


22.776 


41.282 


i 


51.051 


207.394 


1 


79.325 


500.741 


i 


107.600 


921.32 


i 


135.874 


1469.13 








i 


23.169 


42.718 


i 


51.444 


210.597 


i 


79.718 


505.711 


f 


107.992 


928.06 


f 


136.267 


1477.63 


54 ins. 


169.646 


2290.22 


i 


23.562 


44-179 


i 


51.836 


213.825 


i 


80. Ill 


510.706 


i 


108.385 


934-82 


i 


136.659 


i486. 17 


i 


170.431 


2311.48 


i 


23.955 


45-664 


1 


52.229 


217.077 


i 


80.503 


515.725 


1 


108.778 


941-61 


I 


137.052 


1494.72 


i 


171. 217 


2332.83 


i 


24.347 


47-173 


l 


52.622 


220.353 


:•) 


80.896 


520.769 


i 


109.170 


948 - 42 


I 


137.44s 


1503.30 


a 


172.002 


2354-28 


i 


24.740 


48.707 


1 


53.014 


223.654 


8 


8l.28y 


525.837 


i 


109.563 


9SS-2S 


i 


137-837 


1511.90 


55 ins. 


172.788 


2375.83 


8 ins. 


25.133 


50.26s 


17 ins. 


53.407 


226.980 


26 ins. 


81.681 


530.930 


35 ins. 


109.956 


962.11 


44 ins. 


138.230 


1520.53 


i 


173.573 


2397.48 


i 


25.52s 


51.849 


i 


53.780 


230.330 


i 


82.074 


536.047 


i 


110.348 


969.00 


1 


138.623 


1529.18 


i 


174-358 


2419.22 


i 


25.918 


53.456 


i 


54-192 


233.70s 


i 


82.467 


541.189 


i 


110.741 


975-91 


i 


139.015 


1537.86 


i 


175-144 


2441.07 


1 


26.311 


55.088 


I 


54-585 


237-104 


i 


82.860 


546.356 


i 


III. 134 


982.84 


1 


139.408 


1546.55 








^ 


26.704 


56.745 


i 


54-978 


240.528 


A 


83.252 


551.547 


i 


111.527 


989.80 


h 


139.801 


1555.28 


56 ins. 


175.929 


2463.01 


i 


27.096 


58.426 


f 


55.371 


243.977 


5 


83.645 


556.762 


1- 


111.919 


996.78 


f 


140.194 


1564.03 


1 


176.715 


2485.05 


i 


27.489 


60. 132 


4 


55.763 


247.450 


I 


84.038 


562.002 


a 


112.312 


1003.78 


a 


140.586 


1572.81 


1- 


177.500 


2507.19 


i 


27.882 


61.862 1 


i 


56.156 


250.947 


i 


84.430 


567.267 


V 


112. 70s 


1010.82 


1 


140.979 


1581 .61 


a 


178.285 


2529-42 



550 



HANDBOOK FOR MACHINE DESIGNERS AND DRAFTSMEN 



Table 29. — Areas and Circumferences of Circles — Binary 
Divisions —{Continued) 



Table 30. — Areas of Circles of Wire Gage Diameters 



Diam. 



Cir. 



Area 



57 ins. 



S8 ins 



S9 ins. 



60 ins. 



61 ins. 



62 ins. 



63 ins. 



64 ins 



6s ins. 



179.071 
179.856 
180.642 
181.427 

182. 212 
182.998 
183.783 
184.569 

185.354 
186.139 
186.925 
187.710 

188.496 
189.281 
190.066 
190.852 

191.637 
192.423 
193.208 
193 -993 

194-779 
195.564 
196.350 
197.135 

197.920 
198.706 
199.491 
200. 277 

201.062 
201 .847 
202.633 
203.418 

204.204 
204.989 
205.774 
206.560 



66 ins. 



67 ins. 



2551.76 
2574.19 
2596.72 
2619.35 

2642.08 
2664.91 
2687.83 
2710.85 

2733.97 
2757.19 
2780. SI 
2803.92 

2827.44 
2851.05 
2874.76 
2898.56 

2922.47 
2946.47 
2970.57 
2994.77 

3019.07 
3043.47 
3067.96 
3092.56 

3117.2s 
3142.04 
3166.92 
319I.9I 

3216.99 
3242.17 
3267.46 
3292.83 

3318.31 

3343.88 
3369.56 
3395.33 



207.345 
208.131 
208.916 
209.701 

210.487 

211 .272 
212.058 
212.843 

213.628 

214.414 
215.199 
215. 



Diam. 



69 ins. 



3421.20 
3447.16 
3473.23 
3499.39 

3525.66 
3552.01 
3578.47 
3605.03 

3631.68 

3658.44 
3685.29 
3712.24 



Cir. 



Area 



71 ms, 



73 ms. 



74 ms. 



76 ins. 



77 ms. 



79 ms 



80 ins. 



81 ins. 



216.770 
217.55s 
218.341 
219. 126 

219.911 
220.697 
221.482 
222.268 

223.053 

223.838 
224.624 
225.409 

226. 195 
226.980 
227.76s 
228.551 

229.336 
230. 122 
230.907 
231.692 

232.478 
233.263 
234.049 
234-834 

235.619 
236.40s 
237. 190 
237.976 

238.761 
239.546 
240.332 
241. 117 

241.903 
242.688 
243.473 
244.259 

245.044 
245.830 
246.61s 
247.400 

248.186 
248.971 
249.757 
250.542 

251.327 
252.898 



3739.28 
3766.43 
3793.67 
3821.02 

3848.46 
3875.99 
3903.63 
3931.36 



3959. 
3987. 
4015. 
4043. 

4071. 
4099. 
4128. 
4156. 

4185. 
4214. 
4242 . 
4271. 

4300. 
4329. 
4359. 
4388. 

4417. 
4447. 
4476, 
4506. 

4536, 
4566 
4596 
4626 

4656, 
4686, 
4717 
4747 

4778, 
4809, 
4839- 
4870. 

4901 
4932 
4963 
4995 

5026. 
5089, 



Diam. Cir. 



254.469 5153 
256.040 5216, 



82 ms. 



83 ins. 



84 ins. 



85 ins. 



86 ins. 



87 ins. 



89 ins. 



90 ms. 



91 ms, 



92 ins 



94 ms 



96 
97 
98 
99 
100 
101 

102 
103 

104 
105 
106 
107 
108 
109 
110 



Area 



257.611 
259. 181 

260.752 
262.323 

263.894 
265.465 

267.035 
268.606 

270.177 
271.748 

273.319 
274.889 

276.460 
278.031 

279.602 
281.173 

282.743 
284.314 

285.885 
287.456 

289.027 
290.597 

292. 168 
293.739 

295.310 
296.881 

298.451 
300.022 

301.593 

304.734 

307.876 

311.01 

314-159 

317.301 

320.442 

323.584 

326.72s 
329-867 
333.009 
336.150 
339.292 
342.433 
345.575 



5281.02 
5345.62 

5410.62 
5476.00 

5541.78 
5607.95 

5674-51 
5741.47 

5808.81 
5876.55 

5944-66 
6013.21 

6082.13 
6151.44 

6221. IS 
6291.2s 

6361.74 
6432.62 

6503.89 
6575.56 

6647.62 
6720.07 

6792.92 
6866.16 

6939.79 
7013.81 

7088.23 
7163.04 

7238.2s 

7389.83 

7542.98 

7697.70 

7854.00. 

8011.86 

8171.28 

8332.29 

8494 • 87 
8659.01 
8824.73 
8992.03 
9160.88 
9331.32 
9503.32 



No. of 
gage 


Birmingham 
or Stubb's 
iron wire 


Brown and 
Sharpe 


British 
Imperial or 
new British 


United 
States 


Stubb's 
steel 
wire 


American 
Steel & 
Wire Co. 


0000 


.162 


.167 


.126 


.130 




. 122 


000 


.142 


.132 


.108 


. Ill 




. 104 


00 


.113 


. 104 


.095 


.093 




.086 





.091 


.083 


.082 


• 077 




.074 


I 


.071 


.066 


.071 


.062 


.041 


.063 


2 


.064 


.053 


.060 


.056 


.038 


.054 


3 


.053 


.041 


.050 


.049 


.035 


.047 


4 


.045 


.033 


.042 


■ 043 


.034 


.040 


5 


.038 


.026 


.035 


.038 


.033 


.034 


6 


.032 


.020 


.029 


.032 


.031 


.029 


7 


.025 


.016 


.024 


.027 


.031 


.024 


8 


.021 


.013 


.020 


.024 


.031 


.020 


9 


.017 


.010 


.016 


.019 


.030 


.017 


10 


.014 


.008 


.013 


.016 


.028 


.014 


II 


.011 


.0063 


.010 


.013 


.027 


.012 


12 


.009 


.0055 


.009 


.009 


.027 


.009 


13 


.0071 


.0039 


.0063 


.0071 


.026 


.0063 


14 


.0055 


.0031 


.0047 


.0047 


.025 


.0047 


IS 


.0039 


.0024 


.0039 


.0039 


.025 


.003 


16 


.0031 


.0024 


.0031 


.0031 


.024 


.0031 


17 


.0024 


.0016 


.0024 


.0024 


.024 


.0024 


18 


.0016 


.0016 


.0016 


.0024 


.022 


.0016 


19 


.0016 


.0008 


.0016 


.0016 


.021 


.0016 


20 


.0008 


.0008 


.0008 


.0008 


.020 


.0008 


21 


.0008 


.0006 


.0008 


.0008 


.020 


.0008 


22 


.0006 


.0005 


.0006 


.0008 


.019 


.0006 


23 


.0005 


.0004 


.0005 


.0006 


.018 


.OOOS 


24 


.0004 


.0003 


.0004 


.0005 


.018 


.0004 


25 


.0003 


.0002 


.0003 


.0004 


.017 


.0003 


26 


.0002 


.0002 


.0002 


.0003 


.016 


.0002 



For larger integral circles see Table 20. 



INDEX 



i 



I 



Accelerated motion, graphical expression of, 
466 
laws of, 466 
Adapter for T slots, 277 
Addition of binary fractions, 321 
Air, atmospheric, value of one atmosphere of 
pressure, 449 
pressure of, at various altitudes, table 

of, 449 
weight and volume of, 449 
Air, chambers, arrangement of, 247 
charging devices for, 246 
for long suction pipe lines, 247 
Air, compressed, compressor, 449 

at high altitudes, formulas for, 451 

table of, 451 
bearings for, pressure on, 8 
compound, constants for, table of, 
450 
good and bad practice, 450 
power calculations, chart for, 453 
formulas for, 450 
compression curve, index of, formula 
for, 452 

table of, 4S4 
chart for, 455 
plotting, 454 
consmnption of by air-lift pumps, 
formula for, 461 
by pneumatic tools, table of, 460 
by direct lift hoists, table of, 

460 
by direct acting pumps, formula for, 
460 
discharge of through orifices, table of, 

461 
efficiencies defined, 449 
friction of, in pipes, formula for, 

453 

chart for, 456 
heavy pressures, packings for, 459 
high pressure, joints and fittings for, 

266 
intercoolors for, area of surface of, 
chart for, 457 
construction of, 457 
formula for, 457 
isothermal curve, formula for, 456 
light pressures, pistons of, construc- 
tion of, 458 
single stage , constants for, table of, 

450 

power calculations, chart for, 452 
formulas for, 449 
valves for, 459 
hoists, direct lift, consumption of com- 
pressed air by, table of, 460 
-lift pump, construction of, 461 

consumption of compressed air by, 
formula for, 461 



Air-lift pump, economy tests of, charts of, 462 
Allowances and Tolerances, for fits {see Fits). 

for bearings {see Bearings). 
Alloys, 390 

aluminum, S.A.E. specifications for, 392 
design of parts of, 393 
precautions in pouring, 393 
brass castings, S.A.E. specifications for, 

391, 392 
brass rods, S.A.E. specifications for, 391 
bronze, commercial, S.A.E. specifica- 
tions for, 391 
phosphor, S.A.E. specifications for, 391 
tobin, S.A.E. specifications for, 392 
manganese, S.A.E. specifications for, 

392 
casting, U. S. navy specifications for, 

table of, 394 
copper-tin-zinc, strength of, chart of, 

390 
for bearings, 19, 20, 21 
for tubing, S.A.E. specifications for, 

392 
fusible, table of, 394 
German silver, S.A.E. specifications for, 

391 
sheet brass, S.A.E. specifications for, 390 
Aluminum alloys, bars, weight of, table of, 
400 
design of parts of, 393 
precautions in pouring, 393 
S.A.E. specifications for, 392 
Angles and corresponding tapers, table of, 273 
of cams, limiting, 188 
of dovetails, acute, measurement of, 
table of, 276 
obtuse, measurement of, table of, 277 
for gashing worm wheels, 137 
of worm threads, helix, influence of on 
efficiency, 131 
Antilogarithms, table of, 503 
Arcs, circular, lengths of, tables of, 524, 526 
length of, graphical method of finding, 

317 
of large radius, compass for, 317 
radius of, from span and rise, 317 
of contact of belts on pulleys, 56 
Areas of circles, binary divisions, table of, 

549 
and squares of equal area, table of, 533 
decimal divisions, table of, 544 
of wire gage diameters, table of, 550 
small fractional, table of, 548 
whole number, table of, 537 
of circular segments, table of 532 
of diaphragms, balancing, effective, 315 
of irregular figures, finding, graphical 

methods, 469 
of poppet valves, eSective pressure, 316 
of ports and pipes for steam engines, 435 
551 



Areas of segments of boiler heads, net, table 
of, 412 

of steel round and square bars, table of, 
398 

of surfaces of spheres, table of, 536 
Arms, rock, design of, 5 

dimensions, table of, 288 

equal division of arc of vibration, 314 
Atmosphere, pressure of, in various units, 230 
Automobile engines {see Engine gas). 

Babbitt metal, composition of, 19, 20 
pouring, 23 

procedure in making, 19, 20 
proper grades of materials, 21 
Back gears {see Gears, back). 
Balancing, 300 

area of diaphragms, effective, 315 

by drilling, weight of metal removed, 

table of, 302 
center of gravity of counterweights, 466 
cones, 300 

fixture, sensitive, 300 
machine, the Akimoff, 305 
General Electric Co.'s, 302 
the Norton, 305 
the Riddell, 302 
the Westinghouse, 300 
of long drums, 303 
problem, the, 303 
reciprocating parts, 305 
running, technique of, 305 
standing and running, distinction be- 
tween, 303 
Ball bearings {see Bearings, hall). 

expansion drive stud, the, 315 
Balls, surface and volume of, table of, 536 

weight of, table of, 396 
Barometric pressure at various altitudes, 

table of, 449 
Beams, 472 

bending moments and deflections of, 

formulas for, 473 
deflection, coefficients of, table of, 475 
Hodgkinson critized, 479 
of uniform strength, chart for, 479 

approximate outlines of, 481 
properties of I, table of, 477 
reinforced section factors of, table of, 475 

chart for, 476 
safe loads of rolled 1, table of, 474 
strength of, formulas for, 472 
without lateral support, constants for 

loading, table of, 474 
wood, safe loads of, table of, 478 
Bearings, ball, 30 

effect of wear, rust and corrosion on, 30 
friction of shafting fitted with, 50 
fundamental dimensions, 30 
load capacity, 35 



HMB 



552 



INDEX 



Bearings, ball, lubrication essential, 30 

radial, dimensions and loads of, table of, 

36 
typical mountings of, 31 
retaining oil and excluding grit from, 33 
suitable lubricants for, 30 
thrust, dimensions and loads of, table of, 

37 
tjTDical mountings of, 34 
Bearings, knife edge, 29 
Bearings, plain or sliding, 8 

action of high and low speed, 13 
allowances and tolerances for {see Fits, 

running). 
ball-and-socket, 25 
bore finishes, 22 
breaking down point of oil film, 14 

chart for, 14 
cast iron for, 18 
center of pressure, 4 
conditions of film lubrication, 13 
design of, for film lubrication, 15 
evaluation of friction loss and heat 

radiation, 16 
dimensions of, chart for, 15 

table of, 24 
dissipation of heat, 1 2 
effect of film lubrication on wear, 14 
, end play in, 26 

friction of shafting fitted with, 49 
General Electric Co.'s practice, 12, 25 
limiting speed and temperature, 1 7 
main for steam engines, Reynold's 
rules for, 429 
chart of, 430 
materials for, 18, 19, 20, 21 
oil grooves for, 12, 23 
oil retaining grooves for, 26 
pressures on, tables of, 8 
product of pressure and velocity, 1 1 
pouring of babbitt, 23 
proper point for introduction of oil, 13 
ratios of length to diameter, 14 
relation, of speed and pressure, 12 

of speed pressure and temperature, 
12 
ring oiled, 24 
temperature rise in, 12, 13, 18 

charts for, 12, 13 
temperatures of, final, chart for, 17 
Tower's experiments, 13 
water jacketed, 25 
Westinghouse practice, 21 
Bearings, roller, 39 
conical, 42 

Hyatt, dimensions and loads, table 
of, 40 

friction of shafting, fitted with, 50 
Norma, dimensions and loads, table 

of, 40 
thrust, for heavy loads, 42 
Bearings, thrust, pressure on, 8, 11, 27 

ball, dimensions and loads on, table 
of, 37 

typical mountings of, 34 
double-cone spindle, 29 
film lubrication in, 28 



Bearings, thrust, Kingsbury, 28 
multiple disk, 27 

product of pressure and velocity, 1 1 
Schiele curve for, 29 
vulcanized fiber, 27 
Belts, 54 

and rope drives, cost compared, 150 
arc of contact of, table of, 56 

factors for, 54, 55, 344 
Bird's rules for, 54 
comparative power of leather, steel and 

ropes, 60 
comparative power on pulleys of differ- 
ent materials, 56 
constants for driving power of, 54 
idler pulleys, correct use of, 56 
laying out mule pulleys for, 58 
length of, formulas for, 60 
limiting widths for single and double, 54 
main for steam engines, formula for, 429 
examples from practice, 55 
extreme example from practice, 55 
power of, Sellers's practice, chart of, 55 

Barth's practice, chart of, 55 
quarter twist, guide pulleys substituted 
for, 56 
laying out holes through floors for, 56 
lay out of, 56 
shippers for, improved, 61 
steel, 60 

substitutes for quarter twist, 56 
surplus of pulley face over width of, 

chart for, 62 
tandem or riding, 59 
Bevel gears {see Gears, bevel). 
Bevel friction gears {see Gears, friction). 
Blanks, shell, diameters of, 320 
Blocks, hoisting, efficiency of, 152 
Blowers, centrifugal, motors for, sizes of, 
tables of, 350, 351 
calculating power to drive, 350 
Blue-print solution, 322 
Boilers, steam, 405 

braced and stayed surfaces of, pressure 

on, 411 
braces or stays, loads on, table of, 412, 

413 
chimneys for, horse-power of, table of, 

421 
coal for, American, analysis and heating 

value of, table of, 420 
furnaces and flues, circular, 414 

Adamson, 415 
hanging or supporting, 420 
heads, segments of, net areas of, table of, 

412 
horse-power of, 405 

and heating surface, 405 
joints, butt and strap, double-riveted, 
strength of, 408 
quadruple riveted, strength of, 409 
quintuple riveted, strength of, 409 
saw-tooth type, 410 
triple riveted, strength of, 409 
efficiency of, 407, 408 
lap, double-riveted, strength of, 408 
single riveted, strength of, 408 



Boilers, joints, percentage of plate strength 
in, chart for, 418 
of rivet strength in, chart for, 418 
ligaments of, strength of, 410 
loss of coal due to scale, chart of, 419 
manholes and frames of, riveting, 415 
plates for, thickness of, 405 
rivets for, strength of, compared with 
solid plate, chart of, 418 
strength of, 405 
specifications for, 405, 406 
safety valves for, 416 
pop, capacity of, table of, 416 
U. S. steamboat inspection, formula 

for, 419 
Power formula for, 420 
Philadelphia formula for, 420 
saving due to heating feedwater, chart 

of, 419 
standard test piece for materials of, 406 
stays or braces, loads on, tables of, 412, 

413 
crown bar or girder, 414 
gusset, 413 
stay tubes, 414 
stay bolts, pitch of, table of, 411 

specifications for, 406, 407 
steel plates for, specifications for, 405 
strength of, A. S. M. E. rules for, 405 
tube sheets, pressure on, 414 
tubes for, thickness of, 405 
properties of, table of, 421 
resistance of in tube sheets, 421 
specifications for, 407 
working pressure on, formulas for, 407 
Bolts {see also Screws), 210 

boiler stay, pitch of, table of, 411 

loads in, table of, 412 
eye, strength of, chart for, 490 
for cylinder covers, pitch of, 431 
and tank heads, chart for, 434 
foundation, washers for, dimensions of, 

283 
S.A.E. standard, strength of, table of, 

216 
S.A.E. standard, table of, 213 

tap drills for, table of, 213 
stress on, due to initial and applied 
loads, 210 
due to tightening nuts, 218 
steam-engine frame, strength of, 430 
taper, Baldwin Locomotive Works sys- 
tem of, 217 
under action of shock, 493 
U. S. standard, table of, 211, 212 

tap drills for, table of, 211 
V-thread standard, 210 
Whitworth standard, table of, 212 
Bolt cutters, motors for, sizes of, table of, 335 
headers, motors for, sizes of, table of, 335 
Boring mills, motors for, sizes of, table of, 335 
machines, cylinder, motors for, sizes of, 
table of, 335 
motors for, sizes of, table of, 337 
Braces, design of, 6 
Brakes, 175 

automatic load, 178 



INDEX 



553 



I 



Brakes, axial, retarding moment of, formula 
for, 176 
band, retarding moment of, formula for, 
17s, 181 
chart for, 175 
tensions in, formula for, 176 

chart for, 177 
the differential, 1 75 
width of, 17s 
block, retarding moment of, formula for, 

176 
coefficient of friction of, tables of, 175, 

181 
drums, area of surface of, 176 
hoiscing, superior construction of, 1.77 
multiple disc, 178 
Prony, 178 

area of surface of, 179 
design of, 179 
water cooled, 179 
rope, 179 

data for design of, 179 
forms of, 179 
Weston, 178 
Brass, bars, weight of, table of, 400 

casting, S.A.E. specifications for, 391, 

392 
rods, S.A.E. specifications for, 391 
sheet, S.A.E. speciiacations for, 390 
sheets, weight of, table of, 397 
strength of, table of, 472 
tubing, seamless, weight of, table of, 396 
U. S. navy specifications for, table of, 394 
wire, weight of, table of, 395 
Brinell hardness numerals, table of, 376 

and scleroscope hardness numerals, table 

of, 377 
British thermal units and kilogram calories, 

conversion table of, 403 
Bronze gears {see Gears, spur). 

manganese, S.A.E. specifications for, 392 
phosphor, S.A.E. specifications for, 391 
S.A.E. specifications for, 391 
strength of, table of, 472 
tobin, S.A.E. specifications for, 392 
U. S. navy specifications for, table of, 394 
Bulldozers, motors for, sizes of, table of, 335 
Bushel, U. S. and British compared, 495 
Bushings for jigs and fixtures, table of, 285 

Cams, 188 

chart for laying out, 193 

with separate base lines, 194 
conjugate or double, 194 
diameter, determination of, 188 
drum, conical rollers for, 192 

laying out, 188, 192 

objectionable arrangement of, 194 
equalizing spring pressure on, 196 
face, laying out, 188 
formers for, making, 191 
gravity curve for, 193 
high-speed, 193 
inertia of roller, effect of, 194 
levers, correct arrangement of, 195 

of unequal length, 193 
limiting angle of, 188 



Cams, motion, nature of, 188 

of high accuracy, laying out, 194 
of the linotype, 188 
of the monotype, 194 
parabolic base curve for, 193 
precision drawing boardfor laying out, 195 
rollers for, dimensions of, table of, 288 
self -locking for jigs and fixtures, 278 
solution for blackening templets when 

laying out, loi 
springs, use of, for return motion, 188 
studs for rolls, dimensions of, table of, 287 
templets for, making, 191 
two step, 189 
types of, 188 
Cast iron, 361 
bearings, 18 

castings, light, medium and heavy, 361 
composition and properties of, 361 
composition of, for various purposes, 

table of, 362 
influence of constituents on, 361 
malleable, properties of, table of, 365 

stress strain diagrams of, 366 
strength of, table of, 472 
U. S. navy specifications of, 367 
Castings, shrinkage of, table of, 472 
Center of gravity of plain figures, 466 
of counterweights, 466 
of irregular figures, finding, graphical 
method, 466 
Centigrade thermometer scale, defects of, 401 
to Fahrenheit thermometer scales, con- 
version formulas between, 401 
conversion table between, 401 
Centrifugal force, chart of, 465 
formula for, 466 
stress due to, chart of, 67 
Chains, 164 

attachment of, to hoisting drums, 165 
crane and cable, strength of, 164 

types, uses and limiting speeds, table 
of, 165 
Ewart, 166 

direction in which to run 168 
laying out sprockets for, 167 
speeds and working loads on, table of, 
166 
high-speed silent, 171 
hoisting, working loads on, table of, 164, 

174 
leading types illustrated, 164 
link belt, 173 

data for design of, table of, 174 
Morse, 171 

data for design of, table of, 173 
open and stud link, strength of, 164 
roller, 169 
cases for, 170 
crossed, 170 

data for design of, table of, 173 
length of, formulas for, 170 
sprockets for, formulas for, diameter 
of, 170 
for crane, laying out, 164 

idler for, 169 
material for, 170 



Chains, roller, sprockets for, table of diam- 
eters of, 171 
tooth form, 170 
velocity and capacity, formulas for, 

169 
vertical drives, 170 
Charts, method of using without defacement, 
at end of preface 
use of in design, 6 
Chimneys for steam boilers, horse power of, 

table of, 421 
Chords, length of for division of circles, table 

of, 534 
Circles, and squares of equal areas, table of, 

533 

binary divisions, areas and circumfer- 
ences of, table of, 549 

decimal divisions, areas and circumfer- 
ences of, table of, 544 

division of, table of, 531 

lengths of chords for division of, table of, 

534 
of wire gage diameters, area of, table of, 

SSo 
small fractional, areas and circumfer- 
ences of, table of, 548 
whole number, areas and circumferences 
of, table of, 537 
Circtilar arcs, length of, graphical method of 
finding, 317 
tables of, 524, 526 
of large radius, compass for, 317 
radius of, from span and rise, 317 
Circular segments, areas of, table of, 532 
Circtmiferences of circles, binary divisions 
table of, 549 
decimal divisions, table of, 544 
small fractional, table of, 548 
whole number, table of, 537 
Clearances, American railroad, chart of, 323 
between punches and dies, table of, 286 
Clippings and notes, fiHng, 321 
Clutches, claw, 187 
Clutches, friction, ball, 186 
cone, angle of, 185 

axial thrust of, formula for, 185 

chart for, 184 
construction of, 185 
Rockwood practice, formulas for, 

i8s 
contracting band, 186 
design constants for, 185 
dimensions of, chart for, 183 
expanding ring, dimensions of, table 
of, 182 
separating force, formula for, 185 
horse power of, formula for, 182, 185 
Lane, 186 

Weston multiple disk, 186 
dry plate, 186 
running in oil, 186 
Coal, American, analysis and heating value of, 
table of, 420 
loss of, in steam boilers due to scale, 

chart of, 419 
saving of, due to heating feed water of 
steam boilers, chart of, 419 



554 



INDEX 



Cock, plug, Westinghouse construction of, 

280 
Columns, 482 

cast iron, strength of, tables of, 480, 481 
properties of sections of, formulas for, 

482 
steel, strength of, table of, 482 
wood, strength of, table of, 483, 485 
Compass for circular arcs of large radius, 317 
Compressed air {see Air, compressed). 
Compressors, air (see Air compressors). 
Cone pulleys {see Pulleys, cone). 
Constructions and materials for resisting 

shock, 491 
Copper, bars, weight of, table of, 400 

plugs, use of, for measuring hammer 

blows, 353 
sheets and strips, S.A.E. specifications 

for, 391 
sheets, weight of, table of, 347 
strength of, table of, 472 
wire, standard gage for, 229 
strength of, table of, 229 
weight of, table of, 395 
Cosecants and secants, natural, table of, 506 
Cosines and sines, natural, table of, 506 
Counterbores, dimensions of, table of, 289 
Countershafts, speed of, 81 
Counterweights, center of gravity of, 466 
Couplings, Baush universal, 309 
Hooke universal, 308 
shaft, clamp, table of, 279 
flanged, table of, 278 
flexible, table of, 279 
Cranks, reducing overhang of, 4 

ball, dimensions of, 280 
Cross-rails, design of, 2 

Cross sections, A.S.M.E. standard for draw- 
ings, 321 
Cube roots, factoring method of finding, 

523 

of binary fractions, table of, 533 

of whole numbers, table of, 537 
Cubes of binary fractions, table of, 548 

of whole numbers, table of, 537 
Cutters, milling, design of, 359 

rotary, cutting bevel gears with, 119 
Cutting tools {see Tools, cutting). 

Decimal, equivalents of binary fractions, 
table of, 535 
of other than binary fractions, table of, 

535 
of prime number fractions, table of, 

526 
of a foot, inches and fractions reduced 
to, table of, 524 
Design, mechanical principles of, i 
Diagonals and sides of squares, table of, 533 
Diameters and speeds, chart of, 340 

table of, 525 
Diaphragms, efiective balancing area of, 315 
Dies and punches, clearance between, table 
of, 286 
holders for, table of, 286 
punching, dimensions of, 284 
Differential mechanisms, 464 



Dovetails, acute angle, measurement of, table 
of, 276 
obtuse angle, measurement of, table of, 

277 
slides and gibs, dimensions of, table of, 
276 
Drills and drilling, for balancing, weight of 
metal removed, table of, 302 
machine arms, radial, design of, 3 
machines, motors for, sizes of, tables of, 
335, 337 
multiple spindle, motors for, sizes of, 
table of, 337 
power constants for, 324 
power required for, formulas for, 327 

table of, 329, 331 
reamer for taper pins, table of, 275 
speed of, formulas for, 330 

table for, 331 
tap for machine screws, 214 
for pipe threads, 216 
for S.A.E. standard threads, table of, 

213 
for U. S. standard threads, table of, 
211 
torque and thrust of, charts of, 327, 328, 

330 
Drums, attachment of chains to, 165 
approved section of, 166 
brake area of, surface of, 176 
hoisting, superior construction of, 155 
Durability, effect of equal length, wearing 
surfaces on, i 

Eccentric rod, construction of, 274 
Elasticity as a factor in resisting shock, 491 
Ellipses, laying out approximate, 317 
Energy expended and loss of velocity, chart 
of, 79 
of hammer blows, measuring, 352 
of moving bodies, formulas for, 466 
of one pound of various velocities, chart 
of, 78 
Engine, automobile, pressure on bearings, 

10, II 
Engine, gas, 444 

bearings for, dimensions of, tables of, 9 
fly-wheels for, formulas for, 444 
horse power, probable, table of, 444 
parts of, dimensions of, formulas for, 

444 
Engine, steam, 422 

bearings of, Reynold's rules for, 429 
charts of, 430 

pressure on, table of, 8 
belts for, examples from practice, 55 

formula for, 429 
Bilgram diagram for link motion for, 
441 

for slide valves for, 439 
condensing water for, table of, 442 
crossheads, dimensions of , table of, 432 
cylinder cover bolts, pitch of, 431 

strength of, 433 

chart of, 434 
cylinder cover joints for, 433 
cylinder covers for, dimensions of, 430 



Engine, steam, economy of various types of, 

table of, 424 
expansion ratios, actual from theoreti- 
cal, table of, 427 
fly-wheels for, weight of, formula for, 

429 
frame bolts, strength of, 430 
horse power per lb. of m.e.p., table of, 

427 
isothermal expansion curve, laying out, 

427 
link motion for, Bilgram diagram for, 
441 

dimensions of parts of, 431 
main frames for, strength of, 429 
mean forward pressure in, factors 
connecting actual and theoretical, 
table of, 426 

chart of, 426 

table of, 426 
packing boxes for, construction of, 

435 
parts of, dimensions of, formulas for, 

427 
pipe lines, coverings for, 437 
cheap, 439 
drop of pressure in, chart of, 436 
■ formula for, 437 

inclination of, 439 
loss of heat in, covered and un- 
covered, charts for, 438 
piston rings, snap, action of, 431 
dimensions of, chart of, 431 
making patterns for, 431 
piston rod ends, dimensions of, tables 

of, 432 
piston valves for, construction of, 

433 

poppet valves for, construction of, 
441 

opening for given lift, formulas for, 
442 

ports and pipes for, area of, 435 

power calculations of, 426 

reciprocating parts, weight of, for- 
mula for, 429 

slide valves for, Bilgram diagram for, 

439 
friction of, table of, 441 
steam consumption in, calculation of, 
422 
per horse power, table of, 424, 425 
table of, 424 
total weight of, formula for, 429 
Epicyclic gears {see Gears, planetary). 
Equipment, special shop, permissible cost of, 

table of, 319 
Expansion of solids by heat, table of, 404 
Eye and yoke rod ends, dimensions of, tables 

of, 287, 288 
Eye bolts, strength of, chart for, 490 

Fabroil gears {see Gears, spur). 

Face plates, number of T slots in, 277 

Factors, prime, table of, no 

of safety in machine construction, 472 

of IT, SOI 



INDEX 



555 



Fahrenheit to Centigrade, thermometer 
scales, conversion formulas be- 
tween, 401 
conversion table between, 402 
Falling bodies, laws of, 465 

relations of time, velocities and heights, 
tables of, 464 
Fans, centrifugal, motors for, sizes of, tables 
of, 350, 351 
calculating power to drive, 350 
Feed, speed and volume of cut, chart of, 341 
Filing notes and clippings, 321 
Film lubrication (see Bearings, plain or 

sliding). 
Fire streams, effective, table of, 240 
Fits, drive {see Fits, press). 
Fits, press, straight, 2go 

allowances and hoop stresses, chart 

of, 293 
allowances and pressures, chart of, 

292 
Brown & Sharpe practice with, table 

of, 298 
C. W. Hunt Co.'s practice with, 

table of, 299 
General Electric Co.'s practice with, 

chart of, 293 
lubricant for, 290, 296, 297 
suitable pressures for, 290 
Fits, press, taper, 290 

advantage of, 297 

allowances and hoop stresses, chart 

of, 29s 
allowances and pressures, chart of, 

294 
amount of taper, 296 
C. W. Hunt Co.'s practice with, 

table of, 299 
measuring, 296, 297 
Westinghouse practice with, 296 
Fits, running, allowances and tolerances, 
British standard, chart of, 291 
Brown & Sharpe practice with, 

table of, 298 
C. W. Hunt Co.'s practice with, 

table of, 299 
General Electric Co.'s practice with, 
table of, 298 
Fits, shrink. General Electric Co.'s practice 

with, chart of, 293 
Fittings, A.S.M.E. standard pipe, 258 
tables of, 261, 262 
hydraulic, high pressure, 244 
pipe for high pressure air, 266 
friction of, chart for, 240 
formula for, 437 
standard screwed pipe, table of, 265 
Flanges, pipe {see Pipe flanges). 
Flat plates, ribs for strengthening, 431 

strength of> formulas for, 484 
Floating lever, the, 309 
Floor plates, cast iron, construction of, 316 
Force and velocity relations in link work, 316 
Forced fits {see Fits, press). 
Forming tools, design of, 281 
Foundation bolts, washers for, dimensions of, 
283 



Fly wheels, 67 

Allis Chalmers construction of, 70, 72 
arms and hubs of, weight and value of, 

75 
beam action in, 6g 

Benjamin's tests of bursting speeds of, 69 
bursting speed of, formulas for, 68 

table of, 70 
cast in one piece, 69 
centrifugal stress in, chart for, 67 

formula for, 67 
coefiScients for calculating regulating 

power of, table of, 77 
coefficient of steadiness of, 75, 77 
dimensions of, formulas for, 69 
for gas engines, weight of, formula for, 

444 
for high speeds, 69 
for intermittent work, 75 

charts for calculating, 78 
for steam engines, weight of, formula for, 

429 
Fritz construction of, 73 
Haight construction of, 73 
hollow arms for, 75, 105 
joints of high efficiency, 73 
limiting low velocity of, 75 
Mesta construction of, 69 
Newton construction of, 71 
Providence Steam Engine Co.'s construc- 
tion of, 76 
regulating power of, 75 
shrinkage allowance and shrink links for, 

74 
Stanwood construction of, 73 
safe speeds of, table, of, 71 
to carry belts, 73 

velocities of center of gyration, table of, 79 
Westinghouse construction of, 71 
with flanged joints, method of failure of, 

69 
with joints at points of contrary flexure, 
69 
Foot, decimals of, inches and fractions re- 
duced to, table of, 524 
Fourth powers of binary fractions, 548 
Fractions, binary, addition of, 321 
cubes of, table of, 548 
cube roots of, table of, 533 
decimal equivalents of, table of, 535 
fourth powers of, table of, 548 
square roots of, table of, 533 
squares of, table of, 548 
other than binary, decimal equivalents 

of, table of, 536 
prime number, decimal equivalents of, 
table of, 526 
Frames, and supports, design of, 5 

for punching and shearing machines, 

strength of, 485 
for steam engines, strength of, 429 
Friction, clutches {see Clutches, friction). 
ratchet, 186 
coefficient of, table of, 175, 181 
of hydraulic cup leather packing, table 

of, 243 
of high pressure packing boxes, 243 



Friction of steam engine slide valves, table of, 
441 
of shafts fitted with ball bearings, 56 
with plain bearings, 50 
with roller bearings, 50 
Functions, division of, 3 

Gages, 221 

sizes expressed in decimals, 222 
snap, design of, 4 
standard, decimal, table of, 224 
for copper and brass wire, 229 
for sheets and plates, 224 
for steel wire, 224 

for wire and sheet metal, table of nine, 
223 
wire and sheet metal, Westinghouse 
method of abandoning, 221 
Gallon, U. S. and British compared, 495 
Gas engine {see Engine, gas). 

natural, power calculations for com- 
pression of, formulas for, 449 
Gear box construction, 90 
cases, air vents for, 135 
cutting machines, motors for, sizes of, 
table of, 337 
Gears, back, 80 

correct ratio of, 81 
planetary, 89 
ratios for motor drives, 90 
Gears, bevel, 113 

axial thrust of, table of, 123 

cut angle, two practices relating to, 

113 
cutting with rotary cutters, offset 
method of, 119 

parallel depth method of, 121 
dimensions and angles of, formulas 
for, 113 

table of, 114 

of miter, table of 117 
eqtiivalent of spur teeth, chart of, 119 

spur face, chart of, 118 
modified addendum of, 122 
needed shop dimensions of, 116 
selecting from shop lists, 116 
skew, first type of, 123 

second type of, 125 
strength of, charts for, 118, 119 

formulas for, 116 
tooth profiles, laying out, 116 
Gears epicyclic {see Gears, planetary). 
Gears, friction, 126 

coefiScient of friction, tables of, 126 

128 
construction of bevel, 128 
contact pressure on, table of, 126, 128 
materials of, 126 
Rock wood practice with, 128 
transmitting capacity, chart of, 127 

formulas for, 128 

tables of, 128, 129 
variable speed disk drives, 129 
Gears, helical, 138 

choice of helix angle of, 142 
cutters for, chart of, 140 
durability and efficiency of, 138 



556 



INDEX 



Gears, helical, of any helix angle on shafts 

at any angle by calculation, 144 
at right angles by calculation, 140 
by graphics, 144 
of 45 deg. helix angle on shafts at right 
angles by calculation, 138 
by graphics, 140 
table of, 139 
principles of, 140 
real diametrical pitches of, table of, 

145 
relation between v/orm gears and, 

138 

speed ratio of, 138, 141 

with helices of opposite bands, 146 
Gears, miter {see Gears, bevel). 
Gears, planetary, 147 

velocity ratios of various combina- 
tions, 147 
Gears, skew bevel {see Gears, bevel, skew). 
Gears, spiral {see Gears, helical). 
Gears, spur, 93 

Adamson system of, 93 

approximate outlines of, 95 

bronze, strength of, 103 

Brown & Sharpe system of, 93 

chordal pitch, 107 

thickness, table of, 108 

diameters and circular pitches, table 

of, III 

dimensions of. Brown & Sharpe for- 
mulas for, 94 

of parts, charts for, 106 
formulas for, 105, 107 
fabroil, strength of, 103 
Fellows system of, 93, 95 
generation of an involute, 93 
Grant's odontograph, epicycloidal, 98 

involute, 95, 97 
herringbone, limiting speed of, 103 

power of, table of, 104 

strength of, 103 
hollow arms for, 105 
Hunt system of, 93 
interference of involute teeth, 93 
involute systems, table of, 93 
limiting speed of, 103 
Logue system of, 93 
modified involute profiles, 93 
multipliers connecting diameter and 

pitch, table of, 96 
rawhide, strength of, 103 
Sellers system of, 93 
sets of cutters for, 108 
shrouded, strength of, 102 
stub tooth systems, 93 
strength of, chart for, loi 

Lewis formula for, 98 

Marx and Davis formula for, 100 

Mesta rules for, chart for, 102 
teeth of, full size, engravings for, 99 

gaging, 108 
that failed in service, table of, 102 
tooth parts by circular pitch, table of, 
96 

by diametral pitch, table of, 97 
width of face of, 107 



GearSjWonn, 130 

admissible temperatures of, 134 
angles for gashing, table of, 137 
Brown & Sharpe standard, formulas 

for, 130 
capacity of, chart for, 135 

formulas for, 134 
cases, air vents for, 135 
diametral pitch, change gears for cut- 
ting, table of, 131 
durability and eflSciency of, 131 

chart of, 133 
helix angle in relation to durability 

and efficiency of, 131 
high efficiencies obtained by, 133 
Hindley, capacity of, 134 
interference of, 130 
lead and pitch of, 210 
pitch diameters of, table of, 131 
relation between helical gears and, 138 

of circular and normal pitches, table 
of, 132 

of pressure and velocity, chart of, 

134 
thread profile of, 130 
threads, Acme and Brown & Sharpe 

compared, 220 
tooth parts by circular pitch, tables of, 

96, 136 
wire system of measuring, table of con- 
stants for, 136 
wheels, milled, 135 
pitch diameters of, table of, 131 
Geneva stop, the, 313 
Geometrical progression of speeds {see Pulleys, 

cone), 
German silyer, S.A.E. specifications for, 391 
Gibs, slides and dovetails, dimensions of, 

table of, 276 
Grinding limits. Brown & Sharpe practice 
with, table of, 298 
machines, motors for, sizes of, table of, 

337 
Gravity, center of, of counterweights, 466 
of irregular figures, finding, 466 
of plain figures, 466 
laws of falling bodies, 465 
velocity, time and height of fall, rela- 
tions of, tables of, 464 
Guide, the narrow, 2 

Gyration, center of, velocities of in fly-wheels, 
table of, 79 

Hammer blows, measuring energy of, 352 
steam, taper for piston rod ends, 273 

Handles for machine tools, dimensions of, 
280, 284 

Hand wheels, dimensions of, 280 

Hardening steel {see Steel). 

Hardness nvimerals, Brinell, table of, 376 
Brinell and scleroscope, table of, 377 

Heat, 401 

British thermal units and kilogram cal- 
ories, conversion table of, 403 
expansion of solids by, table of, 404 
loss of, in steam pipe lines, charts for, 438 
melting points of substances, table of, 404 



Heat, thermometer scales, conversion 
formulas between, 401 
conversion tables between, 401, 402 
transmission of through tubes, table of, 

404 
treatment of steel {see Steel). 
Heavy loads, moving, power required for, 

chart of, 352 
Heights of fall, velocities and time, relation of, 

table of, 464 
Helical gears {see Gears, helical). 
springs {see Springs, helical). 
Herringbone gears {see Gears, spur). 
Hindley worms {see Gears, worm). 
Hoisting drums {see Drums, hoisting). 
hooks {see Hooks, hoisting). 
rope {see Ropes). 
Hoists, direct lift, consumption of compressed 

air by, table of, 460 
Hooke universal coupling, 308 
Hydraulics and hydraulic machinery, 230 
air chamber charging devices, 246 
chambers, arrangement of, 247 
constants, table of, 230 
grade line, 239 

packing boxes, high pressure, dimensions 
of, 243 
friction of, 243 
cup leather, friction of, 243 

table of, 242 
high pressure, forms of, 242 
leather ring, dimensions of, 242 
pipe joints for high pressure, table of, 268 
press cylinders, thickness of, chart for, 
241 
formula for, 240 
presses, motors for, sizes of, tables of, 

336, 339 
press rams, thickness of, chart for, 241 

formula for, 241 
pumps, power and capacity of, chart of, 

249 
rams, delivery of, table of, 248 
siphons, arrangement of, 248 

discharge of, formula for, 250 
valves and fittings, high pressure, 244 
Hyperbolic logarithms, table of, 505 

Inches and fractions reduced to decimals of a 

foot, table of, 524 
Index rings, design of, 3 
India rubber, properties of, 489 
Indicator paper, metallic, 322 
Inertia, moment of, finding, graphical 
methods, 467, 469 

formulas for, 473 

of rectangles, table of, 474 
Intercoolers for compressed air, construction 

of, 457 
area of surface of, chart for, 457 
formula for, 457 
Iron cast {see Cast iron). 

sheet, U. S. standard for thickness of , 224 

weight of, table of, 397 
wire strength of, table of, 229 

weight of, table of, 395 
wrought, strength of, 472 



INDEX 



557 



Jigs and fixtures, 278 

bushings for, table of, 285 

screws for, table of, 285 

self -locking cams for, 278 
Joints for steam boilers {see Boilers, steam). 

for steam engine cylinder covers, 433 

hydraulic {see Hydraulics). 

knuckle, construction of, 236 

pipe {see Pipe joints) . 

toggle, thrust in, 465 

universal {see Coupling, universal). 
Journals {see Bearings). 

Keys and ke3rways, 50 

dimensions of common, 50, 52 

the Brown & Sharpe, 53 

the Kennedy, 51 

the Nordberg, 50 

the Woodruff, 52 

weakening effect of, on shafts, 52 
Kilogram calories and British thermal units, 

conversion, table of, 403 
Knobs for machine tools, 280, 284 
Knuckle joint, construction of, 276 

dimensions of, table of, 286 

Lathe beds, design of, 3, 5 

Lathes, motors for, sizes of, table of, 335, 337 

Lead and pitch of screws defined, 210 

Lead plugs, for measuring energy of hammer 

blows, 352 
Legs, three desirable, 5 

should not be at ends of frames, 5 
Lever, the floating, 309 

rocking, equal division of arc of vibra- 
tion, 314 
Limits, grinding, Brown & Sharpe practice 

with, table of, 298 
Link motion, parts of, dimensions of, 431 

Bilgram diagram for, 441 
Linkwork, velocity and force relations in, 310 

equal division of arc of vibration, 314 
Loads, heavy, moving, power required for, 

chart of, 352 

Logarithms, anti, table of, 503 

common, table of, 502 

hyperbolic, table of, 505 

Lubricant for press fits, 290, 296, 297 

Lubrication of ball bearings, 30 

of bearings, correct point for introduc- 
ing, 13 
film, breaking down point of film, 14 
conditions of, in bearings, 13 
design of bearings to secure, 15 
effect of, on wear, 14 
in thrust bearings, 28 

Machine parts, minor, 271 

screws {see Screws, machine). 

tools, ball cranks for, dimensions of, 280 
hand wheels for, for dimensions of, 280 
handles for, dimensions of, 280, 284 
in groups, motors for, 347 

tables of, 346 
knobs for, dimensions of, 280, 284 
motors for, calculation of, 338 
motors for, sizes of, table of, 335 



Malleable castings, properties of, table of, 365 

stress strain diagrams from, 366 
Mandrels, split ring expanding, taper for, 273 
Materials and constructions for resisting 
shock, 491 

of construction, strength of, table of, 472 
Mathematical tables, 501 

extending the range of, 501 
Mechanical powers, the, 464 

advantage, rules for, 464 
Mechanics, analytical, 464 
Mechanisms, differential, 464 
Measures and weights {see Weights and 

measures). 
Melting points of substances, table of, 404 
Metallic indicator paper, 322 
Metals, melting points of, table of, 404 

weight and specific gravity of, table of, 

395 
Metric system, British compound units, con- 
version table of, 499 
the case against, 495 
thermal units, conversion table of, 

403 
units of length, weight and capacity, 

conversion tables of, 496 
units of pressure, conversion table of, 

499 
units of work and power, conversion 

table of, 500 
units of value, conversion table of, 500 
Milling cutters, design of, 359 

machines, power constants for, table of, 

331 
motors for, sizes of, table of, 338 
power consumed by, charts of, 333 
tables of, 332, 333, 334 
Mills, boring, motors for, sizes of, table of, 335 
Minor machine parts, 271 
Miter gears {see Gears, bevel). 
Mixed numbers, squares of, finding, 526 

table of, 527 
Moment of inertia, formula for, 473 

of irregular figures, finding, graphical 

methods, 467, 469 
of rectangles, table of, 474 
Motion, accelerated, graphical expression of, 
466 
laws of, 466 
Motors, electric, adjustable speed, range and 
ratings for, table of, 345 
back gears for, ratios of, table of, 90 
for centrifugal fans, calculating power 
of, 350 
sizes of, tables of, 350, 351 
for machine tools in groups, tables of, 
346, 347 
calculation of, 338 
sizes of, table of, 335, 342 
for power presses, sizes of, table of, 339 
for wood working machines, sizes of, 

table of, 339 
selection of, table to assist, 344 
standard, power, pulley sizes and belt 
speeds of, table of, 343 
ratings and data for geared connection, 
table of, 345 



Moving heavy loads, power required for, 

chart of, 352 
Mule pulley stands, laying out, 58 

Naperian logarithms, table of, 505 
Narrow guide, the, 2 

Natural gas, power calculations for, formu- 
las for, 449 
Notes and clippings, filing, 321 
Numbers, table of prime factors of, i ro 
mixed, squares of, finding, 526 
table of, 527 
Nuts {see Screws and bolts). 

Packing boxes, for steam engines, construc- 
tion of, 43 s 
hydraulic high pressure, dimensions of, 

243 
cup leather, friction of, 243 
cup leather, table of, 242 
friction of, 243 

leather ring, dimensions of, 242 
rings, snap piston, action of, 43 1 
dimensions of, chart of, 431 
Paper, metallic indicator, 322 
Pawls, silent, 277 
Pi (x), approximations of, 501 

factors and relations of, table of, 501 
Pillars {see Columns). 

Pins, cotter, S.A.E. standard split, table of,274 
taper, correct use of, 274 

reamer drills for, table of, 275 
Pipe, 251 

Abendroth & Root flanges and fittings, 
table of, 265 
spiral riveted, table of, 260 
brass, seamless, weight of, table of, 396 
cast iron, thickness of, formulas for, 257 

tables for, 259, 260 
copper and brass, S.A.E. specifications 

for, 392 
copper, strength of, formula for, 258 
discharge of water in, at one ft. per sec, 

velocity, tables of, 235, 236 
double extra strong, internal diameters 
of, table of, 239 
table of, 251 

test pressure of, table of, 255 
equation or equalization of, formula for, 
256 
tables for, 258, 259 
extra strong, internal diameters of, table 
of, 239 
table of, 251 

test pressure of, table of, 255 
factor of safety in, 256 
fittings, friction of, chart for, 240 
formula for, 437 
loss of head by friction of water in, 

chart of, 240 
standard screwed, table of, 265 
strength of, bursting, table of, 257 
flanges and fittings, A.S.M.E. standard, 
258 
tables of, 261, 262, 264 
for high pressure hydraulic work, 
table of, 268 



558 



INDEX 



Pipe, flow of water in, chart for, 238 

formulas for, 234 
for steam engines, area of, 435 
hydraulic of large sizes, table of, 252 
joints Rapieff, 262 

for high pressure air, 266 

for high pressure hydraulic work, 246 
table of, 268 

for superheated steam, 267 

gasket, 259 

Van Stone, 267 

Walmaco, 267 
lead, strength of, rule for, 258 
length of, per sq. ft. of surface, table of, 

254 
lines, air, friction of, chart for, 456 
formula for, 453 
flow of water in, chart for, 238 

formula for, 437 
hydraulic grade line of, 239 
loss of head by friction of water in, table 

of, 237 
, markings, A.S.M.E. standard, 267 
precautions in laying, 239 
prevailing velocity of water in, 235 
steam, coverings for, 437 

cheap, 438 
steam, drop of pressure in, chart of, 436 
inclination of, 439 
loss of heat in, charts for, 438 
suction, should have air chambers, 247 
U. S. navy standard, 269 
Pelton Water Wheel Go's, hydraulic, 

table of, 270 
standard block tin, table of, 252 
brass and copper, table of, 252 
British, table of, 254 
cold drawn seamless, bursting pres- 
sure of, table of, 255 
table of, 253 
drawn, bursting pressure of, table of, 

2SS 

table of, 251 

test pressure of, table of, 255 
large O. D., table of, 253 
lead, table of, 252 
square, table of, 252 
welded rectangular, table of, 253 
strength of, bursting, formula for, 251 
collapsing, charts for, 256, 257 
formulas for, 254 . 
threading machines, motors for, sizes of, 

table of, 336 
threads, tap drills for, table of, 216 
Piston rings, snap, action of, 431 
chart of, 431 
rod ends for steam hammers, taper for, 

374 
valves, construction of, 433 
Pitch and lead of screws defined, 210 
of gears, diametral, 94 
circular, 94 
chordal, 107 
Planers, beds for, design of, 5 

motors for, sizes of, tables of, 336, 342 
rotary, motors for, sizes of, table of, 336 
Planetary gears {see Gears, planetary). 



Plates, cast-iron floor, construction of, 316 
face, number of T slots in, 227 
fiat, ribs for strengthening, 431 
strength of, formulas for, 484 
Pneumatic tools, consumption of compressed 

air by, table of, 460 
Polygons, computing, formulas and table 

for, 531 
Ports for steam engines, area of, 435 
Powers, mechanical, the, 464 
Press fits (see Fits, press). 
Presses, hydraulic, cylinders for, thickness of, 
chart for, 241 
formula for, 240 
packing boxes for, dimensions of, 243 
cup leather, friction of, 243 

table of, 242 
forms of, 242 
friction of, 243 

leather ring, dimensions of, 243 
motors for, sizes of, tables of, 336, 339 
rams for, thickness of, chart for, 241 

formula for, 241 
valves and fittings for, 244 
Presses, power, cutting capacity of, formulas 
and chart for, 355 
fly-wheels for, 75 
frames for, strength of, 485 
motors for, table of, 339 
pressure and work of, tables and charts 
of, 349 
Pressure area of poppet valves, effective, 316 
equivalents in various units, table of, 449 
of atmosphere measured in various units, 

230 
per sq. in. converted into ft. head of 
water, table of, 230 
Prony brake (see Brakes). 
Pulleys, 62 

arc of contact of belts on, 56 
cast-iron, dimensions of, chart for, 64 
formulas for, 62 
table of, 63 
strength of, 62, 65 
comparative power of different materials 

for, s6 
for countershafts, correct design of, 65 
guide as substitutes for quarter twist 

belts, 56 
height of crown of, chart of, 62 
idler, correct use of, 56 
loose, lubricating with grease, 65 

self-oiling, 65 
mule, laying out, 58 
of different materials, comparative power 

of, s6 
overhung, design of, 63 
split, correct method of parting, 62 
surplus of face over belt width, chart of, 

62 
tight and loose, correct design of, 65 
Pulleys, cone, 80 

arithmetical solution of, 84 
construction to prevent belt climbing, 

82 
geometrical progression of speeds, 80 
Earth's ratio, 89 



Pulleys, cone, geometrical progression of 
speeds, prevailing ratios, 89 
chart for, 80 
tables for, 85, 88 
graphical solution of, 80 
ratio of back gears, 81 
slide rule solution of, 82 
speed of countershaft for, 81 
. the high power or low ratio, 82 
Pumps, air lift, economy tests of, charts of, 
462 
construction of, 461 
consumption of compressed air by, 
formula for, 461 
Pumps, direct acting, consumption of com- 
pressed air by, formula for, 460 
Punches and dies, clearance between, table of, 
286 
dimensions of, 284 
holders for, table of, 286 
Punching machines, frames for, strength of, 

485 
motors for, sizes of, table of, 336 
power constants for, 348 
pressure and work of, table and charts 

of, 349 

Rack for bar stock, design of, 283 

Railroad clearances, American, chart of, 323 

Rails, cross, design of, 2 

Rams, hydraulic, delivery of, table of, 248 

press, thickness of, formula for, 241 
chart for, 241 
Ratchet, ball friction, 186 
Rawhide gears (see Gears, spur). 
Reamer drills for taper pins, table of, 275 
Reamers, fluted, dimensions of, table of, 289 
Reciprocals, table of, 537 
Relations and factors of tt, 501 
Reservoirs {see Tanks). 

Resilience, importance of, for resisting shock, 
491 

of steel, tables of, 492 
Revolution, solids of, weight of, 319 
Revolutions per minute and linear speeds, 

table of, 525 
Rings, snap piston, action of, 431 

dimensions of, chart of, 431 
Rivets for steam boilers {see Boilers, steam). 

pressure required to drive, chart of, 350 
Rock arms, design of, 5 

dimensions of, table of, 288 

equal division of arc of vibration, 314 
Rod, eccentric, construction of, 274 
Rod ends, taper, piston for steam hammers, 

273 
yoke and eye, dimensions of, table of, 
287, 288 
Roller bearings {see Bearings, roller). 
Rollers for cams, dimensions of, table of, 288 
Rolls, bending and straightening, motors for, 

sizes of, table of, 336 
Roots, cube, of binary fractions, table of, 533 
of whole numbers, table of, 537 
factoring method of finding, 523 
square, finding those not given in 
tables, 501, S33 



INDEX 



559 



Roots, square, of binary fractions, table of, 533 

of whole numbers, table of, S3 7 
Ropes and rope driving, 150 

American and British practice com- 
pared, 150 
and belt drives, cost compared, 150 
^ brakes, 179 {see Brakes), 

m- cotton and manilla, rival claims of, 150 
horse power of, tables of, 151, 152 
speed of, 151 
driving, efficiency of, 163 
drums, superior construction of, 155 
hoisting blocks, efficiency of, 152 
knots, efficiency of, 154 
hitches and bends, 153 
leather and steel belts, comparative 

power of, 60 
manilla, grade for use in power transmis- 
sion, 151 
hoisting, working loads of, 152, 163 
most economical spe^d of, 150 
splicing, 152 
multiple and continuous wrap systems 

compared, 150 
relation of speed and first cost, chart of, 

150 
sag between pulleys, table of, 151 
sheaves, cross section of, 151 
dimensions of arms, chart of, 64 
limiting diameters of, for fibrous rope, 

151 
for wire rope, 160 
tension on slack side, table of, 151 
wire, durability of, 156 

hoisting, working loads of, 163 
materials of, 154 
splicing, 154 
standard, tables of, 157 
Rubber, india, properties of, 489 
Running fits {see Fits, running). 

Safety valves for steam boilers {see Boilers, 

steam). 
Saws, motors for, sizes of, table of, 337 
Scale in steam boilers, loss of coal due to, 

chart of, 419 
Schiele curve for thrust bearings, 29 
Scleroscope and Brinell hardness numerals, 

table of, 377 
Screws, 210 

Acme and Brown & Sharpe threads com- 
pared, 220 
standard, table of, 212 
taps for, 289 
A.S.M.E. machine screw standard 
heads, tables of, 215 
table of, 214 

tap drills for, table of, 214 
Baldwin Locomotive Works system of 

taper, 217 
British Association standard, table of, 

212 
efficiency of, 210 

for jigs and fixtures, tables of, 285 
friction of, 210 

interference of with split nuts, 217 
lead and pitch of, 210 



Screws machine, tap drills for, table of, 214 
, metric standard, table of, 213 
pipe, tap drills for, table of, 216 
S.A.E. lock washer standard, table of, 
216 
standard, strength of, table of, 216 
table of, 213 

tap drills for, table of, 213 
set, holding power of, 218 
stress on, due to initial and applied loads, 
210 
due to tightening nuts, 218 
U. S. standard, constants for bottom 
diameters, table of, 213 
strength of, table of, 211 
table of, 211, 212 
tap drills for, table of, 211 
V thread standard, 210 
Whitworth standard, table of, 212 
wire system of measuring, 218 
Seams for steam boilers {see Boilers, steam). 
Secants and cosecants, natural, table of, 506 
Sections, cross, A.S.M.E. standard for 

drawings, 321 
Segments, circular, area of, table of, 532 
of boiler heads, net area of, table of, 
412 
Set screws, holding power of, 218 
Shaft coupUngs {see Couplings) . 
Shafts, 43 

allowances and tolerances, Brown & 
Sharpe practice with, 297, 298 
C. W. Hunt Go's, practice with, 299 
General Electric Go's, practice with, 

298 
for grinding, Brown & Sharpe prac- 
tice with, 297 
angle of torsion of, chart for, 45 

formulas for, 44 
couplings for, clamp, table of, 279 
flanged, table of, 278 
flexible, table of, 279 
critical speed of, 45 

charts for, 47, 48, 49 
friction of fitted with ball bearings, 50 
with plain bearings, 50 
with roller bearings, 50 
hollow, strength and weight of, chart for, 

45 
strength of, formulas for, 44 
table of, 46 
power required to drive, 51 
strength of, chart for, 43 

formulas for, 43 
weakening effect of key ways on, 52 
Shaping machines, motors for, sizes of, table 

of, 337 
Shear and tension combined, chart of, 485 
Shearing machines, frames for, strength of, 
485 
motors for, sizes of, table of, 336 
power constants for, 348 
pressure and work of, table and charts 
of, 349 
Sheaves, rope, cross-sections of grooves in, 

151 
dimensions of, chart for, 64 



Sheaves, rope, limiting diameters of, for 

fibrous rope, 151 
for wire rope, 160 
Shell blanks, diameters of, 320 
Shippers, improved, for belts, 61 
Shock, bolts under action of, 493 

forms and dimensions of, parts to resist, 

492 
materials and constructions for resisting, 

491 
resilience as a factor in resisting, 491 
Shop equipment, permissible cost of special, 

table of, 319 
Shrinkage of castings, table of, 472 
Shrink fits {see Fits, shrink). 
Shrouded gears {see Gears, spur). 
Sides and diagonals of squares, table of, 533 
Sines and cosines, natural, table of, 506 
Siphons, arrangement and capacity of, 248 
Skew bevel gears {see Gears, bevel). 
Slots, T, 276 

adapter for, 277 
in face plates, number of, 277 
Sellers standard, table of, 277 
Solids of revolution, weight of, 319 
Solution for blackening cam templets when 
laying out, 191 
blue print, 322 
Specific gravity and weight of materials, 

weight of, tables of, 395 
Speeds and diameters, chart of, 340 
table of, 525 
and revolutions per minute, table of, 

525 
feeds and volume of cut, chart of, 341 
geometrical progression of {see Pulleys, 
cone). 
Spheres, surface and volume of, table of, 536 

weight of, table of, 396 
Spiral gears {see Gears, helical). 

springs {see Springs). 
Springs, 197 

common defects in, 208 
elliptic, deflection of chart for, 205 
fiber stress in, 201 
strength and deflection of, formulas 

for, 201 
strength of, chart for, 204 
equalizing pressure of on cams, 196 
flat single leaf, strength and deflection 

of, formulas for, 201 
for use in salt water, 209 
helical, conical, formulas for, 200 
deflection of, chart for, 199 
double or triple, design of, 200 
fiber stress in, 197 

in torsion, deflection of, chart for, 202 
fiber stress in, 200 
strength and deflection of, formulas 

for, 200 
strength of, chart for, 201 
strength and deflection of, formulas 

for, 197 
strength of, chart for, 198 
Pennsylvania Railroad practice with, 208 
phosphor bronze, 209 
preliminary graphic design of, 197 



560 



INDEX 



Springs, spiral, deflection of, chart for, 207 
strength and deflection of, formulas 

for, 203 
strength of, chart for, 206 
steel, heat treatment of, and properties 
given, table of, 387 
properties and temper, chart of, 387 
suitable carbon content, 208, 387 
tempering, 208 

tempering, Gary process of, 209 
use of, for return motion of cams, 188 
Westinghouse practice with, 208 
Sprockets for cable chain, laying out, 164 
for Ewart chain, laying out, 167 
for roller chain, diameter of, formulas 
for, 170 
idler, 169 
material for, 170 
table of, 171 
tooth form, 170 
Spur gears {see Gears, spur). 
Square roots, factoring method of finding, 523 
finding those not given in tables, 501, 533 
of binary fractions, table of, 533 
of whole numbers, table of, 537 
Squares and circles of equal area, table of, 533 
largest in round stock, table of, 523 
of binary fractions, table of, 548 
of mixed numbers, finding, 526 

table of, 527 
of whole numbers, table of, 537 
sides and diagonals of, table of, 533 
Steam boilers (see Boilers, steam). 
engine (see Engine, steam). 
expansion ratios of, actual from theoreti- 
cal, table of, 427 
hammer piston rod ends, taper for, 273 
horse power per lb. of m.e.p., table of, 

427 
mean forward pressure of, chart of, 
426 
factors connecting actual and theo- 
retical, table of, 427 
table of, 426 
pipe joints, 267 

saturated, properties of, table of, 422 
superheated, properties of, table of, 423 
Steel, 368 

alloy, should be heat treated, 381 
bars, hexagon and octagon, weight of, 
table of, 396 
round and square, weight and area of, 
table of, 398 
carbon, properties of, charts of, 379 

S.A.E. specifications for, 378 
castings, S.A.E. specifications for, 381 
specifications of for U. S. navy, 371 
chromium, S.A.E. specifications for, 384 
vanadium, S.A.E, specifications for, 

38s 
colors and temperatures of, tables of, 389 
composition and properties of, digest of, 

368 
flat sizes, weight of, table of, 397 
for cutting tools, carbon percentages in, 

table of, 372 
for resisting shock, 372, 491 



Steel for steam boilers (see Boilers, steam). 
forging of, by presses and hammers, 369 
forgings, specifications of for U. S. 

navy, 371 
hardness tests, Brinell numerals, table 
of, 376 
scleroscope numerals, table of, 377 
heat treatment of, principles of, 374 

S.A.E. standard, table of, 387 
hoops or bands, weight of, table of, 400 
nickel, S.A.E. specifications for, 381 
nickel chromium, S.A.E. specifications 

for, 382 
plates, weight of, table of, 400 
resilience of, table of, 492 
S.A.E. classification of, 377 
sheets, weight of, table of, 397 
silico-manganese, S.A.E. specifications 

for, 386 
specifications, latitude necessary in, 371 
of, for various purposes, table of, 370 
spring, composition of, 208, 387 

heat treatment, Gary process of, 209 
heat treatment of, 208 

properties given, table of, 387 
properties and temper, chart of, 387 
suitable carbon content of, 208 
tempering, 208 
strength of, and hardness numerals, 
table of, 377 
table of, 472 
wire, strength of, table of, 229 
Step bearings (see Bearings, thrust). 
Stop, the Geneva, 313 
Streams, fire, effective, table of, 240 
Struts, design of, 6 
Stud, ball expansion drive, 315 
Studs for cam rolls, dimensions of, table of, 

287 
Stuffing boxes (see Packing boxes). 
Supports and frames, design of, 5 
location of, 5 
three desirable, 5 
Surfaces, wearing, equal and equalized, i, 2 

Tangents and cotangents, natural, table of, 

506 
Tanks, capacity of horizontal cylindrical, 
tables of, 231, 232 
of rectangular, table of, 235 
of vertical cylindrical, table of, 234 
Tapers (see also Dovetails), 271 

and corresponding angles, table of, 273 
bolts, Baldwin Locomotive Works 

standard of, 217 
Brown & Sharpe standard, disks for 
originating, table of, 273 
table of, 271 
for steam hammer piston rods, 273 
Jarno standard, table of, 271 
Morse standard, disks for originating, 
table of, 273 
table of, 272 
of split ring expanding mandrels, 273 
originating, disk method, 273 
pins, correct use of, 274 

reamer drills for, table of, 275 



Tapers, press fits (see Fits, press, taper). 
Reed standard, table of, 272 
Sellers standard, table of, 272 
Standard Tool Co.'s standard, table of, 

272 
total from taper per foot, table of, 274 
Taps, drills for (see Drills, tap). 

for Acme and square threads, 289 
speeds for, 359 
Tempering steel (see Steel). 
Tension and shear combined, chart of, 485 
Test pieces, A.S.T.M. standard, 478 

A.S.M.E. standard for boiler steel, 406 
Thermal units, British and metric, conver- 
sion table of, 403 
Thermometer scales, centigrade, defects of, 
401 
conversion formulas for, 401 
tables for, 401, 402 
Threading and tapping, speeds for, 359 
Thrust and torque of drills (see Drills and 
drilling). 
bearings (see Bearings, thrust) . 
Ties, design of, 6 
Time velocities and heights of fall, relations 

of, tables of, 464 
Toggle joint, thrust in, 465 
Tolerances and allowances (see Fits, press and 

running) . 
Tools, cutting, cube law of, Herbert's, 358 
forms of, Taylor's, 356 
pressure on, formula for, 324 
Taylor's laws for, 324 
when cutting cast iron, chart for, 

32s 
when cutting steel, chart for, 326 
speed, feed and volume of cut, chart 

for, 341 
speeds and feeds in average practice, 
chart of, 359 
in cast iron, Taylor's, table of, 358 
in steel, Taylor's, table of, 357 
steel for, carbon percentages in, table 
of, 372 
forming, design of, 281 
machine (see Machine tools). 
pneumatic, consumption of compressed 

air by, table of, 460 
special, permissible cost of, table of, 

319 
Torque and thrust of drills (see Drills and 

drilling). 
Torsion members, tubular, 3, 44 
Trigonometric functions, definitions of, 506 

formulas connecting, 506 

natural, table of, 506 

signs of, in the four quadrants, 507 
T slots, 276 

adapter for, 277 

in face plates, number of, 227 

Sellers standard, table of, 277 
Tubes for steam boilers (see Boilers, steam). 

for other purposes (see Pipe). 

for torsion members, 3, 44 

Universal coupling, the Baush, 309 
the Hooke, 309 



! 



Valves, hydraulic, high pressure, 244 
piston, construction of, 433 
poppet, construction of, 441 
effective pressure area of, 316 
for air compressors, 459 
opening for given lift, formulas for, 442 
rotary for air compressors, 459 
safety for steam boilers (see Boilers, 

steam). 
slide, Bilgram diagram for, 439 
friction of, table of, 441 
Velocity and energy, loss of, relations of, chart 
of, 79 
and energy of i lb., relations of, chart 

of, 78 
and force relations in linkwork, 310 
height of fall and time, relations of, 
chart of, 464 
Volume of spheres, table of, 536 

Washers for foundation bolts, dimensions of, 
283 
lock, S.A.E. standard, 216 
Water, 230 

capacity of horizontal cylindrical tanks, 
table of, 231, 232 
of rectangular tanks, tables of, 235 
of vertical cylindrical tanks, table of, 

234 
coefficient of discharge of, table of, 230 
condensing for steam engines, table of, 

442 
discharge of, in pipes at i ft. per sec. 

velocity, tables of, 235, 236 
over weirs, formulas for, 239 

tables of, 239 
feet head of, converted into pressure per 

sq. in., table of, 230 
fire streams, effective, table of, 240 



INDEX 

Water, flow of, fundamental formula for, 230 
in pipes, chart for, 238 

formulas for, 234 
in pipe lines, hydraulic grade line, 239 
jets, spouting velocity and discharge 
of, table of, 234 
and horse power of, chart for, 23 1 
loss of head by friction in pipe fittings, 
chart of, 240 
in pipe, table of, 237 
power required to pump, chart of, 249 
precaution in laying pipe lines for, 239 
pressure per sq. in. converted into feet, 

head of, table of, 230 
velocities of, with corresponding heads 

and pressures, 236 
velocity of prevailing in pipe lines, 235 
weight, volume and pressure of, 230 
Wear, effect of equal length wearing surfaces 

on, I 
Weight and specific gravity of materials, 
table of, 395 
of brass, copper and aluminum bars, 

table of, 400 
of flat sizes of steel, table of, 397 
of fly-wheels for steam engines, coeffi- 
cients for, table of, 77 
■ formula for, 429 
of iron, brass and copper wire, table of, 

3QS 
of metal removed by drilUng, table of, 

302 
of reciprocating parts of steam engines, 

formula for, 429 
of seamless brass tubing, table of, 396 
of sheet iron, steel, copper and brass, 

weight of, table of, 397 
of spheres, table of, 396 
of solids of revolution, 319 



561 

Weight of steam engines, total, formula for, 
429 
of steel hexagon and octagon bars, table 
of, 396 
hoops or bands, table of, 400 
plates, table of, 400 
round and square bars, table of, 398 
Weights and measures, 495 

British-metric, conversion tables of com- 
pound units, 499 
of units of length, weight and capac- 
ity, 496 
of units of pressure, 499 
of thermal units, 403 
of units of work and power, 500 
of units of value, 500 
metric system, the case against, 495 
U. S. and British compared, 495 
Weirs, discharge of water over, formulas for, 

239 
table of, 239 
Wheels, hand, dimensions of, 280 

fly {see Fly-wheels). 
Wire, copper, brass, etc., standard gage for, 
229 
gages {see Gages). 
iron, brass and copper, weight of, table 

of, 395 
rope {see Rope). 

sizes and properties of, table of, 225 
steel, standard gage for, 224 
strength of, table of, 229 
Wood working machines, motors for, sizes of, 

table of, 339 
Worm gears {see Gears, worm) . 
Wrenches, dimensions of, 279, 281, 284 

Yoke and eye rod ends, dimensions of, tables 
of, 287, 288 




I 



INDEX TO MAJOR TOPICS 



Page 

Air-lift Pump 461 

Alloys. . . : 390 

.Vreas and Circumferences of Circles 537, 544, 548, 549 

Back Gears and Cone Pulleys 80 

Balancing Machine Parts 300 

Ball Bearings 30 

Beams 472 

Belts and Pulleys 54 

Bevel Gears 113 

Bolts, Nuts and Screws 210 

Brakes 175 

Cams 188 

Cast Iron 361 

Center of Gravity 466 

Centrifugal Force 466 

Chains and Chain Driving 164 

Columns 482 

Compressed Air 449 

Cone Pulleys and Back Gears 80 

Cost of Special Shop Equipment 318 

Cubes 537, 548 

Cube Roots S33. 537 

Decimal Equivalents 526, 535 

Energy 466 

Falling Bodies 465 

Flat Plates 483 

Floating Lever 308 

Fly "Wheels 67 

Friction Clutches 182 

Friction Gears 126 

Gas Engine 444 

Geneva Stop. 313 

Heat 401 

Heat Treatment of Steel 374 

Helical Gears 138 

Hoisting Drums 155 

Hc.sting Hooks and Lifting Eyes 486 

Hydraulics and Hydraulic Machinery 230 

Hyperbolic Logarithms 505 

Keys and Shafts. 43 

Link Motion 440 

Linkwork 310, 314 

Logarithms 502 

Malleable Iron 365 

Mathematical Tables 501 



Page 

Mechanics 464 

Mechanical Principles of Design i 

Metric Conversion Tables 496 

Minor Machine Parts 271 

Miscellaneous Mechanisms, Constructions and Data 308 

Moment of Inertia 467, 473 

Motors for Machine Tools 33s 

Pipe and Pipe Joints ". 251 

Plain or Sliding Bearings 8 

Pneumatic Tools 459 

Polygons 531 

Power Requirements of Tools 324 

Press and Running Fits 290 

Pulleys and Belts 54 

Railroad Clearances 323 

Resilience of Steel 492 

Roller Bearings -. 30 

Ropes and Rope Driving 150 

Running and Press Fits 290 

Safety Valves 416 

Screws, Bolts and Nuts 210 

Shafts and Keys 43 

Sht, Metal and Wire 221 

Shell Blanks 320 

Shock Resistance 491 

Slide Valve 439 

Solids of Revolution 319 

Spheres 396, 536 

Spiral Gears 138 

Spur Gears , 93 

Springs 197 

Squares 526, 533, 537, 548 

Square Roots 533, 537 

Steam 422 

Steam Boilers 405 

Steam Engine 422 

Steel 368 

Strength of Machine Parts 472 

Thermometer Scales 401 

Trigonometric Functions 506 

Universal Coupling 308 

Weight of Materials 395 

Weights and Measures 495 

Wire and Sheet Metal 221 

Worm Gears 130 



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